A new improved estimator of population mean in partial additive randomized response models

Transcription

1 Hacettepe Journal of Mathematics and Statistics Volume 46 () (017), A new improved estimator of population mean in partial additive randomized response models Nilgun Ozgul and Hulya Cingi Abstract In this study, we have developed a new improved estimator for the population mean estimation of the sensitive study variable in Partial Additive Randomized Response Models (RRMs) using two non-sensitive auxiliary variables The mean squared error of the proposed estimator is derived and compared with other existing estimators based on the auxiliary variable The proposed estimator is compared with [19],[5] and [13] estimators in performing a simulation study and is found to be more ecient than other existing estimators using non-sensitive auxiliary variable The results of the simulation study are discussed in the nal section Keywords: Randomized response models; Sensitive question, Auxiliary variable, Eciency, Mean square error Received : Accepted : Doi : /HJMS Introduction Respondents sometimes come across sensitive questions, such as gambling, alcoholism, sexual and physical abuse, drug addiction, abortion, tax evasion, illegal income, mobbing, political view, doping usage, homosexual activities and many others Respondents often do not respond truthfully, or even refuse to answer when asked directly such sensitive questions Obtaining valid and reliable information, the researchers commonly use the randomized response models (RRMs) Starting from the pioneering work of [1], many versions of RRMs have been developed that can deal with both proportion and mean estimations Standard RRMs have been primarily used with surveys that usually require a yes or noresponse to a sensitive question, or a choice of responses from a set of nominal categories Nevertheless, the literature on RRMs is comprised of various studies dealing with situations where the response to a sensitive question results in a quantitative Hacettepe University, Department of Statistics, Beytepe, 06800, Ankara, Turkey, nozgul@hacettepeedutr Corresponding Author Hacettepe University, Department of Statistics, Beytepe, 06800, Ankara, Turkey, hcingi@hacettepeedutr

2 36 variable [5] Quantitative RRMs are used to estimate the mean value of some behavior in a population These RRMs are sub-classied in either additive models or multiplicative models In additive models, respondents are asked to scramble their responses using a randomization device, such as a deck of cards Each of the cards in the deck has a number The numbers in the deck follow a known probability distribution, such as Normal, Chi-square, Uniform, Poisson, Binomial, or Weibull etc The respondent is asked to add his real response to the number listed on card he/she picked, and then report only the sum to the interviewer Multiplicative RRMs are similar to the additive RRMs Again, a deck of cards with known probability distribution is used, but now when the respondents scramble their responses, they are asked to report the product of the real response and the number listed on the selected card The interviewer cannot see the card and can simply record the reported number [18] RRMs can also be categorized by how the respondents are instructed to randomize If all respondents are asked to randomize their response, the model is characterized as a full randomization model If some of the respondents are instructed to randomize their response, the model is characterized as a partial randomization model [1] for the rst time, introduced the randomization method for the proportion of a population characterized by a sensitive variable Later, [], [8] and [0] extended [1] s approach to RRMs by estimating the mean of sensitive quantitative variables Since then, a large number of RRMs have been developed to estimate the mean of quantitative variables [7], [14] proposed additive RRMs while [6], [1], [16], [9], [10] and [11] proposed multiplicative RRMs Later, [17] and [4] proposed mixed RRMs with combining additive and multiplicative techniques In sampling theory, it is known that there is a considerable reduction in Mean Square Error (MSE) equation when auxiliary information is used, in particular when the correlation between the study variable and the auxiliary variable is high [] In recent years, auxiliary information for mean estimation of sensitive variable has been used in RRMs [5], [19] and [13] suggested regression, ratio, regression cum ratio, respectively, using the auxiliary variable for estimating of the quantitative sensitive variable Choice of scrambling mechanism plays an important role in quantitative response models In the RRMs literature, additive models are more eective and user-friendly than multiplicative RRMs and also partial randomization models are more ecient than full randomization models Therefore, in this paper, we focus on additive partial RRMs for quantitative data We propose a new improved estimator for the population mean of the sensitive study variable in partial additive RRMs using two non-sensitive auxiliary variables The remaining part of the paper is organized as follows In section, we present the notations about additive RRMs for quantitative data and introduce various estimators using the auxiliary variable for the unknown mean of a sensitive variable in RRMs In section 3, we introduce the proposed estimator In section 4, the proposed estimator is compared with other existing estimators with a simulation study in RRMs and we obtain specic results suggesting that proposed estimator is more ecient than other estimators Section 5 concludes the paper Notations and Various Existing Estimators in Partial Additive RRMs for Quantitative Data Let Y be the study variable, a sensitive variable which cannot be observed directly Let X be a non-sensitive auxiliary variable which has positive correlation with Y For example, the sensitive study variable Y may be the annual household income and X may be the annual rental value Let a random sample of size n be drawn from a nite

3 37 population For the i th unit (U 1, U,, U N), let y i and x i be the values of the study variable Y and auxiliary variable X, respectively [5] proposed a class of regression estimator for the mean of sensitive variable using a non-sensitive auxiliary variable In their approach, to estimate µ y, a sample of n individuals is selected from the population and each respondent is asked to perform a Bernoulli trial with a probability of success P If this is successful, the respondent then gives the true values of both Y and X In the case of failure, the respondent gives their answers by using the values given in S and R which are the various randomized designs for Y and X variables, respectively Table 1 General Scheme Reported Responses (Z, U) Variables (Y, X) with probability P Randomized Designs (S, R) with probability (1-P) Under the general scheme, given in Table 1, regression estimator in [5] is, z + b (µu ū) c (1) ˆµ DP =, (h 0) h n where z = 1 z n i is the sample mean of the reported responses for the sensitive variable Here, the reported response for the sensitive variable is given by Z = P Y + (1 P )(Y + W ) Here W is the scrambling variable which has pre-assigned distribution W is the scrambling variable with known true mean µ w and known variance Sw ū = 1 n u i n is the sample mean of the reported responses for the non-sensitive auxiliary variable b is suitably selected real constant Here, c and h depend exclusively on the randomized design The variance of ˆµ DP is () ˆµ DP = where S z = N z + b (µu ū) c, h (h 0) (z i µ z) N (u i µ u) N 1 and S u = N (z i µ z)(u i µ u) N 1 are the population variances of z and u, respectively, S zu = is the population covariance between z and u, N 1 N B = Szu is the population regression coecient between z and ue ( z) = µ Su z = 1 z N i and E (ū) = µ u = 1 N N u i are the population means of Z and U, respectively Substituting the population regression coecient B = Szu, minimum variance of unbiased estimator ˆµ DP is S u (3) V ar(ˆµ DP ) min = S z nh ( 1 ρ zu ) where ρ zu = Szu S zs u is the population correlation coecient between Z and U

4 38 [19] proposed a ratio estimator for the mean of sensitive variable using a non-sensitive auxiliary variable for full randomization additive model In their model, the respondent is asked to provide true responses for X The estimator in [19] is (4) ˆµ SR = z ( µx x ) where z is the sample mean of the reported responses for the sensitive variable (Z = Y + W ), E ( x) = µ x = 1 N x i is the known population mean of non-sensitive auxiliary N variable, x = 1 n x i is the sample mean of non-sensitive auxiliary variable n The bias and MSE of ˆµ SR, under rst order of the approximation, are (5) Bias(ˆµ SR) = λµ z ( C x C zx ) and (6) MSE (ˆµ SR) = λµ [ z C z + Cx ] ρ xzc xc z where λ = 1 n 1 Sz, Cz = and C x = Sx are the coecients of variation of Z and N µ z µ x X, respectively, ρ xz is the correlation coecient between X and Z [13] proposed regression-cum-ratio estimator for full randomization additive model Regression-cum-ratio estimator in [13] is (7) ˆµ GRR = [b 1 z + b (µ x x)] ( µx ) x where z is the sample mean of the reported responses for the sensitive variable ( Z = Y + W ), E ( x) = µ x = 1 N x i is the known population mean of non-sensitive auxiliary N variable, x = 1 n x i is the sample mean of non-sensitive auxiliary variable, b 1 and b n are constants The bias and MSE of ˆµ GRR, under rst order of the approximation, are (8) Bias(ˆµ GRR) = (b 1 1) µ z + b 1λµ z { C x C zx } + bλµ xc x (9) and MSE (ˆµ GRR) = (b 1 1) µ z + λ [ b 1µ { z C z + 3Cx } 4ρ zxc zc x + b µ xcx b µ zµ xcx b 1µ { z C } x ρ zxc zc x b 1b µ zµ x { ρzxc zc x C x}] Dierentiating (9) with respect to b 1 and b, the following optimum values which minimize the MSE are (10) b 1(opt) = and (11) b (opt) = µz µ x 1 λc x 1 λ{c x C z (1 ρ zx)} { ( )} ρzxc z 1 + b 1(opt) C x

5 39 Substituting the optimum values of b 1 and b in (10) and (11), the minimum MSE of ˆµ GRR is ( ) ( ) (1) MSE(ˆµ GRR) λµ zcz 1 ρ zx 1 λc x min = λcz (1 ρ zx) + (1 λcx) 3 Suggested Improved Estimator in Partial Additive RRMs Applying the general formulation of [5], we propose an improved estimator for the mean of sensitive variable using two non-sensitive auxiliary variables (X and M) For example, the sensitive study variable Y may be the annual household income and X may be the annual rental value and M may be the number of vehicles in household In our approach, to estimate µ y, the procedure works as follows: the respondent gives the true values of the sensitive variable and non-sensitive variables (Y, X, M) with known probability P, whereas provides the scrambled responses with known probability (1-P) The distribution of the responses is illustrated as: Here, Y is the sensitive variable of Table General Scheme for Proposed Estimator Reported Responses (Z, U, V) Variables (Y, X, M) with probability P Randomized Designs (S, R, L) with probability (1-P) interest with unknown mean µ y and unknown variance S y, X and M are non-sensitive variables with known means µ x and µ m Z is the reported response for the sensitive variable Y which is given by Z = P Y + (1 P )(Y + W ), U and V are the reported responses for the rst non-sensitive variable X and the second non-sensitive variable M, respectively Here W is the scrambling variable which has pre-assigned distribution W is the scrambling variable with known true mean µ w and known variance S w S, R and L are the various randomized designs for Y, X and M variables, respectively The reported responses for auxiliary variables changes to which randomized design adopted Under the general scheme presented in Table, we propose the following improved estimator based on a SRSWOR sample (z 1, u 1, v 1), (z, u, v ),, (z n, u n, v n) of n responses (31) ˆµ NHR = zr c, (h 0) h here ( ) ( ) µ u µ v (3) z R = k 1 z (h 0) k ū + (1 k )µ u k 3 v + (1 k 3)µ v where z = 1 n z i is the sample mean of the reported responses for sensitive variable n and ū = 1 n n u i and v = 1 v n i are the sample means of reported responses for n rst and second auxiliary variables, respectively E (ū) = µ u = 1 N u i and E ( v) = N µ v = 1 N v i are the population means of rst and second reported auxiliary variables, N respectively k 1, k and k 3 are the constants Here, c and h values for partial additive RRMs obtained as: µ (33) z = P µ y + (1 P ) (µ y + µ w) µ z = µ y + (1 P )µ w

6 330 from (33) c and h values are given: (34) µ y = µ z (1 P )µ w Here, c = (1 P )µ w, h = 1 To obtain the MSE equation for the proposed estimator, we dene the following relative error terms [3] (35) e 0 = (36) such that ( z µz) (ū µu) ( v µv), e 1 =, e = µ z µ u µ v E (e 0) = E (e 1) = E (e ) = 0; E ( e 0) = λc z, E ( e 1) = λc u, E ( e ) = λc v, E (e 0e 1) = λρ zuc zc u, E (e 0e ) = λρ zvc zc v, E (e 1e ) = λρ uvc uc v, N (v i µ v) where Cv = S v, S µ v =, ρ zu = Szu, ρ zv = Szv, ρ uv = Suv v N 1 S zs u S zs v S us v Expressing (3) in terms of e's in (35) and retaining terms in e's up to rst order, we have, (37) ( ) ( ) µ u µ v z R µ z = k 1µ z (1 + e 0) µ z k µ u (1 + e 1) + (1 k )µ u k 3µ v (1 + e ) + (1 k 3)µ v = k 1µ z (1 + e 0) (1 + k e 1) 1 (1 + k 3e ) 1 µ z = k 1µ z (1 + e 1) (1 k e 1 + ) (1 k 3e + ) µ z = (k 1 1)µ z + k 1µ z (e 0 k e 1 k 3e k e 0 e 1 k 3e 0e + k k 3 e 1e ) Taking expectation of both sides of (37) and using notations in (36), the bias equation of the estimator z R (38) Bias ( z R) = (k 1 1)µ z k 1λµ z (k ρ zuc zc u + k 3ρ zvc zc v k k 3ρ uvc uc v) using (31) and (38), we obtain the bias equation of the estimator ˆµ NHR (39) Bias (ˆµ NHR) = (k 1 1)µ z k 1λµ z (k ρ zuc zc u + k 3ρ zvc zc v k k 3ρ uvc uc v) (1 P )µ w Retaining terms in e's up to rst order, taking the square of both sides of (37) and expectation and using notations in (36), the MSE equation of the estimator z R (310) E( z R µ z) =E ( (k 1 1) µ z + k1µ ze 0 + k1k µ ze 1 + k1k 3µ ze k1k µ ze 0e 1, k1k 3µ ze 0e +k1k k 3µ )} ze 1e MSE ( z R) =(k 1 1) µ z + λk1s z + λk1s z ( k Cu + k3c v k ρ zuc zc u k 3ρ zvc zc v + k k 3ρ uvc uc v) using (31) and (310), we obtain the MSE equation of the estimator ˆµ NHR MSE (ˆµ NHR) =(k 1 1) µ z + λk1s z + λk1s z ( k Cu + k3c v k ρ zuc zc u (311) k 3ρ zvc zc v + k k 3ρ uvc uc v) N (u i µ u) (v i µ v) where S uv = is the covariance between u and v N 1

7 331 Dierentiating (311) with respect to k 1, k and k 3, and then by setting the resulting equations to zero, we obtain the following equations: (31) (313) MSE (ˆµ NHR) k 1 = (k 1 1) µ z + λk 1Sz + λk 1µ ( z k Cu + k3c v k ρ zuc zc u k 3ρ zvc zc v + k k 3ρ uvc uc v) = 0 MSE (ˆµ NHR) k = k C u ρ zuc zc u + k 3ρ uvc uc v = 0 (314) MSE (ˆµ NHR) k 3 = k 3C v ρ zvc zc v + k ρ uvc uc v = 0 Solving the (31), (313) and (314) simultaneously, we get the optimum values which minimize themse equation (315) k 1(opt) = λc z (1 λc z R zuv) where R zuv = ρ zu + ρ zv ρ zuρ zvρ uv 1 ρ uv (316) k (opt) = ρzu (Cz/Cv) ρzvρuv (Cz/Cu) 1 ρ uv (317) k 3(opt) = ρzv (Cz/Cv) ρzuρuv ( C u / CzC v ) 1 ρ uv Substituting the optimum values of k 1, k and k 3 in (311), the minimum MSE of ˆµ NHR is [ ( R (318) MSE min (ˆµ NHR) = λsz zuv 1 λc z 1 ) 1 λcz Rzuv ( ) R zuv 1 λc ] z / Cz 1 λcz Rzuv MSE and mean equations change depending on the specied models We specied three partial additive models In the rst model M1, the additive model is applied for the sensitive variable while the direct method is utilized for the non-sensitive auxiliary variables {Z=PY+(1-P)(Y+W), U=X, V=M} In the second model M, additive model is applied for the sensitive variable and both of two non-sensitive auxiliary variables {Z=PY+(1-P)(Y+W), U=PX+(1-P)(X+T), V=PM+(1-P)(M+K)} In the third model M3, the additive model is applied for the sensitive variable and the rst non-sensitive auxiliary variable while the direction method is utilized for second non-sensitive auxiliary variable {Z=PY+(1-P)(Y+W), U=PX+(1-P)(X+T), V=M} Here W, T and K are the scrambling variables which have pre-assigned distributions W is the scrambling variable with known true mean µ w and known variance S w in S design, T is the scrambling variable with known true mean µ t and known variance S t in R design, K is the scrambling variable with known true mean µ k and known variance S k in L design [15] Mean equations which will be used in MSE equation in (318) for M1 is obtained as below: Mean equation of z is (319) µ z = P µ y + (1 P ) (µ y + µ w) µ z = µ y + (1 P )µ w

8 33 Mean equation of u is (30) µ u = P µ x + (1 P )µ x µ u = µ x Mean equation of v is (31) µ v = P µ m + (1 P )µ m µ v = µ m Variance equation of z which will be used in MSE equation in (318) for M1 is obtained as, (3) Sz = P E ( Y ) + (1 P )E { (Y + W ) } µ z = Sy + (1 P )Sw + ( P (1 P )µ w Sz = Sy + (1 P ) µ w C w + P ) The correlation equation between Z and U which will be used in MSE equation in (318) for M1 is obtained as S yx (33) ρ zu = S x S y + (1 P ) µ w (Cw + P ) for S zu = S yx, S z = S y + (1 P ) µ w (C w + P ), S u = S x The correlation equation between Z and V which will be used in MSE equation in (318) for M1 is obtained as (34) ρ zv = S ym S x S y + (1 P ) µ w (C w + P ) for S zv = S ym, S z = S y + (1 P ) µ w (C w + P ), S v = S m The correlation equation between U and V which will be used in MSE equation in (318) for M1 is obtained as (35) ρ uv = Sxm S xs m = ρ xm for S uv = S xm, S u = S x, S v = S m Mean, variance and correlation equations which will be used in MSE equation in (318) for M and M3 is obtained similarly given as before Mean, variance and correlation equations for three models are given in Table 3 4 Simulation Study In this section, a simulation study is presented to show the performance of the proposed estimator in comparison to other estimators using the auxiliary variable for partial additive RRMs The proposed estimator,ˆµ NHR, is compared with ˆµ DP in [5], ˆµ SR in [19], and ˆµ GRR in [13] It is known that RRMs based on the auxiliary variable are practically indistinguishable and always perform better than RRMs in which auxiliary variables are not used For this reason, the estimators without using auxiliary variables are not included in the simulation study [5] We generate three nite populations of size from multivariate normal distribution Three populations have theoretical means of [Y, X, M] as µ = [5, 5, 5] and have dierent variance-covariance matrices The populations are generated as the levels of correlation between the variables The correlation levels are considered as low, medium and high The covariance matrices and the correlations are presented in (4), (43) and (44) The scrambling variable W is considered to be a normal random variable with mean equal to zero and standard deviation is equal to 030 The scrambling variables, T and K, are normal random variables with mean equal

9 333 Table 3 Special Partial Additive Randomized Models S R L Mean Variance and Correlation Equations ( µ z = µ y + (1 P )µ w Sz = Sy + (1 P ) µ w C w + P ) M1 Y+W X M µ u = µ x ρ zu = S yx S x S y + (1 P ) µ w (C w + P ) µ v = µ m ρ zv = S ym S m S y + (1 P ) µ w (C w + P ) c = (1 P )µ w, h = 1 ρ uv = Sxm = ρ xm S xs m µ z = µ y + (1 P )µ w ( Sz = Sy + (1 P ) µ w C w + P ) M Y+W X+T M+K µ u = µ x + (1 P )µ t ρ zu = S yx + P (1 P )µ wµ t {S y + (1 P )µ w (C w + P ) } {S x + (1 P )µ t (C t + P )} µ v = µ m + (1 P )µ k ρ zv = c = (1 P )µ w, h = 1 ρ uv = ( µ z = µ y + (1 P )µ w Sz = Sy + (1 P ) µ w C w + P ) S my + P (1 P )µ wµ k {S y + (1 P )µ w (C w + P ) } {S m + (1 P )µ k (C k + P )} S xm + P (1 P )µ tµ k {S x + (1 P )µ t (C t + P )} {S m + (1 P )µ k (C k + P )} M3 Y+W X+T M µ u = µ x + (1 P )µ t ρ zu = S yx + P (1 P )µ wµ t {S y + (1 P )µ w (C w + P ) } {S x + (1 P )µ t (C t + P )} µ v = µ m ρ zv = c = (1 P )µ w, h = 1 ρ uv = S ym S m {S y + (1 P )µ w (C w + P ) } S xm S m {S x + (1 P )µ t (C t + P )} to zero and standard deviations are equal to 00 We use the simulation studies of [5] and [13] to determine the parameters that are to be easier to compare The variance-covariance matrix dene as: S yi S yxi S ymi (41) = S xyi S xi S xmi, i = 1,, 3 i S myi S mxi S mi The variance-covariance matrices and the correlation coecients for each population are given below Population I There are low correlations between the sensitive variable and non-sensitive auxiliary variables in population I The variance-covariance matrix for population I is (4) = , ρ yx = 030, ρ ym = 05 Population II There are medium correlations between the sensitive variable and non-sensitive auxiliary variables in population II The variance-covariance matrix for population II is (43) = Population III , ρ yx = 060, ρ ym = 05

10 334 There are high correlations between the sensitive variable and non-sensitive auxiliary variables in population III The variance-covariance matrix for population III is (44) = , ρ yx = 090, ρ ym = 070 The process is repeated 5000 times and for dierent sample sizes: n = 50, 100, 00, 300, 500 The value of the design parameter P changes from 010 to 090 with an increment of 01 We observe small dierences in eciency with almost each value of the design parameter when auxiliary variable is utilized in partial additive RRMs in our simulation study Thus, we only present the simulation results for P=00 That means when partial additive model is utilized, 00 percent of the respondents give direct answers, the rest of the respondents use the scrambling devices The performance of the estimators is measured by the simulated mean square error as: (45) MSE (ˆµ) = (ˆµ i µ y) 5000 where ˆµ is the estimate of µ y on the i th sample Simulation results are summarized in Tables 4, 5 and 6 In Tables 4-6, theoretical and empirical MSE values of the estimators according to degree of the correlation between the sensitive and non-sensitive variables are given for specied models, respectively For three models, in all circumstances, regardless of both degree of correlation and sample size, the proposed estimator is always more ecient than ˆµ DP,ˆµ SR and ˆµ GRR estimators For population I and II, where there are low and medium correlations between the variables, respectively, the dierences between the MSE values of the proposed estimator and the other estimators are small For population III, where there are high correlations between the variables, the MSE values of the proposed estimator are relatively smaller than the MSE values of other estimators We can say that when the degree of the correlation between the variables increases, the eciency of the proposed estimator increases We also note that the MSE values of the estimators are smaller when the sample size increases and that is an expected result

11 335 Table 4 Theoretical and empirical MSEs of the estimators according to degree of the correlation between the sensitive and non-sensitive variables under Model 1 for P=00 Population I Population II Population III Estimators MSE MSE MSE n Theoretical Empirical Theoretical Empirical Theoretical Empirical ˆµ DP ˆµ SR ˆµ GRR ˆµ NHR ˆµ DP ˆµ SR ˆµ GRR ˆµ NHR ˆµ DP ˆµ SR ˆµ GRR ˆµ NHR ˆµ DP ˆµ SR ˆµ GRR ˆµ NHR ˆµ DP ˆµ SR ˆµ GRR ˆµ NHR Table 5 Theoretical and empirical MSEs of the estimators according to degree of the correlation between the sensitive and non-sensitive variables under Model for P=00 Population I Population II Population III Estimators MSE MSE MSE n Theoretical Empirical Theoretical Empirical Theoretical Empirical ˆµ DP ˆµ SR ˆµ GRR ˆµ NHR ˆµ DP ˆµ SR ˆµ GRR ˆµ NHR ˆµ DP ˆµ SR ˆµ GRR ˆµ NHR ˆµ DP ˆµ SR ˆµ GRR ˆµ NHR ˆµ DP ˆµ SR ˆµ GRR ˆµ NHR

12 336 Table 6 Theoretical and empirical MSEs of the estimators according to degree of the correlation between the sensitive and non-sensitive variables under Model 3 for P=00 Population I Population II Population III Estimators MSE MSE MSE n Theoretical Empirical Theoretical Empirical Theoretical Empirical ˆµ DP ˆµ SR ˆµ GRR ˆµ NHR ˆµ DP ˆµ SR ˆµ GRR ˆµ NHR ˆµ DP ˆµ SR ˆµ GRR ˆµ NHR ˆµ DP ˆµ SR ˆµ GRR ˆµ NHR ˆµ DP ˆµ SR ˆµ GRR ˆµ NHR

13 5 Conclusion In the RRMs literature, there are several estimators based on a non-sensitive auxiliary variable In this paper, we propose a new improved estimator based on two non-sensitive auxiliary variables for the population mean of a sensitive variable in partial additive RRMs The proposed estimator is more ecient than other existing estimators in all circumstances, regardless of which the model is applied The proposed estimator can be considered reliable and may lead the researcher to nd a suitable estimator for RRMs The estimation of the mean of a sensitive variable can be improved by using more nonsensitive auxiliary variables It is proved that RRMs based on two or more auxiliary variables are certainly more ecient than those with one auxiliary variable We show that the eciency of the proposed estimator can be quite distinctive if the correlation between the study and the auxiliary variables is high Additionally, the additive RRMs are more ecient and user friendly than the multiplicative RRMs Thus, we substitute our proposed estimator to three specic partial additive RRMs and we compare these newly-generated models In the future work, the study can be extended by combining the additive and multiplicative RRMs for the proposed estimator using more than one auxiliary variable based on dierent sampling methods 337 References [1] Bar-Lev SK, Bobovitch E, Boukai B A note on randomized response models for quantitative data Metrika 60, 55-60, 004 [] Cingi, H Ornekleme Kuram Ankara: 3Baski, Bizim Buro Basimevi, 009 [3] Cingi, H and Kadilar, CAdvances in Sampling Theory-Ratio Method of Estimation Bentham Science Publishers,11, 009 [4] Diana G, Perri PFNew Scrambled Response Models For Estimating The Mean of A Sensitive Quantitative CharacterJournal Of Applied Statistics,37 (11), , 010 [5] Diana G, Perri PF A class of Estimators for Quantitative Sensitive DataStatistical Papers,5, , 011 [6] Eichhorn, B H and Hayre, L SScrambled Randomized Response Methods for Obtaining Sensitive Quantitative Data Journal of Statistical Planning and Inference, 7 (1), , 1983 [7] Gjesvang, CR, Singh,S An improved Randomized Response Model:Estimation of Mean Journal of Applied Statistics, 36 (1), , 009 [8] Greenberg BG, Kuebler RR, Abernathy JR, Horvitz DGApplication of the randomized response technique in obtaining quantitative data Journal of American Statistical Association, 66, 43-50, 1971 [9] Gupta S, Gupta B, Singh S Estimation of sensitivity level of personal interview survey questions J Stat Plan Inference,100, 39-47, 00 [10] Gupta, SN and Shabbir, J Sensitivity estimation for personal interview survey Questions Statistica,64, , 004 [11] Gupta, S N and Shabbir,J On the Estimation of Population Mean and Sensitivity in Two-stage Optional Randomized Response Model Journal of Indian Society of Agricultural Statistics, 61, , 007 [1] Gupta S, Shabbir J, Sousa R and Sehra S Mean and sensitivity estimation in optional randomized response modelcommunications in Statistics, 140, , 010 [13] Gupta, S, Shabbir, J, Sousa, R, Real, P C Estimation of the mean of a sensitive variable in the presence of auxiliary information Communications in Statistics, Theory and Methods, 41, 1-1, 01 [14] Hussain Z, Shabbir, JImproved Estimation Procedures for the Mean of Sensitive Variable using Randomized Response Model PakJStatist5(), 05-0, 009 [15] Ozgul, N Proportion and Mean Estimators in Randomized Response Models, PhD Thesis Hacettepe University, Ankara, 013

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