Open access peer-reviewed chapter

Estimation of Means of Two Quantitative Sensitive Variables Using Randomized Response Technique

Written By

Amod Kumar

Submitted: July 20th, 2021Reviewed: October 18th, 2021Published: April 6th, 2022

DOI: 10.5772/intechopen.101269

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Abstract

I propose an improved randomized response model for the simultaneous estimation of population means of two quantitative sensitive variables by using blank card option that make use of one scramble response and another fake response. The properties of the proposed estimator have been analysed. To judge the performance of the proposed model, I have considered a real data set and it is to be pointed out that the proposed model is more efficient in terms of relative efficiencies and privacy protection of respondents as well. Suitable recommendations have been made to the survey practitioners.

Keywords

  • randomized response technique
  • two quantitative sensitive variables
  • estimation of two means\\
  • blank card
  • privacy protection

1. Introduction

Reliability of data is compromised when sensitive topics on embarrassing or illegal acts such as students taking drugs, drunk driving, abortion, family income, tax evasion etc. are required in direct method of data collection in sample survey. Survey on human population has established the fact that the direct question about sensitive characters often results in either refusal to respond or falsification of the answer. To overcome this difficulty and ensure confidentially of respondents, Warner [1] initiated a technique which is called as randomized response technique (RRT). For estimating π, the population proportion of respondents, a simple random sample of size nrespondents selected from the population Nwith replacement. Each respondent selected in the sample has a random device which consists two statements “I belong to sensitive group A” and “its compliment Ac”. The respondent answers of sensitive or non-sensitive questions depending on the outcome of the random device which is unobservable to the sampler. Greenberg et al. [2] adjusted the Warner [1] model with respect to efficiency and respondent’s cooperation by suggesting unrelated question randomized response model, where the sensitive question was combined with an unrelated (non-sensitive) question.

Greenberg et al. [3] extended the Greenberg et al. [2] model to estimate the population mean of quantitative sensitive variable, such as income, tax dodging etc. In their model, each respondent selected in the sample with replacement was given a random device which presents two outcomes Yand Xwith probabilities Pand 1Prespectively, where Yis the true quantitative sensitive variable and Xis non-sensitive independent variable. Later, Eichhorn and Hayre [4] introduced a new multiplicative randomized response model for estimating the population mean of quantitative sensitive variable.

Under simple random sampling with replacement (SRSWR) scheme, Perri [5] modified Greenberg et al. [3] technique to obtain the estimator of population mean μYby using a blank card option, if a blank card is selected then the respondents are requested to use Greenberg et al. [3] model. In his model, the observed response θPis given by:

θP=YwithprobabilityP1YwithprobabilityP2Blank CardwithprobabilityP3E1

Perri [5] proposed an unbiased estimator of the population mean μY

μ̂P=θ¯PP2+P31PμXP1+P3PE2

with variance

Vμ̂P=σθP2nP1+P3P2E3

where θ¯P=1/ni=1nθPiand

σθP2=P1+P3PσY2+μY2+P2+P31PσX2+μX2P1+P3PμY+P2+P31PμX2

Many different suggestions have been made for the use of these blank cards by various authors including Bhargava and Singh [6], Singh et al. [7], Batool et al. [8], Singh [9] and Singh et al. [10, 11] among others. Furthermore in addition, the theory of randomized response technique to estimate the population parameters of sensitive characteristics was extended by Narjis and Shabbir [12, 13].

Recently, Ahmed et al. [14] have introduced the idea to estimate the means of two quantitative sensitive variables simultaneously by using one scramble response and other face response. Let Y1iand Y2ibe the two values of quantitative sensitive variables with means μY1μY2and variances σY12σY22respectively connected with the ithunit in the populationN. The parameters of interest are μY1μY2which are to be estimated. Each respondent selected in the sample with replacement is asked to produce two fake values of scramble variables S1and S2from two known distributions. Let S1and S2be the independent scramble variables with known means θ1θ2and variance γ20γ02respectively, which help to maintain the protection of respondents. Ahmed et al. [14] defined the scramble response as:

Z1i=S1Y1i+S2Y2iE4

Each respondent selected in the sample is also requested to draw a card from the deck which consist two types of cards, similar to Warner [1] model but has different type of outcomes. Let Pbe the probability of cards bearing the statements in the deck, “the selected respondent to report scramble response as S1” and 1Pis the probability of cards bearing the statement in the deck, “the selected respondent to report scramble response as S2”. Thus, the second response from the ithrespondent given as:

Zi=S1withprobabilityPS2withprobability1PE5

where Pθ1γ20θ1γ20+θ2γ02

Ahmed et al. [14] proposed unbiased estimators of population means μY1and μY2respectively, and are given as:

μ̂AY1=1θ2+1Pγ02+θ22Z¯1θ2Z¯21Pθ1γ022γ20E6

and

μ̂AY2=θ1Z¯2Pγ20+θ12+1Pθ1θ2Z¯11Pθ1γ022γ20E7

where Z¯1=1ni=1nZ1iand Z¯2=1ni=1nZ2i

In follow up of above works and motivated by Ahmed et al. [14], I adopt Perri [5] method and proposes a new improved randomized response model by introducing blank card option for estimation of population means of two quantitative sensitive variables. For example, Y1may stand for the respondents’ income and Y2may stand for the respondents’ expenditure, Y1denotes the import in millions and Y2denotes the export in millions etc. I have demonstrated the efficacious performance of the proposed randomized response model over the Ahmed et al. [14] model along with privacy protection of respondents.

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2. Proposed model

In the proposed model, I have considered the similar supposition as it is the case of Ahmed et al. [14] procedure with the modification that the second response of Warner [1] method is replaced with Perri [5] blank card method. Proceeding on the lines of Ahmed et al. [14] as given in their model, the first observed response is given by:

Z1i=S1Y1i+S2Y2iE8

Noted that by mixing of two quantitative sensitive variables with two scramble variables will make more comfortable to respondent about providing information because it make very hard to guess the true value of two quantitative sensitive variables to an interviewer.

Here I differ from the existing randomized response model available in the literature, in that, the second response is replaced with Perri [5] procedure but has different outcomes. Each selected respondent in the sample provided a random device which consists three type of cards bearing the statements (i) green cards with the statement: report scramble variable S1, (ii) red cards with the statement: report scramble variable S2and (iii) yellow card with no statement (blank cards) with probabilities P1, P2and P3respectively such that i=13Pi=1. Thus, the second response ZAiin the proposed model from ithrespondent is given by:

ZAi=S1withprobabilityP1S2withprobabilityP2BlankcardwithprobabilityP3E9

If a blank card is selected, the respondents are requested to use Ahmed et al. [14] second response. Thus, the second response ZAican be rewritten as:

ZAi=S1withprobabilityP1S2withprobabilityP2S1withprobabilityPS2withprobability1PwithprobabilityP3E10

where PP2+P3θ1γ02P2θ2γ20P3θ1γ02+θ2γ20

Taking expectation on both sides of (Eq. (8)), I have

EZ1i=ES1Y1i+S2Y2i=θ1μY1+θ2μY2E11

With the help from Eqs. (8) and (10), I generate a new response Z2ias:

Z2i=Z1iZAi=S12Y1i+S1S2Y2iwithprobabilityP1S1S2Y1i+S22Y2iwithprobabilityP2S12Y1i+S1S2Y2iwithprobabilityPS1S2Y1i+S22Y2iwithprobability1PwithprobabilityP3E12

Taking expectation on both sides of (Eq. (12)), I get

EZ2i=P1+P3Pγ20+θ12μY1+θ1θ2μY2+P2+P31Pθ1θ2μY1+γ02+θ22μY2E13

Using the method of moments on Eqs. (11) and (13), I have:

θ1μ̂Y1+θ2μ̂Y2=1ni=1nZ1iE14

and

P1+P3Pγ20+θ12+P2+P31Pθ1θ2μ̂Y1+P1+P3Pθ1θ2+P2+P31Pγ02+θ22μ̂Y2=1ni=1nZ2iE15

Eqs. (14) and (15) can be rewritten as:

θ1,θ2P1+P3Pγ20+θ12P1+P3Pθ1θ2+P2+P31Pθ1θ2,+P2+P31Pγ02+θ22μ̂Y1μ̂Y2=Z¯1Z¯2E16

Applying Cramer’s rule on Eq. (16), I obtain

Δ=θ1,θ2P1+P3Pγ20+θ12+P2+P31Pθ1θ2,P1+P3Pθ1θ2+P2+P31Pγ02+θ22
=θ1P1+P3Pθ1θ2+P2+P31Pγ02+θ22θ2P1+P3Pγ20+θ12+P2+P31Pθ1θ2
=P2+P31Pθ1γ02P1+P3Pθ2γ20E17
Δ1=Z¯1,θ2Z¯2,P1+P3Pθ1θ2+P2+P31Pγ02+θ22
=P1+P3Pθ1θ2+P2+P31Pγ02+θ22Z¯1θ2Z¯2E18

and

Δ2=θ1Z¯1P1+P3Pγ20+θ12+P2+P31Pθ1θ2,Z¯2
=θ1Z¯2P1+P3Pγ20+θ12+P2+P31Pθ1θ2Z¯1E19

Thus, the estimators of the population mean μY1and μY2are respectively given by:

μ̂Y1=Δ1Δ=P1+P3Pθ1θ2+P2+P31Pγ02+θ22Z¯1θ2Z¯2P2+P31Pθ1γ02P1+P3Pθ2γ20E20

and

μ̂Y2=Δ2Δ=θ1Z¯2P1+P3Pγ20+θ12+P2+P31Pθ1θ2Z¯1P2+P31Pθ1γ02P1+P3Pθ2γ20E21

I have the following theorems.

Theorem 1:μ̂Y1is an unbiased estimator of the population mean μY1.

Eμ̂Y1=μY1E22

Proof:Taking expectation on both sides of Eq. (20), I have

Eμ̂Y1=P1+P3Pθ1θ2+P2+P31Pγ02+θ22EZ¯1θ2EZ¯2P2+P31Pθ1γ02P1+P3Pθ2γ20
=P1+P3Pθ1θ2+P2+P31Pγ02+θ221/ni=1nEZ1iθ21/ni=1nEZ2iP2+P31Pθ1γ02P1+P3Pθ2γ20
=P1+P3Pθ1θ2+P2+P31Pγ02+θ22i=1nθ1μY1+θ2μY2θ2i=1nP1+P3Pγ20+θ12μY1+θ1θ2μY2+P2+P31Pθ1θ2μY1+γ02+θ22μY2nP2+P31Pθ1γ02P1+P3Pθ2γ20

After simplification, I get

=P2+P31Pθ1γ02P1+P3Pθ2γ20μY1P2+P31Pθ1γ02P1+P3Pθ2γ20=μY1.

which completes the proof.

Theorem 2:μ̂Y2is an unbiased estimator of the population mean μY2.

Eμ̂Y2=μY2E23

Proof:Taking expectation on both sides of Eq. (21), I have

μ̂Y2=θ1EZ¯2P1+P3Pγ20+θ12+P2+P31Pθ1θ2EZ¯1P2+P31Pθ1γ02P1+P3Pθ2γ20

Similarly, following the pattern as given in Theorem 1, I obtain

=P2+P31Pθ1γ02P1+P3Pθ2γ20μY2P2+P31Pθ1γ02P1+P3Pθ2γ20=μY2.

hence, it is proved.

Theorem 3:The variance of the unbiased estimator μ̂Y1is given by:

Vμ̂Y1=P1+P3Pθ1θ2+P2+P31Pγ02+θ222σZ12+θ22σZ222θ2P1+P3Pθ1θ2+P2+P31Pγ02+θ22σZ1Z2nP2+P31Pθ1γ02P1+P3Pθ2γ202E24

where σZ12=γ20σY12+μY12+γ02σY22+μY22+θ12σY12+θ22σY22+2θ1θ2σY1σY2,

σZ22=σY12+μY12P1+P3Pγ40+4γ30θ1+6γ20θ12+θ14+P2+P31Pγ20+θ12γ02+θ22+σY22+μY22P2+P31Pγ04+4γ03θ2+6γ02θ22+θ24+P1+P3Pγ20+θ12γ02+θ22+2σY1σY2+μY1μY2P1+P3Pθ2γ30+3γ20θ1+θ13+P2+P31Pθ1γ03+3γ02θ2+θ23P1+P3Pγ20+θ12+P2+P31Pθ1θ2μY1+P1+P3Pθ1θ2+P2+P31Pγ02+θ22μY22

and

σZ1Z2=σY12+μY12P1+P3Pγ30+3γ20θ1+θ13+P2+P31Pθ2γ20+θ12+σY22+μY22P1+P3Pθ1γ02+θ22+P2+P31Pγ03+3γ02θ2+θ23+2σY1σY2+μY1μY2P1+P3Pθ2γ20+θ12+P2+P31Pθ1γ02+θ22θ1μY1+θ2μY12P1+P3Pγ20+θ12+P2+P31Pθ1θ2μY1+P1+P3Pθ1θ2+P2+P31Pγ02+θ22μY2

The proof is given in Appendix.

Theorem 4:The variance of the unbiased estimator μ̂Y2is given by:

Vμ̂Y2=θ12σZ22+P1+P3Pγ20+θ12+P2+P31Pθ1θ22σZ122θ1P1+P3Pγ20+θ12+P2+P31Pθ1θ2σZ1Z2nP2+P31Pθ1γ02P1+P3Pθ2γ202E25

Proof:The proof is similar as given in Theorem 3.

In the next section, I discuss a privacy protection measure to compare the respondent’s privacy protection and efficiency for the considered model.

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3. Privacy protection measure

A number of measures have been introduced in the literature to estimate the performance of competitive strategies taking into account both efficiency and respondent privacy protection. For a discussion on privacy protection measures for randomized response survey of stigmatizing character, see Lanke [15], Leyseiffer and Warner [16], Bhargava and Singh [17] and among other. These measures of privacy protection are based on the qualitative characters.

When dealing with quantitative sensitive variable, the respondent privacy is conserved by asking interviewees to algebraically scramble the true response by means of a coding mechanism. Respondent’s privacy protection measures for quantitative sensitive variable have been investigated by Diana and perri [18] and Zhimin et al. [19] which is based on the square of correlation coefficient i.e. ρ201. Later, Diana and Perri [20] introduced the new measure of privacy protection of respondents by using auxiliary variable. These measures are normalized with zero (one) denoting maximum (minimum) privacy protection. Recently, Singh et al. [11] considered the case when no auxiliary variable is available in the procedure and studied the normalized measure of respondent privacy. This normalized measure allows researchers to attain a trade-off between efficiency and privacy. Moreover, it is worth remarking that if one procedure is more efficient than other, then it will be less protective. Thus, all the provided measures using the randomized procedure for the privacy protection, they have concluded for a measure of respondent’s privacy protection having a trade-off between these two aspects.

τ=1ρ2E26

The values of τcloser to 1 indicates more privacy protection and greater cooperation may be expected using randomized response models while τcloser to zero denotes that the privacy protection is completely violated. Now, I use this normalized measure for comparing the trade-off between efficiency and privacy protection.

In the proposed model, there are two quantitative sensitive variables Y1iand Y2iassociated with the second observed response Z2i. Following Section 2, I compute the square of correlation coefficients between the second observed response Z2iand quantitative sensitive variables Y1iand Y2irespectively, and are given as:

ρy1iZ2i2=P1+P3Pγ20+θ12σY12+θ1θ2σY1σY2+P2+P31Pθ1θ2σY12+γ20+θ22σY1σY22σY12σZ22E27

and

ρy2iZ2i2=P1+P3Pγ20+θ12σY1σY2+θ1θ2σY22+P2+P31Pθ1θ2σY1σY2+γ20+θ22σY222σY22σZ22E28

where σZ22is given in Theorem 3.

Now, I define the measure of respondent’s privacy protection associated with the proposed second response Z2ias:

τPJ=1ρyJiZ2i2,J=1,2E29

I also define the square of correlation coefficients for the Ahmed et al. [14] response model. In the case of Ahmed et al. [14] model, there are also two quantitative sensitive variables associated with the second observed response. Thus, the square of correlation coefficients between the second observed response and quantitative sensitive variables Y1iand Y2iare respectively given by:

ρy1iZ2i2=Pγ20+θ12σY12+θ1θ2σY1σY2+1Pθ1θ2σY12+γ20+θ22σY1σY22σY12σZ22E30
ρy2iZ2i2=Pγ20+θ12σY1σY2+θ1θ2σY22+1Pθ1θ2σY1σY2+γ20+θ22σY222σY22σZ22E31

where

σZ22=σY12+μY12Pγ40+4γ30θ1+6γ20θ12+θ14+1Pγ20+θ12γ02+θ22+σY22+μY221Pγ04+4γ03θ2+6γ02θ22+θ24+Pγ20+θ12γ02+θ22+2σY1σY2+μY1μY22γ30+3γ20θ1+θ13+1Pθ1γ03+3γ02θ2+θ23Pγ20+θ12+1Pθ1θ2μY1+1θ2+1Pγ02+θ22μY22

I also define the measure of respondent’s privacy protection for Ahmed et al. [14] as:

τAJ=1ρyJiZ2i2,J=1,2E32

In the next section, I investigate the performance of the proposed model with respect to Ahmed et al. [14] model in terms of relative efficiency and privacy protection under different parametric situations.

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4. Efficiency vs privacy protection

The relative efficiency (RE) of the proposed estimators μ̂Y1and μ̂Y2over Ahmed et al. [14] the estimators μ̂AY1and μ̂AY2are respectively given by:

REJμ̂YJμ̂AYJ=Vμ̂AYJVμ̂YJ,J=1,2E33

To have a possible trade-off between relative efficiency and privacy protection of respondents, I consider the parametric values in this manner that the relative efficiencies are maximum and expect greater privacy protection of respondents. I decided to take P=0.6, μY1=25–45 with a step 5 and μY2=35–55 with a step 5, five values of θ1and θ2, equal to 2–10 and 4–16 with a increment 2 and 3 respectively, σY1=7, σY2=5, γ20=2, γ02=9, γ30=1.5, γ03=1.2, γ40=3.2and γ04=3.5. I have also chosen different values of probabilities Pii=123and presented in Tables 1 and 2.

P1P2P3θ1θ2μY1μY2RE1τP1τA1
0.200.200.602425352.030.97620.9765
4730401.720.95720.9570
61035451.650.93990.9395
81340501.620.92770.9272
101645551.600.92060.9201
0.150.150.702425351.740.97630.9765
4730401.520.95720.9570
61035451.470.93980.9395
81340501.450.92760.9272
101645551.440.92050.9201
0.100.100.802425351.470.97640.9765
4730401.330.95710.9570
61035451.300.93970.9395
81340501.290.92750.9272
101645551.280.92040.9201
0.050.050.902425351.220.97640.9765
4730401.160.95710.9570
61035451.150.93960.9395
81340501.140.92740.9272
101645551.130.92030.9201

Table 1.

Relative efficiency of the proposed estimator μ̂Y1with respect to Ahmed et al. [14] estimator μ̂AY1and privacy protection of the τP1and τA1.

P1P2P3θ1θ2μY1μY2RE2τP2τA2
0.200.200.602425351.980.97620.9765
4730401.700.95720.9570
61035451.640.93990.9395
81340501.620.92770.9272
101645551.600.92060.9201
0.150.150.702425351.710.97630.9765
4730401.510.95720.9570
61035451.470.93980.9395
81340501.450.92760.9272
101645551.440.92050.9201
0.100.100.802425351.450.97640.9765
4730401.330.95710.9570
61035451.300.93970.9395
81340501.290.92750.9272
101645551.280.92040.9201
0.050.050.902425351.210.97640.9765
4730401.160.95710.9570
61035451.150.93960.9395
81340501.140.92740.9272
101645551.130.92030.9201

Table 2.

Relative efficiency of the proposed estimator μ̂Y2with respect to Ahmed et al. [14] estimator μ̂AY2and privacy protection of the τP2and τA2.

Tables 1 and 2 show how the proposed model works in term of efficiency along with privacy protection. For the situation under investigate it emerges that the proposed model based on blank card method is more efficient than Ahmed et al. [14] model. Hence, the finding results, which are worth discussing, are described in the following points.

  1. It is observed from Tables 1 and 2 that the proposed estimators μ̂YJJ=12almost equally efficient in term of efficiencies and privacy protection of respondents.

  2. From Tables 1 and 2, it may be observed that the values of relative efficiencies of the proposed estimators μ̂YJJ=12with respect to Ahmed et al. [14] estimators μ̂AYJJ=12are more than 1 for all case.

  3. The behaviour of the estimators in Tables 1 and 2, indicates that the highest efficiency 2.03 attains when Pi=0.20i=12and Pi=0.60with corresponding values of θ1=2and θ2=4while the minimum efficiency 1.13 attains when Pi=0.05i=12and Pi=0.90with corresponding values of θ1=10and θ2=16.

  4. It is also seen that the values of relative efficiencies are decreasing as the values of θJand μYJJ=12increase for the fix values of Pii=123.

  5. However, it is observed that for the fix values of θ1, θ2, μY1and μY2the values of REii=12are decreasing as the values of Pii=12decrease.

  6. Furthermore, it can be interpreted that with the increase in the values of Pii=12while decrease in the values of P3, θ1, θ2, μY1and μY2there is a increasing pattern in the values of REii=12.

  7. From Tables 1 and 2, it is clear that the measure of privacy protection of proposed randomized response model and Ahmed et al. [14] model is closer to one for all the cases, which indicate maximum privacy protection of respondents.

  8. It is further observed that the degree of privacy protection of proposed randomized response model and Ahmed et al. [14] model decreasing with the values of θJand μYJJ=12increase.

  9. From Tables 1 and 2, it may also be seen that the value of respondent’s privacy protection are showing an increasing trend with the increase in the values of Pii=12while decrease in the values of P3, θ1, θ2, μY1and μY2.

  10. However, it is visible that the proposed model is more efficient than Ahmed et al. [14] model but less protective only when θ1=2and θ2=4. The model which provides more efficiency yields less privacy protection. Hence, I conclude that

Vμ̂AYJ>Vμ̂YJ,J=1,2

and

τAJ>τPJ,J=1,2

Hence, I conclude that small difference in efficiency may procure substantial improvement in privacy protection of respondent. Thus, our comparisons underline the good performance, in terms of efficiency and respondent’s privacy protection.

  1. Therefore, the proposed randomized response model under the blank card method may be declared to be best for estimating the mean of two quantitative sensitive variables and thus may be recommended to the survey practitioners whenever they deal with extremely sensitive characteristics.

To judge the performance of the proposed model, I consider a real data CO124 of N = 124 units of Sarndal et al. [21]. A random sample of size n = 30 units are drawn from the CO124 population. Let Y1, Y2and Xbe the import, export and military expenditure in the state of U.S. during the year 1983, 1983 and 1981 respectively. The parametric ranges of quantitative sensitive variables Y1and Y2and non-sensitive variable Xhave been found by using t–test and chi–square test, which are terms as μY1331.60567.66, μY2242.56440.30, μX43171.16, σY1256.03432.17, σY2214.47362.02and σX138.90234.46. The relative efficiencies have been computed for these parameters combinations and presented in Tables 37.

P1P2P3θ1θ2σY1σY2μY1μY2P
0.10.20.30.40.5
0.200.70.10412602203402505.905.485.104.774.46
3903005.875.455.084.754.45
4403505.865.455.084.764.46
4904005.865.465.104.784.49
5404405.945.545.184.864.56
3102703402505.294.934.614.324.06
3903005.314.954.634.344.08
4403505.354.994.674.384.12
4904005.405.044.724.434.17
5404405.515.144.824.534.26
3603203402504.844.534.243.993.76
3903004.894.574.294.033.80
4403504.954.634.354.093.86
4904005.024.704.414.153.92
5404405.144.814.524.254.01
0.150.750.10412602203402509.178.317.576.946.39
3903009.038.207.496.876.34
4403508.938.137.446.846.32
4904008.878.097.416.836.31
5404408.928.157.486.906.39
3102703402508.047.326.706.165.70
3903008.027.316.706.185.72
4403508.037.336.736.215.75
4904008.067.376.776.255.80
5404408.177.486.896.375.91
3603203402507.236.606.075.605.20
3903007.266.656.115.655.25
4403507.326.716.185.715.31
4904007.406.786.255.795.38
5404407.546.926.385.925.50
0.100.800.104126022034025017.1014.7712.9211.4210.20
39030016.5914.3812.6211.1910.02
44035016.1614.0612.3811.019.88
49040015.8113.8012.1910.889.79
54044015.6913.7512.1810.909.83
31027034025014.6012.6811.169.928.90
39030014.4012.5411.069.868.86
44035014.2412.4411.009.838.85
49040014.1212.3810.979.828.86
54044014.1612.4511.069.928.97
36032034025012.8011.189.898.837.96
39030012.7511.179.898.857.99
44035012.7411.189.938.908.04
49040012.7511.229.988.968.11
54044012.8811.3610.139.118.26
0.050.850.104126022034025049.1037.3429.5424.0920.10
39030046.4935.5628.2923.1719.43
44035044.1533.9827.1822.3818.85
49040042.0932.6126.2321.7118.37
54044040.7631.7825.7121.3818.16
31027034025040.4631.0224.7320.3017.06
39030039.0930.1224.1419.8916.77
44035037.8429.3223.5919.5316.53
49040036.7028.6023.1319.2316.34
54044036.0328.2322.9419.1616.34
36032034025034.2426.4721.2617.5814.86
39030034.5526.0521.0017.4214.77
44035032.9225.6720.7917.3014.72
49040032.3425.3520.6117.2214.69
54044032.0725.2720.6317.3014.81
0.100.850.054126022034025018.5017.1015.8614.7713.79
39030017.9216.5915.4214.3813.45
44035017.4216.1615.0514.0613.18
49040017.0115.8114.7513.8012.96
54044016.8515.6914.6713.7512.93
31027034025015.7514.6013.5812.6811.88
39030015.5114.4013.4212.5411.76
44035015.3114.2413.2912.4411.69
49040015.1614.1213.2012.3811.64
54044015.1814.1613.2612.4511.72
36032034025013.7612.8011.9411.1810.50
39030013.7012.7511.9111.1710.50
44035013.6712.7411.9211.1810.52
49040013.6612.7511.9411.2210.57
54044013.7912.8812.0811.3610.71

Table 3.

Relative efficiency of the proposed estimator μ̂Y1with respect to Perri [5] estimator μ̂Pwhen θ1=4and θ2=1.

P1P2P3θ1θ2σY1σY2μY1μY2P
0.10.20.30.40.5
0.200.700.10412602203402501.581.441.311.201.10
3903001.491.361.241.141.05
4403501.431.311.201.101.01
4904001.391.271.161.070.98
5404401.351.231.131.040.96
3102703402501.561.421.311.201.11
3903001.491.361.251.151.06
4403501.431.311.201.111.02
4904001.401.281.171.081.00
5404401.361.241.141.050.97
3603203402501.541.411.301.201.11
3903001.481.361.251.151.07
4403501.431.321.211.121.04
4904001.401.291.181.091.01
5404401.361.251.151.070.99
0.150.750.10412602203402502.692.392.141.931.74
3903002.522.252.021.821.64
4403502.402.141.931.741.57
4904002.312.071.861.681.53
5404402.222.001.831.631.48
3102703402502.592.322.081.881.71
3903002.462.201.981.791.63
4403502.362.111.911.731.57
4904002.282.051.851.681.53
5404402.211.991.801.631.48
3603203402502.512.252.041.851.68
3903002.402.161.951.771.62
4403502.322.091.891.721.57
4904002.262.031.841.671.53
5404402.191.981.791.631.49
0.100.800.10412602203402505.464.643.993.473.04
3903005.054.303.713.242.85
4403504.734.053.503.062.70
4904004.493.853.352.932.59
5404404.273.683.202.822.49
3102703402505.114.373.783.312.92
3903004.794.113.573.122.76
4403504.553.913.402.992.64
4904004.353.753.272.882.55
5404404.173.603.152.782.47
3603203402504.814.143.603.172.81
3903004.573.943.443.032.69
4403504.383.783.302.912.59
4904004.223.663.202.832.52
5404404.073.533.102.742.44
0.050.850.104126022034025016.9812.719.907.956.53
39030015.3211.549.047.286.01
44035014.0010.618.366.775.61
49040012.959.887.826.375.30
54044012.039.237.356.015.03
31027034025015.2811.549.067.336.06
39030014.0610.678.426.835.67
44035013.069.977.906.445.36
49040012.239.397.486.125.12
54044011.498.867.095.834.89
36032034025013.8510.568.366.815.67
39030012.969.927.886.445.38
44035012.209.387.486.135.14
49040011.568.937.155.884.94
54044010.968.506.845.644.76
0.100.850.05412602203402505.965.465.024.644.30
3903005.505.054.654.303.99
4403505.144.734.374.053.76
4904004.874.494.153.853.59
5404404.624.273.963.683.43
3102703402505.555.114.714.374.06
3903005.204.794.434.113.82
4403504.934.554.213.913.64
4904004.714.354.033.753.50
5404404.514.173.873.603.37
3603203402505.214.814.454.143.86
3903004.954.574.243.943.67
4403504.734.384.063.783.53
4904004.564.223.923.663.42
5404404.394.073.793.533.31

Table 4.

Relative efficiency of the proposed estimator μ̂Y2with respect to Perri [5] estimator μ̂Pwhen θ1=4and θ2=1.

P1P2P3θ1θ2σY1σY2μY1μY2P
0.10.30.50.70.9
0.200.70.10412602203402500.981.011.111.831589.20
3903000.981.021.121.891728.00
4403500.981.021.131.951854.80
4904000.981.021.142.001968.00
5404400.981.021.152.052064.70
3102703402500.981.011.101.741446.10
3903000.981.011.111.801568.00
4403500.981.011.121.861684.80
4904000.981.011.131.911793.90
5404400.981.011.131.961890.00
3603203402500.991.011.091.681343.80
3903000.991.011.101.731448.40
4403500.991.011.111.781552.40
4904000.991.011.121.831653.10
5404400.991.011.121.881744.10
0.150.750.10412602203402500.991.021.121.841605.30
3903000.991.021.131.911746.40
4403500.991.021.141.971875.50
4904000.991.021.152.021990.80
5404400.991.021.162.072089.90
3102703402500.991.011.111.761459.00
3903000.991.011.121.821582.7
4403500.991.021.121.871701.50
4904000.991.021.131.921812.50
5404400.991.021.141.981910.80
3603203402500.991.011.101.701345.50
3903000.991.011.111.751460.70
4403500.991.101.121.801566.40
4904000.991.011.121.851668.60
5404400.991.021.131.901761.50
0.100.800.10412602203402500.991.021.131.861617.30
3903000.991.021.141.921760.20
4403500.991.021.151.981890.80
4904000.991.031.152.042007.60
5404400.991.031.162.092108.50
3102703402500.991.021.121.771468.60
3903000.991.021.121.831593.70
4403500.991.021.131.881713.90
4904000.991.021.141.941826.20
5404400.991.021.151.991926.10
3603203402500.991.021.101.701362.50
3903000.991.021.111.761469.80
4403500.991.021.121.811576.60
4904000.991.021.131.861680.10
5404400.991.021.131.911774.40
0.050.850.10412602203402501.001.031.141.871626.40
3903001.001.031.151.931770.50
4403501.001.031.151.991902.30
4904001.001.031.162.052020.20
5404401.001.031.172.102122.40
3102703402501.001.021.121.781475.70
3903001.001.021.131.841601.90
4403501.001.021.141.891723.00
4904001.001.031.141.951836.40
5404401.001.031.152.001937.50
3603203402501.001.021.111.711368.40
3903001.001.021.121.761476.60
4403501.001.021.121.821584.20
4904001.001.021.131.871688.50
5404401.001.021.141.921783.90
0.100.850.05412602203402500.991.021.131.861625.60
3903000.991.021.141.931769.60
4403500.991.031.151.991901.30
4904000.991.031.162.042019.10
5404400.991.031.172.102121.20
3102703402500.991.021.121.771475.10
3903000.991.021.131.831601.20
4403500.991.021.131.891722.20
4904000.991.021.141.941835.50
5404400.991.031.152.001936.50
3603203402500.991.021.111.711367.90
3903000.991.021.111.761476.00
4403500.991.021.121.811583.50
4904000.991.021.131.861687.80
5404400.991.021.141.911783.10

Table 5.

Relative efficiency of the proposed estimator μ̂Y1with respect to Ahmed et al.[14] estimator μ̂AY1when θ1=4and θ2=1.

P1P2P3θ1θ2σY1σY2μY1μY2P
0.10.30.50.70.9
0.200.70.10412602203402500.811.172.066.145899.30
3903000.811.172.086.246099.10
4403500.811.172.096.336289.70
4904000.811.172.106.426466.80
5404400.811.172.116.456491.60
3102703402500.821.162.046.075933.10
3903000.811.162.056.176103.60
4403500.811.162.066.266272.90
4904000.811.172.086.346435.10
5404400.811.172.096.386461.80
3603203402500.821.162.016.015971.40
3903000.821.162.036.106115.70
4403500.821.162.046.196264.40
4904000.821.162.066.276411.20
5404400.811.162.076.316438.30
0.150.750.10412602203402500.891.292.296.856603.20
3903000.891.292.316.966833.40
4403500.891.302.327.077052.40
4904000.891.302.337.177255.60
5404400.891.302.357.217289.80
3102703402500.901.282.256.746619.70
3903000.901.282.276.866818.50
4403500.901.292.296.977015.00
4904000.901.292.317.077202.90
5404400.891.292.327.127240.80
3603203402500.901.272.226.656643.30
3903000.901.282.246.766813.50
4403500.901.282.266.876987.80
4904000.901.282.286.977159.30
5404400.901.292.297.037198.80
0.100.800.10412602203402500.981.422.527.567317.70
3903000.981.422.547.707579.10
4403500.981.422.567.827827.10
4904000.981.432.577.938056.60
5404400.981.432.597.998102.10
3102703402500.981.402.477.427310.90
3903000.981.412.507.567539.20
4403500.981.412.527.687763.90
4904000.981.412.547.807978.10
5404400.981.422.557.878029.50
3603203402500.981.392.437.307314.40
3903000.981.392.457.437512.00
4403500.981.402.487.567713.20
4904000.981.402.507.677910.40
5404400.981.412.527.757964.70
0.050.850.10412602203402501.061.542.758.288040.70
3903001.061.552.788.448333.90
4403501.061.552.808.588611.10
4904001.061.562.828.708866.90
5404401.061.562.838.778925.60
3102703402501.061.522.698.108004.10
3903001.061.532.728.268263.00
4403501.061.532.748.408516.70
4904001.061.542.778.538757.70
5404401.061.552.798.628824.70
3603203402501.061.502.637.947982.30
3903001.061.512.668.098208.60
4403501.061.512.698.248437.60
4904001.061.522.728.378661.30
5404401.061.532.748.478732.70
0.100.850.05412602203402500.991.452.648.077968.00
3903000.991.462.668.218258.10
4403500.991.462.688.358532.40
4904000.991.462.698.478785.50
5404400.991.472.718.538842.80
3102703402500.991.442.587.897934.80
3903000.991.442.618.058190.50
4403500.991.452.638.198441.30
4904000.991.452.658.318679.70
5404400.991.462.678.398745.00
3603203402500.991.422.537.757915.80
3903000.991.432.567.898139.10
4403500.991.432.598.038365.30
4904000.991.442.618.168586.30
5404400.991.452.638.258655.90

Table 6.

Relative efficiency of the proposed estimator μ̂Y2with respect to Ahmed et al.[14] estimator μ̂AY2when θ1=4and θ2=1.

P1P2P3θ1θ2σY1σY2μY1μY2RE1RE2
0.850.100.05412602203402503.56883.5201
3903003.57013.5221
4403503.57123.5241
4904003.57223.5258
5404403.57233.5255
3102703402503.56913.5227
3903003.57023.5241
4403503.57123.5255
4904003.57213.5269
5404403.57223.5265
3603203402503.56953.5251
3903003.57043.5260
4403503.57123.5270
4904003.57203.5281
5404403.57213.5276
0.800.150.0541260220340250116.44105.89
390300117.42106.52
440350118.22107.12
490400118.87107.66
540440119.17107.59
310270340250115.91106.64
390300116.84107.08
440350117.63107.54
490400118.30107.94
540440118.65107.87
360320340250115.48107.35
390300116.34107.64
440350117.10107.97
490400117.77108.31
540440118.15108.18
0.750.200.0541260220340250355.18328.81
390300363.11332.28
440350369.70335.58
490400375.11338.62
540440378.32338.29
310270340250348.93332.66
390300356.59335.15
440350363.24337.70
490400368.94340.18
540440372.59339.68
360320340250343.89336.31
390300351.03338.03
440350357.49339.94
490400363.22341.91
540440367.14341.23
0.700.250.1041260220340250637.70649.04
390300661.97658.47
440350682.56667.49
490400699.81675.82
540440711.27675.11
310270340250616.17658.67
390300639.28665.62
440350659.84672.73
490400677.80679.63
540440690.56678.47
360320340250599.19667.92
390300620.45672.86
440350640.14678.31
490400657.98683.88
540440671.33682.30
0.650.300.0541260220340250894.781048.40
390300941.371067.30
440350981.731085.40
4904001016.201102.10
5404401040.801101.10
310270340250851.251066.00
390300894.751080.20
440350934.311094.70
490400969.531108.80
540440996.201106.90
360320340250817.761083.00
390300857.091093.40
440350894.291104.70
490400928.641116.30
540440955.851113.70
0.600.350.05412602203402501098.701512.50
3903001168.001544.30
4403501229.001574.90
4904001281.701603.20
5404401321.401602.10
3102703402501032.301539.30
3903001095.801563.70
4403501154.501588.50
4904001207.601612.70
5404401249.601610.30
360320340250917.001565.30
390300981.161583.60
4403501042.501603.40
4904001099.601623.60
5404401146.601620.10
0.550.400.05412602203402501250.102029.90
3903001338.802077.90
4403501418.002124.00
4904001487.202167.00
5404401541.102166.20
3102703402501163.402065.80
3903001243.702103.40
4403501318.802141.50
4904001387.502178.50
5404401443.502176.20
3603203402501099.002101.10
3903001169.702129.90
4403501238.502160.80
4904001303.502192.10
5404401358.602188.40
0.500.450.5412602203402501359.402591.10
3903001463.502658.30
4403501557.202723.00
4904001639.702783.20
5404401705.702783.60
3102703402501256.302635.20
3903001349.602688.70
4403501437.602742.80
4904001518.702795.20
5404401586.302794.00
3603203402501180.702678.70
3903001262.202720.60
4403501341.902765.20
4904001418.102810.20
5404401483.802807.20
0.450.500.5412602203402501437.803188.60
3903001553.503277.70
4403501658.203363.40
4904001751.003443.30
5404401826.703445.70
3102703402501322.003238.70
3903001424.903310.70
4403501522.603383.30
4904001613.103453.60
5404401689.903454.50
3603203402501237.803288.40
3903001327.103345.90
4403501415.103406.70
4904001499.603467.70
5404401573.503466.80
0.400.550.5412602203402501494.203816.00
3903001618.503929.40
4403501731.404038.20
4904001831.804139.70
5404401915.004145.40
3102703402501368.803869.00
3903001478.703961.90
4403501583.604055.30
4904001681.004145.50
5404401764.704150.00
3603203402501278.003922.10
3903001373.103997.50
4403501467.104076.80
4904001557.604155.90
5404401637.804158.80
0.350.600.5412602203402501535.204468.10
3903001665.704607.70
4403501784.704741.50
4904001890.704866.30
5404401979.504876.50
3102703402501402.304520.00
3903001517.504636.00
4403501627.504752.20
4904001730.004864.20
5404401819.004873.80
3603203402501306.704573.00
3903001406.004668.60
4403501504.404768.30
4904001599.304867.60
5404401684.004876.00

Table 7.

Relative efficiency of the proposed estimators μ̂Y1and μ̂Y2with respect to Ahmed et al. [14] estimators μ̂AY1and μ̂AY2respectively when θ1=4, θ2=1and P=0.9.

The behaviour of the estimators in Tables 37 indicate that the proposed estimators perform better than Perri [5] and Ahmed et al. [14] estimators in terms of efficiency.

  1. When the proposed estimators μ̂Y1and μ̂Y2are compared with Perri [5] estimator μ̂P, the estimator μ̂Y2gives lesser efficiency than the estimator μ̂Y1. Also, it is clear from Tables 5 and 6 that when the proposed estimators μ̂Y1and μ̂Y2are compared with Ahmed et al. [14] estimators μ̂AY1and μ̂AY2, the estimator μ̂Y1gives lesser efficiency than the estimator μ̂Y2.

  2. From the simulation results in Tables 5 and 6, it can be interpreted that the values of relative efficiencies are coming out to be near 1 when P= 0.1, this is the cost to be paid for perturbing the data, so that privacy of respondents is protected.

  3. Further, it is observed that for the fix values of Pi, θi, μYiand σYi(i= 1, 2, 3) the value of relative efficiencies are decreasing in Tables 3 and 4 while in Tables 5 and 6 it is increasing as the values of P increase.

The rest of the results can be read out from the given tables.

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5. Conclusions

The main objective of this paper is to estimate the population means of two quantitative sensitive variables. It is to be pointed out that the proposed model is more efficient in terms of relative efficiencies and respondent’s privacy protection. Therefore, these results advocate that the proposed technique is appreciatively favourable in obtaining the truthful response from the respondents.

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Proof:Given that ES1=θ1and ES2=θ2. Following Singh [9], I define

γrs=ES1θ1rS2θ2sE34

Then due to independence of the scramble variables, I have

ES12=γ20+θ12E35
ES13=γ30+3γ20θ1+θ13E36
ES14=γ40+4γ30θ1+6γ20θ12+θ14E37
ES22=γ02+θ22E38
ES23=γ03+3γ02θ2+θ23E39
ES24=γ04+4γ03θ2+6γ02θ22+θ24E40
ES1S2=θ1θ2E41
ES12S22=γ20+θ12γ02+θ22E42
ES13S2=γ30+3γ20θ1+θ13θ2E43

and

ES1S23=θ2γ03+3γ02θ2+θ23E44
Vμ̂Y1=P1+P3Pθ1θ2+P2+P31Pγ02+θ222i=1nVZ1i+θ22i=1nVZ2i2θ2P1+P3Pθ1θ2+P2+P31Pγ02+θ22i=1ncovZ1iZ2in2P2+P31Pθ1γ02P1+P3Pθ2γ202
=P1+P3Pθ1θ2+P2+P31Pγ02+θ222σZ12+θ22σZ222θ2P1+P3Pθ1θ2+P2+P31Pγ02+θ22σZ1Z2nP2+P31Pθ1γ02P1+P3Pθ2γ202E45

where, the variance σZ12is given by:

σZ12=EZ1i2EZ1i2
=ES1Y1i+S2Y1i2ES1Y1i+S2Y1i2
=ES12Y1i2+S22Y2i2+2S1S2Y1iY2iθ1μY1+θ2μY22
=γ20+θ12σY12+μY12+γ02+θ22σY22+μY22
+2θ1θ2σY1σY2+μY1μY2θ1μY1+θ2μY22
=γ20σY12+μY12+γ02σY22+μY22+θ12σY12+θ22σY22+2θ1θ2σY1σY2E46

The variance σZ22is given by:

σZ22=EZ2i2EZ2i2
=P1+P3PES12Y1i+S1S2Y2i2+P2+P31PES1S2Y1i+S22Y2i2P1+P3Pγ20+θ12μY1+θ1θ2μY2+P2+P31Pθ1θ2μY1+γ02+θ22μY22
=P1+P3PES14Y1i2+S12S22Y2i2+2S13S2Y1iY2i+P2+P31PES12S22Y1i2+S24Y2i2+2S1S23Y1iY2iP1+P3Pγ20+θ12μY1+θ1θ2μY2+P2+P31Pθ1θ2μY1+γ02+θ22μY22
=P1+P3Pγ40+4γ30θ1+6γ20θ12+θ14σY12+μY12+γ20+θ12γ02+θ22σY22+μY22+2γ30+3γ20θ1+θ13θ2σY1σY2+μY1μY2+P2+P31Pγ20+θ12γ02+θ22σY12+μY12+γ04+4γ03θ2+6γ02θ22+θ24σY22+μY22+2θ1γ03+3γ02θ2+θ23σY1σY2+μY1μY2P1+P3Pγ20+θ12μY1+θ1θ2μY2+P2+P31Pθ1θ2μY1+γ02+θ22μY22

After simplification, this gives

σZ22=σY12+μY12P1+P3Pγ40+4γ30θ1+6γ20θ12+θ14+P2+P31Pγ20+θ12γ02+θ22+σY22+μY22P2+P31Pγ04+4γ03θ2+6γ02θ22+θ24+P1+P3Pγ20+θ12γ02+θ22+2σY1σY2+μY1μY2P1+P3Pθ2γ30+3γ20θ1+θ13+P2+P31Pθ1γ03+3γ02θ2+θ23P1+P3Pγ20+θ12+P2+P31Pθ1θ2μY1+P1+P3Pθ1θ2+P2+P31Pγ02+θ22μY22E47

and the covariance σZ1Z2between Z1iand Z2iis given by:

σZ1Z2=covZ1iZ2i=EZ1iZ2iEZ1iEZ2i
=P1+P3PES1Y1i+S2Y2iS12Y1i+S1S2Y2i+P2+P31PES1Y1i+S2Y2iS1S2Y1i+S22Y2iθ1μY1+θ2μY2P1+P3Pγ20+θ12μY1+θ1θ2μY2+P2+P31Pθ1θ2μY1+γ02+θ22μY22
=P1+P3PES13Y1i2+2S12S2Y1iY2i+S1S22Y2i2+P2+P31PES12S2Y1i2+2S1S22Y1iY2i+S23Y2i2θ1μY1+θ2μY2P1+P3Pγ20+θ12μY1+θ1θ2μY2+P2+P31Pθ1θ2μY1+γ02+θ22μY22
=P1+P3Pγ30+3γ20θ1+θ13σY12+μY12+2γ20+θ12θ2σY1σY2+μY1μY2+θ1γ02+θ22σY22+μY22+P2+P31Pγ20+θ12θ2σY12+μY12+2θ1γ02+θ22σY1σY2+μY1μY2+γ03+3γ02θ2+θ23σY22+μY22θ1μY1+θ2μY2P1+P3Pγ20+θ12μY1+θ1θ2μY2+P2+P31Pθ1θ2μY1+γ02+θ22μY22

After some algebra, I get

σZ1Z2=σY12+μY12P1+P3Pγ30+3γ20θ1+θ13+P2+P31Pθ2γ20+θ12+σY22+μY22P1+P3Pθ1γ02+θ22+P2+P31Pγ03+3γ02θ2+θ23+2σY1σY2+μY1μY2P1+P3Pθ2γ20+θ12+P2+P31Pθ1γ02+θ22θ1μY1+θ2μY12P1+P3Pγ20+θ12+P2+P31Pθ1θ2μY1+P1+P3Pθ1θ2+P2+P31Pγ02+θ22μY2E48

Finally, substituting the expressions given in Eqs. (13), (14) and (15) in Eq. (12), I get the variance of the estimator μ̂Y1as given in Eq. (24).

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Written By

Amod Kumar

Submitted: July 20th, 2021Reviewed: October 18th, 2021Published: April 6th, 2022