Mean squared error for \n
\r\n\tThe most phenomena of colloids in the surrounding ecosystem appear in the blue color of the sky, which due to the scattering of light by colloidal particles in the air, and this phenomenon is named Tyndall effect. Also, seawater seems blue due to scattering of light by the colloidal particle present in seawater. In our body, blood is a colloidal solution. Some examples in our daily life include milk, cream, gelatin, colored glass, butter, jelly, and river mud. Colloids are useful for human health, commercial, and industrial use. Colloids and colloidal systems are essential and play a significant role in our life. Important applications of colloids are in medicines, sewage disposal, purification of water, cleansing action of soap, the formation of the delta, industry, mining, smoke precipitation, photography, electroplating, artificial rain, agriculture, the rubber industry, and others.
\r\n\r\n\tThis book aims to gather recent researches by outstanding experts in the field of colloids (chemistry, physics, biology, medicine, nanotechnology, pharmacology, and others), which include colloid synthesis, modification, and application.
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In 1989 he received his Ph.D. in analytical environmental chemistry from Assiut University, Egypt. His research interest has been analytical and environmental chemistry with special emphasis on soil and water pollution monitoring, control and treatment; bioindicators for water pollution monitoring; wastewater impact; advanced oxidation treatment; photocatalysis, nanocatalyst, nanocomposite and adsorption techniques in water and wastewater treatment. Prof. Rashed supervised several M.Sc. and Ph.D. thesis in the field of pollution, analytical and environmental chemistry. He participated as an invited speaker in 30 international conferences worldwide. Prof.Rashed acts as editor-in-chief and an editorial board member in several international journals (30) in the fields of chemistry and environment. His society membership includes several national and international societies. 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Venkateswarlu",coverURL:"https://cdn.intechopen.com/books/images_new/371.jpg",editedByType:"Edited by",editors:[{id:"58592",title:"Dr.",name:"Arun",surname:"Shanker",slug:"arun-shanker",fullName:"Arun Shanker"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}}]},chapter:{item:{type:"chapter",id:"70252",title:"Competitive Advantage Leadership in the Ultra Modern Era",doi:"10.5772/intechopen.90080",slug:"competitive-advantage-leadership-in-the-ultra-modern-era",body:'It goes without saying that our environment is constantly changing. In today’s twenty-first century, the changes are unpredictable resulting in a situation where adaptive institutions (those who perceive changes as a necessity) are the ones to survive and compete. One of the examples is the case of Nokia that the fall of Nokia is caused by the failure of its leader to accept changes in the market preference towards Android phones. The twenty-first century is regarded as a global leadership century where a notion of limitless border among nations, countries, and people should be comprehended. This provides the opportunities for everyone to become a leader and to work across the varied skills and cultures. The era of the rapid, yet disruptive world demands a new leadership style with managerial abilities that proactively manages the changes and takes the advantage over the opportunities. This type is the combination of managerial and entrepreneurial leadership.
In [1], a new leadership approach, i.e., entrepreneurial, is necessary for today’s dynamic and competitive environment (including market atmosphere). This is to enhance the performance of an institution as well as its capacity to adapt and to take the advantages over the opportunities to manage its sustainability. Entrepreneurial leadership is expected to be applicable in a hypercompetitive and disruptive environment. Entrepreneurial leadership refers to an individual possessing the quality to lead and continuously adapt with the ever-changing development of the environment and market in order to take the advantages for his or her organization. Recently, the concept of entrepreneurial leadership complements the idea of managerial leadership that has been implemented and developed in several business institutions and nonprofit organizations.
The above phenomenon emphasizes the idea of the necessity for organizational management, where several of its employees have double position, i.e., as a manager and as an entrepreneur leader. With being said, organization leaders are urged to possess “prime” managerial and entrepreneurial abilities as a strategic element for the success of the organization. The aim of this book is to provide insight and the key concepts grounding from several studies on leadership. This book discusses programs and activities that are able to improve the managerial and entrepreneurial leadership, and maintain the quality of these two styles of leadership in sync and continuously.
Human capital, in an institution or a company, serves as the most strategic resource. In [2], human capital refers to the way of dominating and developing knowledge, skill, ability, capacity, and innovation in an organization. Furthermore, in [3], the key factors in the implementation of human capital comprise of people and skill, create value for the organization—people as assets, remuneration, training, development, and career progression to promote the success of the performance of the organization or company. In [4], the interrelation between human and performance is reflected by the career opportunities and satisfying rate. These two aspects are central to the series of particular model for optimum performance.
Research in [2] issues the roles of human capital strategic, i.e., competence, compensation, and career, towards the performance of employees. The purpose of this research is to explore the extent to which the three independent variables, i.e., competence (X1), compensation (X2), and career (X3), influence the dependent variable, i.e., employee performance (Y). The result of those research is found:
the contribution of competence, compensation, and career towards the performance of employees is 57.5%. In addition, the percentage of 42.5% is influenced by other factors;
the factor of competence (X1), compensation (X2), and career (X3) simultaneously significantly contribute to the employee performance (Y);
the factor of competence (X1) significantly contributes to the employees’ performance the most in comparison to the variable of compensation and career;
it is revealed that the factor of career (X3) is the second variable that significantly contributes the most on the employee performance; and
the factor of compensation is not significantly contributes on the employees performance.
Another research in [5] conducted research on human capital in private company. It is aimed at exploring the extent to which the variable of training (X1) and competence (X2) contributes to the employee performance. The results of the research are:
the contribution of competence, compensation, and career towards the performance of employees is 73.8%; on addition, 27.2% is influenced by other factors;
the factors of training and competence simultaneously significantly contribute to the employee performance; and
the factor of competence (X2) contributes to the employee performance the most in comparison to the variable of training.
The result of the aforementioned studies can serve as reference in enhancing the performance of the organization in today’s global era. This can be done by:
designing training programs in enhancing the performance of the leader and all member of the organization. The focuses of this program are as follows:
• designing workshops to meet the demand of the tasks and condition of the leader, and organization staffs in the digital era;
• promoting an environment that supports the utilization of technology to ensure the conduct of quality workshop;
• ensuring a professional training transfer, this signifies that the mode of the training is not only in a traditional face-to-face mode. In the digital era, it is necessary to develop once experience and knowledge by exchanging ideas between organization leaders, the leaders with staff. This should be carried on a regular basis and continuously in the workplace. Furthermore, the format of the development program can take form in digital training, mentorship, and coaching;
• designing skill equivalency programs for the leaders and organization staffs through competency tests based on the needs of the standard of education and industry;
• implementing technology for the assessment of employees; and
competency development programs can also be implemented to enhance the performance of the leaders and staffs. The programs are able to:
• improve the empathy of an individual towards his or her career, which drives the person to perform to their maximum capacity;
• raise the initiative of leaders and organization staffs to maintain their performance and to carry out their duties without even being ordered;
• promote a sense of providing sincere service continuously for the users;
• improve the interpersonal understanding to cooperate with other organizations on the basis mutual trust;
• improve the effectiveness of personal competencies, i.e., integrity and good behavior according to the codes of conducts or principles of morality;
• improve the abilities in implementing the principles of human resources management to contribute to the success of the organization;
• improve the ability to manage interaction with customers and other organization in providing quality services to ensure the success of the organization;
• improve the ability to express ideas to the employees and leaders in dealing with issues, such as the provision of legal assistance, health insurance, and daily need services (cooperation of the employees);
• improve one’s initiative by providing spaces and opportunities for leader and staffs to express their ideas in every event; and
• promote a situation that integrates the core value of honesty and responsibility.
compensation development programs in enhancing the performance of the organization staffs are as follows:
• increasing remuneration to the leaders and organization employees for their position and contribution in attaining the goals of the organization;
• increasing monthly allowance for the aforementioned staffs. The allowance is given based on the performance of the staffs;
• providing incentives for all the organization staffs as an award for their maximum performance;
• increasing indirect compensation by providing specific programs, i.e., health and life insurances for the employees; and
• maximizing the benefits of the pension fund as compensation from the organization to its retirement.
career development programs in enhancing the performance of the organization staffs be able to:
• enhancing the effectiveness of human resources career planning in organization. This focuses on the identification of the career path for the advancement of the leaders and employees;
• maximizing the effectiveness of individual career planning focusing on the responsibility of the individual for his or her career;
• improving the effectiveness of career development by facilitating the development of an individual’s work performance and achievement in attaining career goals, and
• enhancing career development programs by integrated, transparent, and objective career development of the employees and institution.
The result of the research on the aspects of human capital above represents the efforts in developing the ability of human resources, i.e., employees of an institution. The efforts are, by nature, managerial attempts that are mandatory for a leader to manage to increase the competitiveness of the institution in a globalized era.
The character of leadership in the future is different from the quality of leadership before. The most notable trait is the way the leaders uses their power to empower their subordinates; this term is well-known as empowering and enabling [6]. The importance of empowering and enabling concept reflects the changes from the idea of managing knowledge, and people were not integrated to managing people with knowledge to be integrated. In enforcing characteristic of leadership based on the above concept, and in [6], it mentions several abilities that should be possessed by the leader of an organization in this competitive era, such as:
abilities to determine and formulate built shared vision of the organization;
abilities to determine coordination pattern between fellow leaders and subordinates, e.g., integrating the potential of the subordinates in coordinating a multiple task-focused team;
abilities to cope with conflicts in a fair and calm manner (acknowledge and test mental models);
possessing broad knowledge to determine a better policy based on the dynamics and complexity outside the institutional system of the organization;
encourage a challenging atmosphere to the employees in promoting innovation, creativity, and risk-taking attitude; and
abilities to design a concept of a complex issues and conceptualize it into innovation and grounding for the learning in an organization (conceptualize, inspire, learning, and action).
Generally, the term leadership is defined as an individual with the capability to mobilize, synchronize, and lead a group and to express ideas in which other people accept it. A leader is central to promoting changes and is responsible for directing every works and dealing with obstacles to ensure the clarity of the vision of an institution. Another obligation of a leader is to conceptualize an organizational climate that promotes the independence of the employees and, at the same time, the employees aware of their responsibility. In [7], the leadership is closely related to the ability to motivate and influence subordinates, and a successful leader is the one who is able to direct and influence subordinates. Furthermore, a leader with this quality is able to carry out his or her managerial roles based on the pre-determined goals. According to [8], a good leader possesses several characteristics as follows:
balanced responsibility: the term balance in this topic refers to the balance between the responsibility of a person for works and subordinates;
positive role model: role model encompasses the responsibility, behavior, and achievement that people expect to be possessed by a person with a certain position. In other words, a leader must be a good role model for his or her subordinates;
good communication skill: a good leader should be able to express is or her ideas concisely and clearly. The way the leader communicates must be done in an appropriate manner;
positive influencer: a good leader positively influences his or her subordinates. This is considered as an art of using the authorities of the leader to direct and change the perspective of the subordinates towards a particular point of view; and
convincing: a successful leader is able to take advantage of his or her communication skill to convince the others and direct them to fulfill their responsibilities.
The tendency of future leadership heads towards teaching organization in which it functions to anticipate changes and diverse knowledge, skill, and ability of human capital and to enhance the performance of an institution. The success of an institution in the competitive, globalized era is determined by the momentum of the institution to change or adapt with its business or industrial environment. The above discussion shows that the competence of leadership is central to the efforts of the organization in achieving competitiveness and gaining an advantage in this globalized era. A leader is urged to be responsible for directing every attempt in coping with issues and ensuring the clarity of the vision. Another obligation of a leader is to conceptualize an organizational climate that promotes the independence of the employees and, at the same time, the employees aware of their responsibility.
One of the most notable achievements of modern sociology of leadership is the development of role theory. Every member of society, according to this theory, has a certain position and plays a certain role. This indicates that leadership is viewed as an aspect of role of differentiation. Furthermore, this signifies that leadership is defined as an interaction between an individual with his or her members of the group. Ideologically, a leader or a manager has some characteristics, i.e., strong, persistent, and having broad knowledge, that outweigh other people [9]. Although leaders are few in numbers, their contribution is central to the success of attaining the goal of their organization. The vision of a leader is determined from the direction of the organization. Even though a leader is not the only contributing factors to the success of an organization, its absence results in the organization being static and has no clear target.
In the history of human civilization, the sustainability of an organization is a dynamic movement that is determined by a small group of people as who manage the organization. It can be said that the advancement of people comes from a few numbers of people with a specialty that eager to be a pioneer, creator, and expert in an organization. This group of people is leaders, and this fact signifies that leadership is a key to good management. In carrying out leadership duties, a leader is responsible not only for his or her superior and for the success of the organization, but also for the internal problem of the institution. This encompasses the responsibility for developing and guiding human resources through training and development. From the external factor, a leader is responsible for the social aspect of a society or public accountability. The basic leadership theory discusses two factors, i.e.
factors involving in the conceptualization of a leader and
characteristics base of leadership.
Research on two aforementioned factors is more suitable than the theory. However, the theory of leadership is also intriguing as it helps to define and to formulate some issues in several studies. Some literature on leadership reveals that the theories of leadership are influenced by research in [10] the background of famous people that implement inheritance-based leadership. Further studies point out that every member of a society has different intelligence, energy, and moral power. The individual is also lead by a superior person.
In the further development, some theorists develop a view of the emergence of great leaders as a product of time, space, and temporal situation. In [7], two developed hypotheses of leadership are:
the quality of a leader and leadership that is determined by a situation of a group; and
the quality of an individual in coping with a momentary situation is a product of successful leadership that previously addressed a similar issue.
Two theories, i.e., trait theory and situational theory explain leadership as an effect of a single authority and power. The interactive effect between individual and situational factors seems to receive little attention. With that being said, exploring the concept of leadership should encompass:
the characteristics of effective, intellectual, and individual behaviors; and
the specific condition of an individual in the implementation of leadership.
Another opinion [10] asserts that aspects which should be focused on in order to understand the concept of leadership are:
characteristics and motives of a leader as an ordinary person;
conceiving that a leader should pay attention to his or her subordinates as they follow the leader;
a leader should play his or her leadership roles; and
institutional correlation which should involve leader and subordinates.
The above opinions can be categorized as a personal-situational theory of leadership. This is because the view discusses not only the present situation but also the interaction among individuals or between leaders and their groups. A theory of leadership that is developed according to the above three theories is interaction-expectation theory. This theory develops the role of leadership by referring to three basic variables, i.e., attitude, interaction, and sentiment. It is assumed that an increase in the frequency of interaction and participation is closely related with the increase in sentiment or feeling of happiness and the clarity of norm of groups. The higher the position of an individual, the more appropriate the activities with the norm of groups. In addition, the interaction is more distributed, and more groups of members are convinced to interact.
A study in [11] reports that to support corruption eradication in an organization, a managerial leader should possess several qualities such as smart, sociable, confident, honest, trustworthy and responsible, encouraging members to perform and obtain optimal satisfaction, expressing thoughts concisely, passionate, valuing morality, focusing on internal supervision, anticipatory followed by good corporate governance, and not developing centralized leadership. A managerial, centralistic leader is a person who prioritizes forcing which impacts on legal aspects, such as:
vague rules and conditions where it cause ambiguous results for every violation committed by all members of organization;
inconsistent implementation of sanctions for violations;
over-early investigation, fact-discovery, and conclusion-making which are susceptible to biased verdict and law implementation;
vague law procedures, subjective perspective on a case, and lack of credibility; and
lack of law justice and equality.
The above research results serve as a guideline institutions or organization in improving its competitiveness. This is also to help the implementation of corruption prevention/eradication, i.e., prioritizing the organizational culture and the implementation of strict “organizational ethics” in an organization. A managerial organization leader is obliged to implement clean and good corporate governance in relation to the implementation of organizational culture and ethics. The following are some organization ethics that should be improved:
every human resource in an organization must follow his or her job description accurately and respond it immediately;
employees in all work units of an organization should implement their jobs based on a certain analysis. The employees are also demanded to pay attention to the details of the works;
employees in all work units of an organization should ensure transparent and responsible processes and the management and provision of facilities;
all leaders in every work unit should consider the impact or the outcome of the improvement of organization performance in every decision-making process; and
all leaders in every work unit are encouraged to implement empowering and enabling approach in organizing the operational works. The examples of these approaches are mentoring, coaching, team learning (learning with others), or team working. These are carried out together with fellow leaders or employees or between the leaders and the employees.
Managerial leadership is an important role demanded of managers due to their managerial works (technical works). In [12], typical activity patterns of managerial works are the people as the responsible managerial party working in a long period and often bringing the project at home. Their mind continually finds and analyzes new information. This is an automatic process, and they are always reminded to finish it under any circumstances. They keep receiving requests of information, assistance, briefing, and authority from the subordinates, co-workers, leaders, and other people excluded in the organization. This managerial activity is different from general perceptions that consider a manager is only making plans, arranging activities, and then waiting for an exception of normal operations that need much attention.
Manager activities are varied, yet it is often interrupted because of the high activity frequency and intensity so that talks are unfinished, and significant activities are interspersed with unimportant ones, requiring a quick change of feeling. A considerable number of managers are very reactive instead of being proactive. Most of their time is spent by carefully analyzing the problems of the organization/business as well as developing plans to deal with it. The remaining time is used to read letters or documents, handle administrative works, read reports from subordinates, write reports/memos, read journals or technical publications, and news related to managerial activities.
In [12], it claims that general manager network frequently reaches hundreds of people, either the internal or the external parties of the organization. Network interaction is necessary for information needs inside or outside the organization, which later affect the manager performance and career. The ability to develop network and cooperation is essential to support changes and innovations that can be successfully carried out. Developing network in organizations is performed with the following ways:
having a talk with the people in formal or informal occasions in them organization;
serving a particular committee, stakeholder, and task force;
joining a group of community leaders, advisory board, and social club; and
being active in conducting and attending several conferences, workshops, seminars, trade and industry exhibitions, profession association forum, education forum, science and technology, and others.
The managers should constantly build networks. They constantly interact and communicate to gain information from different sources, such as:
written sources (disposition, memo, recording, letter, report, written order/instruction, and contract);
planned and/or unplanned meetings; and
in the site observations.
In performing interaction and communication, the main choice of the managers is by using “oral communication”; via telephone and face-to-face meeting to obtain information exchanges and efforts to influence people. Managers prefer current information, which is from face-to-face contact with the ones who have information access, including those outside the managers’ work sub-unit. Rumors are new information on the latest event and development; written reports, meanwhile, are summary-like information of the old event. New and detailed information helps a manager; information gained from facts, opinions, gossips, behaviors towards interests, and things related to him as a manager. Oral communication gives an effect, i.e., strengthened by intonation, movement, expression, and other nonverbal communication. Managers, in oral communication, also needs jokes and discusses job-unrelated topics, such as sports, arts, hobbies, and the like. Moreover, social activity and informal communication be able to facilitate them to develop and maintain the effectiveness of the strategic network relationship.
Managers, in decision-making, normally need supports and authorities from various parties, hierarchy levels, and organization sub-units. For this reason, they should consult with the subordinates, co-workers, and leaders regarding the vital decision to be taken. Every person can be the initiator of the decision-making that it does not only depend on people who make the last option among action alternatives. Some parties, managers, and functional specialists oftentimes disagree with the solution of the issues that will be decided due to different perspectives, assumptions, values, impacts, and consequences. In addition, decision-making is a complicated process that it takes months to years as a result of interruptions, issues deflected by the opponents, crisis, or it is returned to the initiators to be fixed based on critics and suggestions also note that strategic decisions which have impact on the organizations/institutions can be performed, depending on the influencing ability, the initiators persistence, and relative authority of the people who are responsible for the decision-making. Managerial decisions are not always on a large-scale, complex, having a big change, and going through a long political process that spends a lot of time; this rarely happens to managerial problems. Most managerial problems that should quickly be decided are related to: operations; work plans; giving authority by utilizing funds for office stationery and equipment purchases; wage increase for the employees; and work achievement allowance. This decision has its solution and low risk, and the decision-maker should consult with some people as well as should immediately make the decision due to the critical deadline; although this decision is not too important, it still needs the adequate technical capability, quick risk analysis, and action. A delay in decision-making, which is considered less significant as they want to get more information, will provide a negative impact on the operations of the organizations and their managers.
The most strategic decisions are made outside the process of formal planning and are formulated in a gradual, flexible, and intuitive manner. Besides, executive leaders set a strategy to reach general work goals and plans so that it is still possible to accept suggestions. The strategy is then perfected and becomes an agreement in the organization to be socialized and implemented in accordance with the situation and condition of each work unit. In [13], it mentions nine duties to be in a managerial position, as follows:
supervising: working together with the subordinates to improve performances, analyze strengths and weaknesses, give training, develop skills, plan works, and establish work achievements;
planning and organizing: formulating short-term plan; project implementation, budget plan; evaluating organizational structures; determining resources allocation and use; arranging short-term operation goals based on long-term plans; and supporting and developing operation policies and procedures;
making decisions—decision forms: making a business decision with no doubt in an unstructured situation; provide deviation authority towards procedures of unusual cases or new conditions excluded in the rules/procedures;
monitoring indicators—monitoring forms: internal strengths/weaknesses; external opportunities/threats; performance indicators; company’s finances and properties; market conditions; economic, social, cultural, and political circumstances; security defense; and technology;
controlling—controlling forms: work plans; budget plans; completion schedules of production/work along with goods and services distribution; productivity; goods and services quality; and operations effectiveness;
responding issues: response forms towards issues: responding to complaints; communicating to advance relationship and cooperation; negotiating; promoting to build and maintain the image of universities and products; and developing and maintaining trust from all parties;
coordinating—coordination forms: coordinating and communicating with other work units of the internal organization in which managers do not have direct control; fulfilling the set work plans related to other work units of the internal or external organizations; overcoming problems to achieve goals and performances; fostering work relationship with strategic partners; and resolving conflicts among important individuals due to the misunderstanding in executing the work;
consulting—consultation forms: technological development in particular organization that influence universities activities and performances; introducing new technology and techniques; acting as an expert; acting as a consultant; and giving solutions of another party’s problems; and
administrating—administration forms: basic administrative activities; availability of information about the procedures and practices of work implementation; analyzing essential information for the organization; and arranging detailed (complete, accurate, secure, and easy-to-find administration, archive, and documentation).
In general, there are 10 manager roles to be applied in institution/organization:
formal leaders’ roles: managerial roles for a formal leader are by performing particular legal and social duties (signing letters, contracts/MOU, and official permits); leading the meeting and ceremonial events; and participating in ceremonies and rituals;
leaders’ roles: the roles as a leader include: being responsible for the functions of sub-units that are integrated to be a system to carry out the duties to reach organization goals; present guidelines to the subordinates; motivating the subordinates to make an achievement; and creating a conducive condition for the well-running duties;
connector: the roles of a connector are establishing and maintaining network relationship with individuals or groups outside the managers work unit, either in the internal or external organization; the nature of the connector’s roles (composing a new contract/MOU; giving assistance as a basic aspect to help each other if needed);
observer: the roles as an observer constantly find information, such as reading reports and memos; attending a meeting and briefing; doing an observation; disseminating vertical and horizontal information; analyzing information to get business opportunities and threats; and conducting an information internalization that is considered significant;
disseminator: the roles as a disseminator included: developing and maintaining special access towards information sources inside or outside the organization; and forwarding vertical or horizontal information that is considered necessary;
spokesperson: the roles as a spokesperson are formulating important information to be delivered; delivering a statement about information values to parties outside the work unit of the internal and external organizations; following the procedures (middle-level and lower-level managers should report to their leaders; chief executives should report to their leaders or the owners); lobbying when dealing with the leaders and outside parties; and mastering actual knowledge regarding organization and its environment;
entrepreneur: the roles as an entrepreneur are as follows: initiating and designing changes; utilizing opportunities for the importance of business that supports organization; developing new products; developing new equipment; re-organizing formal structures; supervising the changes and improvements; and delegating to the subordinates as in line with their ability;
person who addresses problems: handling unexpected and sudden crisis, for instance: conflicts among the subordinates; losing a good and potential subordinate; fire/accidents; natural disasters; riots; strikes; and subordinate apathy (hopeless);
Allocator of resources: the roles as a resource allocator are allocating money, personnel, materials, equipment, facilities, and services; and controlling developing controlling strategies, coordinating resources activities, and integrating strategic achievements of the resource; and
negotiator: the roles as a negotiator are as follows: making a commitment; involving in several negotiations (contract between organization’s employees and management); subordinates demand; contract with customers and suppliers; giving special/important works to a good and potential staff; and other ad hoc negotiations.
The dynamics of businesses/industries lately have an impact on the strategic behaviors of organization leaders, including university leaders. Uncertainty and rapidity of changes in science and technology affect the behavior changes of businessman, CEOs of business and industry. Around the year of 2000, the concept of entrepreneurial leaders emerges because of the increase of business dynamics—which then comes up as an identification of entrepreneurial personalities and characteristics, followed by entrepreneurial education; then, contemporary works on entrepreneurial motivation arise. A new leadership type is required, i.e., business leadership to encounter the increased competitiveness and uncertainty in this dynamic market [1]. Such a type of leadership is “entrepreneurial leaders.” Does entrepreneurial leadership exist?
In [14], it defines entrepreneurial leadership as an ability to take risks, have innovations, focus on duties, bear personal responsibilities, and have economic orientations; this presents the initial definition of entrepreneurial leadership. He argues that leadership quality is more than just being an administrator or manager. As such, this implies that leaders regulate the company with energy, confidence, persistence, and learning ability. In era before digital, the interesting question about entrepreneurial leadership as a new paradigm. People in this era of intense competition correlate entrepreneurs with leadership functions, i.e., should provide visions to develop products, services, and new organizations. A leader should also be entrepreneurial; they also opine that entrepreneurial leadership is associated with concept and idea, and it is frequently related to nonorganizational issues. Conversely, entrepreneurial leadership tends to have characteristics, e.g., tending to be individual-behaved, visionary, problem-solver, decision-maker, risk-taker, and strategic initiative-taker.
In [15], the concept of entrepreneurial leadership is clarified and the empirical measure of significant factors to build up entrepreneurial leadership is developed. Here are the definitions of entrepreneurial leadership that have several characteristics: creating visionary scenarios; motivating and playing the cast of characters for discovery; and exploitation of strategic values creation in an organizational environment. Entrepreneurial leaders can learn and develop knowledge, and in [16], it is believed that “the ability to develop, share, and improve knowledge will replace asset ownership and control as the main source of competitive excellence.” Entrepreneurial leaders can increase their knowledge and promptly share it with their organizations. Fast learning and its sharing enable the entrepreneurial leaders and their organizations to get better quality than other organizations. In [1], the entrepreneurial leaders should always cope with a sudden change as well as understand that the dynamic market condition requires them to move beyond the increase that provides the value added (incremental). In the dynamic market, entrepreneurial changes need a real-time knowledge generation and strategic experiments, facilitating all organizations to identify opportunities and create a momentum to manage those opportunities. Entrepreneurial leaders understand the implication of resources and competences. A strategic implementation in a competition demands the entrepreneurial leaders to develop a unique combination between resources and competencies/abilities, which are defined as the main competence that creates competitive excellence in the market. In [16], it is also noted that leader’s ability to develop and exploit their “main competence” will give significant results for the success level of the company.
According to [17], entrepreneurial leadership competencies are defined a special leadership ability required to successfully leading competitive and challenging activities, vital things that show up in the success of entrepreneurial activities, either in a new business or organization. Nevertheless, in [18], there is not a sufficient information about the process of entrepreneurial learning and where people learn these entrepreneurial leadership competencies. Moreover, as explained in [19], it might hope that the process of entrepreneurial learning is improved by the availability of a role model (example), private network expansion and strengthening, and temporary internship.
On that ground, a clear description of “entrepreneurial leaders” refers to people who have an appropriate “entrepreneurial” approach and ability to keep oneself taking part in the development of a rapidly changing situation or market as well as to exploit opportunities to get benefits for the organization faster than others. The presentation of this concept introduction causes some organizations to try to develop concepts in the business area. In [20], it specifically indicates entrepreneurial leadership as an entrepreneurship research field that is ignored, and in [21], it identifies a missing link in traditional entrepreneurship education, which makes it not conducive to leadership learning. In learning contemporary entrepreneurship, the recurring concept is that “entrepreneurship is interdisciplinary.” The world is now experiencing an entrepreneurial revolution, which is more intense than an industrial revolution in the twentieth century.
The concept of leadership in entrepreneurship is a combination of two words: leadership and entrepreneurship. Therefore, one can contextualize that leadership in entrepreneurship involves gist in both concepts. In [22], it brings up the facts that entrepreneurship is a process of creating something different from others through:
time use;
provision of financial capital and risk along with psychological and social risks;
compensation receipt;
satisfaction; and
personal freedom.
In this case, entrepreneurship is related to the invention or creation of new ideas that are different from others. An entrepreneur should have the characteristics of confidence, results-oriented, risk-taking, leadership, originality, and future-oriented [23]. By referring to the definitions of entrepreneurship and the characteristics of an entrepreneur, entrepreneurship is a process of inventing or creating new products/services, indicated by the behavior of willing to make time, providing financial capital or risk and psychological/social risks, receiving compensation, and having the satisfaction and personal freedom.
Leadership is generally an ability to mobilize, harmonize, activate groups, and deliver ideas to make other people understand, acknowledge, and follow him as a leader. In [24], leadership qualities include being intelligent, sociable, confident, truthful, trustworthy and responsible, persistent, good at expressing thoughts, and having high moral standards. In [25], the leadership refers to the process of the collective briefing that is impactful on mutual willingness to carry out the desired business to achieve the common goals. Furthermore, in [26], the indicators of entrepreneurial leadership are the high ability to motivate, the ability to influence other people, set direction, have a good communication skill, develop changes, address strategic resources, encourage people to be competitive, and good at finding opportunities. Entrepreneurial leadership is leadership characters that are strongly correlated with once concepts and ideas in driving his business to arrive at optimal results (profits). Those concepts and ideas include visions and missions, decision-making, applied strategies, and the like. Leadership characters in entrepreneurship raise inspiration and motivation for employees and business partners to gain business opportunities, in addition to the managerial direction that becomes their competencies to be a role model in entrepreneurship civilization. Thus, each business unit requires leadership in entrepreneurship. This condition often emerges issues in strict competition along with rapid communication and information technology advancement. Many company units are collapsed lately.
In [27], it is argued that entrepreneurial leadership is one’s ability in the process of collective briefing that has an effect on mutual willingness to always invent or create new ideas by giving motivation to be oriented towards the future to look for opportunities, developing changes, directing to be results/achievements-oriented, taking risks, and being persistent. This indicates that entrepreneurial leadership and creativity are variables that cannot be ignored as the dominant factors to predict the level of innovative behavior of product producers in a company. By referring in [28], the innovative behavior of product/service producers has become a fundamental aspect to realize an optimal performance achievement in business activities. This is because such behavior plays a crucial role in improving their product competitiveness as well as encouraging them to do more different things, even within limitations. Hence, the innovative behavior of product producers will try to use their limitations by optimizing new ideas to create new, different, more useful, and more valuable products/services. The results of those study provide further implications that human resources (employees) who are innovative and responsive to the advancement of science and technology must be prepared earlier through educational institutions/schools (early childhood education/kindergarten, elementary schools, junior and senior high schools, and universities). These institutions play a strategic role and cannot be set aside in forming and preparing innovative and responsive behavior of the young generation towards the advancement of science and technology through the character development of entrepreneurial leadership and creativity in the curriculum (hidden curriculum) and noncurriculum (extracurricular).
In [29], higher education institution plays a crucial role and is very potential to educate and prepare the next generation of entrepreneurs leadership in a country, and the key to the success of entrepreneurship leadership development among students (young generation) is that the higher education institution should do the following things:
encouraging students to take active and independence activities, by having a hand on experience in entrepreneurship, such as establishing a business while studying through start-up capital program facilitated by higher education institution/university;
facilitating them to have an internship in companies that partners with the university; and
business mentoring by various entrepreneurs facilitated by university through seminars, workshops, focus group discussion (FGD), etc.
Competitive advantage leadership is leadership that has managerial and entrepreneurial abilities supported by human capital (Figure 1). The foundations of leadership that must be possessed are:
move the organization and its job descriptions effectively;
able to analyze tasks and operational details;
carry out tasks openly and responsibly;
mastering facility management so to be able to provide organizational operational facilities effectively and efficiently;
ability to make appropriate decisions by anticipating the impact of decisions on the performance of the organization and its members;
always empower the organization and its personnel and carry out continuous monitoring; and
always provide couching and counseling to his subordinates.
Competitive advantage leadership architecture (by Soehari TD, 2019).
These basic leadership abilities must be followed by managerial and entrepreneurial abilities, while the managerial abilities that must be possessed are: supervising; planning; organizing; making-decision; monitoring indicator; controlling; response issues; coordinating; consulting; and administrating. While the entrepreneurial abilities that must be possessed are:
having strong concepts and ideas, and able to move business effectively and efficiently;
having a vision and mission that is supported by the ability to make appropriate decisions and ability to implement the outlined strategy;
having the ability to inspire and motivate employees and business partners to obtain opportunities and, at the same time, utilize these business opportunities to achieve business goals; and
the ability to communicate and use IT to support business operations.
The important factors of human capital are:
competence;
training and development;
organization and career path; and
compensation.
Four important factors of human capital are used to support basic leadership, managerial leadership, and entrepreneurial leadership so as to create business competitive advantage.
Erlang distribution is a continuous probability distribution with wide applicability, primarily due to its relation to the exponential and gamma distributions. The Erlang distribution was developed by Erlang [1] to examine the number of telephone calls that could be made at the same time to switching station operators. This distribution can be expressed as waiting time and message length in telephone traffic. If the duration of individual calls are exponentially distributed then the duration of succession of calls is the Erlang distribution. The Erlang variate becomes gamma variate when its shape parameter is an integer (for details see Evans et al. [2]). Bhattacharyya and Singh [3] obtained Bayes estimator for the Erlangian queue under two prior densities. Haq and Dey [4] addressed the problem of Bayesian estimation of parameters for the Erlang distribution assuming different independent informative priors. Suri et al. [5] used Erlang distribution to design a simulator for time estimation of project management process. Damodaran et al. [6] obtained the expected time between failure measures. Further, they showed that the predicted failure times are closer to the actual failure times. Jodra [7] showed the procedure of computing the asymptotic expansion of the median of Erlang distribution.
\nThe probability density function of an Erlang variate is given by
\nwhere \n
In this chapter, Erlang distribution is considered. Some structural properties of Erlang distribution have been obtained. The parameter estimation of Erlang distribution is obtained by employing the maximum likelihood method of estimation, method of moments and Bayesian method of estimation in different sections of this chapter. In Bayesian approach, the parameters are estimated by using Jeffrey’s and Quasi priors under different loss functions (Figure 1).
\nPdf’s of Erlang distribution for different values of lambda and k.
\n
The gamma distribution is a generalized form of the Erlang distribution.
If the shape parameter k is 1, then Erlang distribution reduces to exponential distribution.
If the scale parameter is 2, then Erlang distribution reduces to Chi-square distribution with 2 degrees of freedom.
Thus from the above descriptions, we can say that exponential distribution and Chi-square distribution are the sub-models of Erlang distribution.
\nIn this chapter, we have used different approaches for parameter estimation. The first two methods come under the classical approach which was founded by Fisher in a series of fundamental papers round about 1930.
\nThe alternative approach is the Bayesian approach which was first discovered by Reverend Thomas Bayes. In this chapter, we have used two different priors for the parameter estimation. Also three loss functions are used which are discussed in their respective sections. A number of symmetric and asymmetric loss functions used by various researchers; see Zellner [8], Ahmad and Ahmad [9], Ahmad et al. [10], etc.
\nThese methods of estimations are elaborated in their respective sections accordingly.
\nThe most general method of estimation is known as maximum likelihood (MLH) estimators, which was initially formulated by Gauss. Fisher in the early 1920 firstly introduced MLH as general method of estimation and later on developed by him in a series of papers. He revealed the advantages of this method by showing that it yields sufficient estimators, which are asymptotically MVUES. Thus the important feature of this method is that we look at the value of the random sample and then select our estimate of the unknown population parameter, the value of which the probability of getting the observed data is maximum.
\nSuppose the observed data sample values are \n
Similarly the definition applies when the random sample comes from a continuous population but in that case \n
Since the principle of maximum likelihood consists in finding an estimator of the parameter which maximizes the likelihood function for variation in the parameter. Thus if there exists a function \n
Since \n
a form which is more convenient from practical point of view.
\nThe MLH estimation of the rate parameter of Erlang distribution is obtained in the following theorem:
\nTheorem 2.1: Let \n
Proof: The likelihood function of random sample of size n having Erlang density function Eq. (1) is given by
\nThe log likelihood function is given by
\nDifferentiating Eq. (3) w.r.t. \n
One of the simplest and oldest methods of estimation is the method of moments. The method of moments was discovered by Karl Pearson in 1894. It is a method of estimation of population parameters such as mean, variance, etc. (which need not be moments), by equating sample moments with unobservable population moments and then solving those equations for the quantities to be estimated. The method of moments is special case when we need to estimate some known function of finite number of unknown moments.
\nSuppose \n
In general \n
e.g., \n
where \n
Then by the method of moments \n
The MM estimation of the rate parameter of Erlang distribution is obtained in the following theorem:
\nTheorem 2.2: Let \n
Proof: If the numbers \n
where \n
The rth moment of two parameter Erlang distribution about origin is given by
\nUsing Eq. (5) in Eq. (6), we have
\nIf r = 1 in Eq. (7), we get
\nIf r = 2, then Eq. (7) becomes
\nThus the variance is given by \n
When we divide \n
On taking the square roots of Eq. (8), we have the coefficient of variation
\nThe rate parameter \n
If r = 1, in Eq. (5), then
\nAlso if r = 1 in Eq. (7), then
\nThus,
\n\n
Nowadays, the Bayesian school of thought is garnering more attention and at an increasing rate. This thought of statistics was given by Reverend Thomas Bayes. He first discovered the theorem that now bears his name. It was written up in a paper “An Essay Towards Solving a Problem in the Doctrine of Chances.” This paper was found after his death by his friend Richard Price, who had it published posthumously in the Philosophical Transactions of the Royal Society in 1763. Bayes showed how inverse probability could be used to calculate probability of antecedent events from the occurrence of the consequent event. His methods were adopted by Laplace and other scientists in the nineteenth century. By mid twentieth century interest in Bayesian methods was renewed by De Finetti, Jeffreys and Lindley, among others. They developed a complete method of statistical inference based on Bayes’ theorem. Bayesian analysis is to be used by practitioners for situations where scientists have a priori information about the values of the parameters to be estimated. In everyday life, uncertainty often permeates our choices, and when choices need to be made, past experience frequently proves a helpful aid. Bayesian theory provides a general and consistent framework for dealing with uncertainty.
\nWithin Bayesian inference, there are also different interpretations of probability, and different approaches based on those interpretations. Early efforts to make Bayesian methods accessible for data analysis were made by Raiffa and Schlaifer [11], DeGroot [12], Zellner [13], and Box and Tiao [14]. The most popular interpretations and approaches are objective Bayesian inference and subjective Bayesian inference. Excellent expositions of these approaches are with Bayes and Price [15], Laplace [16], Jeffrey’s [17], Anscombe and Aumann [18], Berger [19, 20], Gelman et al. [21], Leonard and Hsu [22], De-Finetti [23]. Modern Bayesian data analysis and methods based on Markov chain Monte Carlo methods are presented in Bernardo and Smith [24], Robert [25], Gelman et al. [26], Marin and Robert [27], Carlin and Louis [28]. Good elementary introductions to the subject are Ibrahim et al. [29], Ghosh [30], Bansal [31], Koch [32], Hoff [33].
\nIn Bayesian statistics probability is not defined as a frequency of occurrence but as the plausibility that a proposition is true, given the available information. The parameters are treated as random variables. The rules of probability are used directly to make inferences about the parameters. Probability statements about parameters must be interpreted as “degree of belief.” We revise our beliefs about parameters after getting the data by using Bayes’ theorem. This gives our posterior distribution which gives the relative weights we give to each parameter value after analyzing the data. The posterior distribution comes from two sources: the prior distribution and the observed data. This means that the inference is based on the actual occurring data, not all possible data sets that might have occurred.
\nIn this section, posterior distribution of Erlang distribution is obtained by using Jeffrey’s prior and Quasi prior. The rate parameter of Erlang distribution is estimated with the help of different loss functions. For parameter estimation we have used the approach as is used by Ahmad et al. [34], Ahmad et al. [35], etc. Some important prior distributions and loss functions which we have used in this article are given as below:
\nPrior distribution is the basic part of Bayesian analysis which represents all that is known or assumed about the parameter. Usually the prior information is subjective and is based on a person’s own experience and judgment, a statement of one’s degree of belief regarding the parameter.
\nAnother important feature of the Bayesian analysis is the choice of the prior distribution. If the data have sufficient signal, even a bad prior will still not greatly influence the posterior. We can examine the impact of prior by observing the stability of posterior distribution related to different choices of priors. If the posterior distribution is highly dependent on the prior, then the data (the likelihood function) may not contain sufficient information. However, if the posterior is relatively stable over a choice of priors, then the data indeed contains significant information.
\nPrior distribution may be categorical in different ways. One common classification is a dichotomy that separated “proper” and “improper” priors.
\nA prior distribution is proper if it does not depend on the data and the value of integral \n
In this chapter, we have used two different priors Jeffrey’s prior and Quasi prior which are given below:
\nAn invariant form for the prior probability in estimation problems is given by Jeffery’s [36].
\nThe general formula of the Jeffreys prior, which is defined by
\nwhere \n
In this situation, the Jeffreys prior is given by
\nJeffrey suggested a thumb rule for specifying non-informative prior for parameter \n
Rule 1: if \n
Rule 2: if \n
Under linear transformation, rule 1 is invariant and under any power transformation of \n
When there is no more information about the distribution parameter, one may use the Quasi density as given by \n
The concept of loss function is old as Laplace and was reintroduced in statistics by Abraham Wald [37]. In statistics, typically a loss function is used for parameter estimation, and the event in question is some function of the difference between estimated and true values for an occurrence of data. In the context of economics, loss function is usually economic cost. In optimal control, the loss is the penalty for failing to achieve a desired value.
\nThe word “loss” is used in place of “error” and the loss function is used as a measure of the error or loss. Loss function is a measure of the error and presumably would be greater for large error than for small error. We would want the loss to be small or we want the estimate to be close to what it is estimating. Loss depends on sample and we cannot hope to make the loss small for every possible sample but can try to make the loss small on the average. Our objective is to select an estimator that makes this error or loss small, also which makes the average loss (risk) small and ideally select an estimator that has the small risk.
\nIn this chapter, we have used three different Loss Functions which are as under:
\nThe concept of precautionary loss function (PLF) was introduced by Norstrom [38]. He introduced an alternative asymmetric loss function and also presented a general class of precautionary loss functions as a special case. These loss functions approach infinitely near the origin to prevent underestimation, thus giving conservative estimators, especially when low failure rates are being estimated. These estimators are very useful when underestimation may lead to serious consequences. A very useful and simple asymmetric precautionary loss function (PLF) is
\nThe loss function proposed by Al-Bayyati [39] is an asymmetric loss function and is given by
\nwhere \n
The idea of LINEX loss function (LLF) was founded by Klebanov [40] and used by Varian [41] in the context of real estate assessment. The formula of LLF is given by
\nwhere \n
Let \n
and the likelihood function Eq. (2) given as below
\nJeffreys’ non-informative prior for \n
where \n
On solving the above expression, we have
\nBy using the Bayes theorem, we have
\nUsing Eqs. (2) and (10) in Eq. (11), we get
\nwhere ρ is independent of λ and
\nUsing the value of ρ in Eq. (12)
\nLet \n
The Quasi for \n
By using the Bayes theorem, we have
\nUsing Eqs. (2) and (14) in Eq. (15), we have
\nwhere \n
By using the value of \n
In this section, parameter estimation of Erlang distribution is done by using Jeffreys’ prior under different loss functions. The procedure of calculating the Bayesian estimate is already defined in Section 3. The estimates are obtained in the following theorems:
\nTheorem 4.1: Assuming the loss function \n
Proof: The risk function of the estimator \n
Using Eq. (13) in Eq. (18), we get
\nOn solving the above expression, we get
\nMinimization of the risk with respect to \n
Theorem 4.2: Assuming the loss function \n
Proof: The risk function of the estimator \n
On substituting Eq. (13) in Eq. (20), we have.
\nOn solving the above expression, we get
\nMinimization of the risk with respect to \n
Theorem 4.3: Assuming the loss function \n
Proof: The risk function of the estimator \n
Using Eq. (13) in Eq. (22), we have
\nOn solving the above expression, we get
\nMinimization of the risk with respect to \n
In this section, parameter estimation of Erlang distribution is done by using QUASI prior under different loss functions. The procedure for obtaining the Bayesian estimate is available in Section 3. The estimates of parameter are obtained in the following theorems.
\nTheorem 5.1: Assuming the loss function \n
Proof: The risk function of the estimator \n
Using Eq. (17) in Eq. (24), we have
\nOn solving the above expression, we get
\nMinimization of the risk with respect to \n
Remark: Replacing d = 1 in Eq. (25), the same Bayes estimator is obtained as in Eq. (19).
\nTheorem 5.2: Assuming the loss function \n
Proof: The risk function of the estimator \n
By using Eq. (17) in Eq. (26), we have.
\nOn solving the above expression, we get
\nMinimization of the risk with respect to \n
Remark: Replacing d = 1 in Eq. (27), the same Bayes estimator is obtained as in Eq. (21).
\nTheorem 5.3: Assuming the loss function \n
Proof: The risk function of the estimator \n
Using Eq. (17) in Eq. (28), we have.
\nOn solving the above expression, we get
\nMinimization of the risk with respect to \n
Remark: Replacing d = 1 in Eq. (29), the same Bayes estimator is obtained as in Eq. (23).
\nThe concept of entropy was introduced by Claude. Shannon [42] in the paper “A Mathematical theory of Communication.” This concept of Shannon’s entropy is the central role of information theory, sometimes referred as measure of uncertainty. Shannon entropy provides an absolute limit on the best possible lossless encoding or compression of any communication, assuming that the communication may be represented as a sequence of independent and identical distributed random variables. Entropy is typically measured in bits, when the log is to the base 2, and nats, when the log is to the base n.
\nShannon’s definition of entropy, when applied to an information source, can determine the minimum channel capacity required to reliably transmit the source as encoded binary digit. The entropy of a random variable is defined in terms of its probability distribution and can be shown to be a good measure of randomness or uncertainty. For deriving the entropy of probability distributions, we need the following two definitions that are more discussed in Cover et al. [43].
\nDefinition (i): The entropy of the discrete random variable defined on the probability space is given by
\nIt is obvious that \n
Definition (ii): The entropy of the continuous random variable defined on the real line is given by
\nIn this section, entropy estimation of two parameter Erlang distribution is discussed which given as below.
\nTheorem 6.1: Let \n
Proof: Shannon’s entropy for a continuous random variable is defined as
\nUsing Eq. (1) in Eq. (30), we have
Now
\nAlso
\nSubstitute the value of Eqs. (32) and (33) in Eq. (31), we have.
\nFor model selection the approach of Akaike information criterion (AIC) and Bayesian information criterion (BIC) based on entropy estimation are used. The Akaike information criterion (AIC) was introduced by Hirotsugu Akaike [44] and proposed it as a measure of goodness of fit of an estimated statistical model. It is a measure of the relative quality of a statistical model for a given set of data. It has been found in information theory that it offers a relative estimate of the information lost when a given model is used to represent the process that generates the data.
\nThe AIC is not a test of the model in the sense of hypothesis testing; rather it is a test between models—a tool for model selection. Given a data set, several competing models may be ranked according to their AIC, with the one having the lowest AIC being the best.
\nThe formula for AIC is given by
\nwhere K is the number of parameters and \n
AICC was first introduced by Hurvich and Tsai [45] and its different derivations were proposed by Burnham and Anderson [46]. AICC is AIC with a correction for finite sample sizes and is given by
\nBurnham and Anderson [47] strongly recommended that we should use AICC instead of AIC when the sample size is small or if K is large. Since AICC converges to AIC as the sample size is getting large. Using AIC, instead of AICC, when the sample size is not many times larger than K2, increases the probability of selecting models that have too many parameters, i.e., of over fitting. The probability of AIC over fitting can be substantial, in some cases.
\nThe Bayesian information criterion (BIC) also known as Schwarz Criterion is used as a substitute for full calculation of the Bayes’ factor since it can be calculated without specifying prior distribution. In BIC, the penalty for additional parameters is stronger than that of the AIC.
\nThe formula for the BIC is given by
\nwhere K is the number of parameters, n is the sample size and \n
The AIC and BIC of two parameter Erlang distribution are obtained in this section, which are given below.
\nThe Shannon’s entropy of two parameter Erlang distribution is given by
\nAlso
\nComparing Eqs. (35) and (36), we have
\nThe AIC and BIC methodology attempts to find the model that best explains the data with a minimum of their values, we have.
\n\n\n
or
\nand
\nor
\nWe have generated the data for Erlang distribution of different sample sizes (15, 30 and 60) in R Software for each pairs of \n
The flexibility and potentiality of the Erlang distribution is compared with its sub models, which is examined by using different criterions like AIC, BIC and AICC with the help of the following illustration.
\nIllustration I:
\nWe provide the compatibility of the Erlang distribution (ED) with their sub-models; Chi square and exponential distributions. For this purpose, we generated the data set for Erlang distribution of large sample size (i.e., 200) in R Software for each pairs of \n
Illustration II:
\nThe data set is taken from Lawless [48]. The observations involves the number of million revolutions between failures for each of 23 ball bearings, the individual bearings were inspected periodically to determine whether “failure” had occurred. Treating the failure times as continuous, the 23 failure times are:
\n(17.88, 28.92, 33.00, 41.52, 42.12, 45.60, 48.40, 51.84, 51.96, 54.02, 55.56, 67.80, 68.64, 68.64, 68.88, 84.12, 93.12, 98.64, 105.12, 105.84, 127.92, 128.04, 173.40)
\nWe primarily studied the maximum likelihood (MLH) estimation and Bayesian estimation to estimate the rate parameter of Erlang distribution. In Bayesian method, we use Jeffreys’ prior and Quasi prior under three different loss functions. These methods are compared through simulation technique and the results are presented in the Tables 1 and 2 respectively.
\nN | \nk | \n\n\n | \n\n\n | \n\n\n | \n\n\n | \n\n\n | \n||
---|---|---|---|---|---|---|---|---|
c = −1 | \nc = 1 | \na = 0.5 | \na = 1.0 | \n|||||
15 | \n1 | \n0.5 | \n0.15350 | \n0.16179 | \n0.10530 | \n0.18428 | \n0.13350 | \n0.12597 | \n
2 | \n1.0 | \n0.51869 | \n0.54673 | \n0.43100 | \n0.58896 | \n0.47443 | \n0.44430 | \n|
30 | \n1 | \n0.5 | \n0.03480 | \n0.03238 | \n0.03238 | \n0.03266 | \n0.04191 | \n0.03259 | \n
2 | \n1.0 | \n0.06850 | \n0.04258 | \n0.07049 | \n0.02085 | \n0.03077 | \n0.02171 | \n|
60 | \n1 | \n0.5 | \n0.03910 | \n0.03320 | \n0.02856 | \n0.03488 | \n0.03110 | \n0.03060 | \n
\n | \n2 | \n1.0 | \n0.10349 | \n0.09918 | \n0.08981 | \n0.10244 | \n0.09398 | \n0.09200 | \n
Mean squared error for \n
ML = maximum likelihood estimate, p = precautionary LF, A = Al-Bayyati’s LF, l = LINEX LF.
N | \nk | \n\n\n | \nd | \n\n\n | \n\n\n | \n\n\n | \n\n\n | \n||
---|---|---|---|---|---|---|---|---|---|
c = −1 | \nc = 1 | \na = 0.5 | \na = 1.0 | \n||||||
15 | \n1 | \n0.5 | \n0.5 | \n0.15350 | \n0.14710 | \n0.11081 | \n0.18979 | \n0.13901 | \n0.13148 | \n
1.0 | \n0.15350 | \n0.16179 | \n0.10530 | \n0.18428 | \n0.13350 | \n0.12597 | \n|||
2 | \n1.0 | \n0.5 | \n0.51869 | \n0.51528 | \n0.43951 | \n0.59747 | \n0.48294 | \n0.45281 | \n|
1.0 | \n0.51869 | \n0.54673 | \n0.43100 | \n0.58896 | \n0.47443 | \n0.44430 | \n|||
30 | \n1 | \n0.5 | \n0.5 | \n0.03480 | \n0.03286 | \n0.02179 | \n0.02206 | \n0.21630 | \n0.02200 | \n
1.0 | \n0.03480 | \n0.03238 | \n0.03238 | \n0.03266 | \n0.04191 | \n0.03259 | \n|||
2 | \n1.0 | \n0.5 | \n0.06850 | \n0.05828 | \n0.06361 | \n0.01397 | \n0.02389 | \n0.01483 | \n|
1.0 | \n0.06850 | \n0.04258 | \n0.07049 | \n0.02085 | \n0.03077 | \n0.02171 | \n|||
60 | \n1 | \n0.5 | \n0.5 | \n0.03910 | \n0.03206 | \n0.03206 | \n0.03534 | \n0.03156 | \n0.03106 | \n
1.0 | \n0.03910 | \n0.03320 | \n0.02856 | \n0.03488 | \n0.03110 | \n0.03060 | \n|||
2 | \n1.0 | \n0.5 | \n0.10349 | \n0.09641 | \n0.09021 | \n0.10284 | \n0.09439 | \n0.09241 | \n|
\n | \n\n | \n1.0 | \n0.10349 | \n0.09918 | \n0.08981 | \n0.10244 | \n0.09398 | \n0.09200 | \n
Mean squared error for \n
ML = maximum likelihood estimate, p = precautionary LF, A = Al-Bayyati’s LF, l = LINEX LF.
From the results obtained in Tables 1 and 2, we observe that in most of the cases, Bayesian estimator under Al-Bayyati’s loss function has the smallest mean squared error (MSE) values for Jeffrey’s prior and Quasi prior as compared to other loss functions and the maximum likelihood estimator. Thus we can conclude that Bayes estimator under Al-Bayyati’s loss function is efficient when the loss parameter C is −1.
\nAlso we estimated the unknown parameters of sub-models of Erlang distribution. The Akaike information criterion (AIC), Bayesian information criterion (BIC), and the corrected Akaike information criterion (AICC) are used to compare the candidate distributions. The best distribution corresponds to lower −logL, AIC, BIC, AICC statistics value.
\nFrom the results obtained in Tables 3 and 4, we observe that the Erlang distribution is a competitive distribution as compared to its sub-models (i.e., exponential distribution and Chi-square distribution). In fact, based on the values of the AIC, BIC and AICC criteria, it shows clear picture that the Erlang distribution provides the best fit for these data among all the models considered.
\nModel | \nParameter estimate | \nStandard error | \nMeasures | \n|||
---|---|---|---|---|---|---|
\n\n | \nAIC | \nBIC | \nAICC | \n|||
Exponential | \n\n\n | \n0.04203 | \n315.6853 | \n633.3705 | \n636.6689 | \n581.9245 | \n
Chi-square | \nk = 1.090 | \n0.0586 | \n312.3096 | \n625.6346 | \n628.9329 | \n633.3902 | \n
Erlang | \nk = 2 | \n289.9327 | \n581.8654 | \n585.1637 | \n625.6543 | \n|
\n | \n\n\n | \n0.0594 | \n\n | \n\n | \n\n | \n\n | \n
AIC, BIC and AICC criterion for different sub-models of ED.
Model | \nParameter estimate | \nStandard error | \nMeasures | \n|||
---|---|---|---|---|---|---|
\n\n | \nAIC | \nBIC | \nAICC | \n|||
Exponential | \n\n\n | \n0.002877 | \n121.4324 | \n244.8648 | \n246.0003 | \n245.0248 | \n
Chi-square | \nk = 2.9420 | \n1.1745 | \n177.5835 | \n357.1670 | \n358.3025 | \n357.327 | \n
Erlang | \nk = 1 | \n113.0298 | \n230.0597 | \n232.3307 | \n230.5212 | \n|
\n | \n\n\n | \n0.01683 | \n\n | \n\n | \n\n | \n\n | \n
AIC, BIC and AICC criterion for different sub-models of ED.
In this paper we have generated three types of data sets with varying sample sizes for Erlang distribution. These data sets were simulated and behavior of the data was checked in case of parameter estimation for Erlang distribution in R Software. By the virtue of the data analysis we are able predict the estimate of rate parameter for Erlang distribution under three different functions by using two different prior distributions. With the help of these results we can also do comparison between loss functions and the priors.
\nAlso the comparison of Erlang distribution with its sub-models was carried out. The results acquired in Tables 3 and 4, it shows the clear picture that Erlang distribution performs better as compared to its sub-models. Thus we can say that Erlang distribution is efficient as compared to its sub-models (i.e., exponential distribution and Chi-square distribution) on the basis of the above procedures.
\nThe authors acknowledge editor of this journal for encouragement to finalize the chapter. Further, the authors acknowledge profound thanks to anonymous referee for giving critical comments which have immensely improved the presentation of the chapter. Also I extend my sincere thanks to all the authors whose papers I have consulted for this work.
\nThe authors declare that there is no conflict of interests regarding the publication of this chapter.
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\n\n4.1 The Corresponding Author represents and warrants that the Chapter does not and will not breach any applicable law or the rights of any third party and, specifically, that the Chapter contains no matter that is defamatory or that infringes any literary or proprietary rights, intellectual property rights, or any rights of privacy. The Corresponding Author warrants and represents that: (i) the Chapter is the original work of themselves and any Co-Author and is not copied wholly or substantially from any other work or material or any other source; (ii) the Chapter has not been formally published in any other peer-reviewed journal or in a book or edited collection, and is not under consideration for any such publication; (iii) they themselves and any Co-Author are qualifying persons under section 154 of the Copyright, Designs and Patents Act 1988; (iv) they themselves and any Co-Author have not assigned and will not during the term of this Publication Agreement purport to assign any of the rights granted to IntechOpen under this Publication Agreement; and (v) the rights granted by this Publication Agreement are free from any security interest, option, mortgage, charge or lien.
\n\nThe Corresponding Author also warrants and represents that: (i) they have the full power to enter into this Publication Agreement on their own behalf and on behalf of each Co-Author; and (ii) they have the necessary rights and/or title in and to the Chapter to grant IntechOpen, on behalf of themselves and any Co-Author, the rights and licenses expressed to be granted in this Publication Agreement. If the Chapter was prepared jointly by the Corresponding Author and any Co-Author, the Corresponding Author warrants and represents that: (i) each Co-Author agrees to the submission, license and publication of the Chapter on the terms of this Publication Agreement; and (ii) they have the authority to enter into this Publication Agreement on behalf of and bind each Co-Author. The Corresponding Author shall: (i) ensure each Co-Author complies with all relevant provisions of this Publication Agreement, including those relating to confidentiality, performance and standards, as if a party to this Publication Agreement; and (ii) remain primarily liable for all acts and/or omissions of each such Co-Author.
\n\nThe Corresponding Author agrees to indemnify and hold IntechOpen harmless against all liabilities, costs, expenses, damages and losses and all reasonable legal costs and expenses suffered or incurred by IntechOpen arising out of or in connection with any breach of the aforementioned representations and warranties. This indemnity shall not cover IntechOpen to the extent that a claim under it results from IntechOpen's negligence or willful misconduct.
\n\n4.2 Nothing in this Publication Agreement shall have the effect of excluding or limiting any liability for death or personal injury caused by negligence or any other liability that cannot be excluded or limited by applicable law.
\n\n5. TERMINATION
\n\n5.1 IntechOpen has a right to terminate this Publication Agreement for quality, program, technical or other reasons with immediate effect, including without limitation (i) if the Corresponding Author or any Co-Author commits a material breach of this Publication Agreement; (ii) if the Corresponding Author or any Co-Author (being an individual) is the subject of a bankruptcy petition, application or order; or (iii) if the Corresponding Author or any Co-Author (being a company) commences negotiations with all or any class of its creditors with a view to rescheduling any of its debts, or makes a proposal for or enters into any compromise or arrangement with any of its creditors.
\n\nIn case of termination, IntechOpen will notify the Corresponding Author, in writing, of the decision.
\n\n6. INTECHOPEN’S DUTIES AND RIGHTS
\n\n6.1 Unless prevented from doing so by events outside its reasonable control, IntechOpen, in its discretion, agrees to publish the Chapter attributing it to the Corresponding Author and any Co-Author.
\n\n6.2 IntechOpen has the right to use the Corresponding Author’s and any Co-Author’s names and likeness in connection with scientific dissemination, retrieval, archiving, web hosting and promotion and marketing of the Chapter and has the right to contact the Corresponding Author and any Co-Author until the Chapter is publicly available on any platform owned and/or operated by IntechOpen.
\n\n6.3 IntechOpen is granted the authority to enforce the rights from this Publication Agreement, on behalf of the Corresponding Author and any Co-Author, against third parties (for example in cases of plagiarism or copyright infringements). In respect of any such infringement or suspected infringement of the copyright in the Chapter, IntechOpen shall have absolute discretion in addressing any such infringement which is likely to affect IntechOpen's rights under this Publication Agreement, including issuing and conducting proceedings against the suspected infringer.
\n\n7. MISCELLANEOUS
\n\n7.1 Further Assurance: The Corresponding Author shall and will ensure that any relevant third party (including any Co-Author) shall, execute and deliver whatever further documents or deeds and perform such acts as IntechOpen reasonably requires from time to time for the purpose of giving IntechOpen the full benefit of the provisions of this Publication Agreement.
\n\n7.2 Third Party Rights: A person who is not a party to this Publication Agreement may not enforce any of its provisions under the Contracts (Rights of Third Parties) Act 1999.
\n\n7.3 Entire Agreement: This Publication Agreement constitutes the entire agreement between the parties in relation to its subject matter. It replaces and extinguishes all prior agreements, draft agreements, arrangements, collateral warranties, collateral contracts, statements, assurances, representations and undertakings of any nature made by or on behalf of the parties, whether oral or written, in relation to that subject matter. Each party acknowledges that in entering into this Publication Agreement it has not relied upon any oral or written statements, collateral or other warranties, assurances, representations or undertakings which were made by or on behalf of the other party in relation to the subject matter of this Publication Agreement at any time before its signature (together "Pre-Contractual Statements"), other than those which are set out in this Publication Agreement. Each party hereby waives all rights and remedies which might otherwise be available to it in relation to such Pre-Contractual Statements. Nothing in this clause shall exclude or restrict the liability of either party arising out of its pre-contract fraudulent misrepresentation or fraudulent concealment.
\n\n7.4 Waiver: No failure or delay by a party to exercise any right or remedy provided under this Publication Agreement or by law shall constitute a waiver of that or any other right or remedy, nor shall it preclude or restrict the further exercise of that or any other right or remedy. No single or partial exercise of such right or remedy shall preclude or restrict the further exercise of that or any other right or remedy.
\n\n7.5 Variation: No variation of this Publication Agreement shall be effective unless it is in writing and signed by the parties (or their duly authorized representatives).
\n\n7.6 Severance: If any provision or part-provision of this Publication Agreement is or becomes invalid, illegal or unenforceable, it shall be deemed modified to the minimum extent necessary to make it valid, legal and enforceable. If such modification is not possible, the relevant provision or part-provision shall be deemed deleted.
\n\nAny modification to or deletion of a provision or part-provision under this clause shall not affect the validity and enforceability of the rest of this Publication Agreement.
\n\n7.7 No partnership: Nothing in this Publication Agreement is intended to, or shall be deemed to, establish or create any partnership or joint venture or the relationship of principal and agent or employer and employee between IntechOpen and the Corresponding Author or any Co-Author, nor authorize any party to make or enter into any commitments for or on behalf of any other party.
\n\n7.8 Governing law: This Publication Agreement and any dispute or claim (including non-contractual disputes or claims) arising out of or in connection with it or its subject matter or formation shall be governed by and construed in accordance with the law of England and Wales. The parties submit to the exclusive jurisdiction of the English courts to settle any dispute or claim arising out of or in connection with this Publication Agreement (including any non-contractual disputes or claims).
\n\nLast updated: 2020-11-27
\n\n\n\n
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