Generalized Difference-Cum Exponential Class of Estimators for Estimating Population Parameters of the Sensitive Variable

We propose a generalized class of estimators to estimate general population parameters of susceptible research variables using additional auxiliary information in SRSWOR. The population constants such as the coefficient of variation, the population mean, the standard deviation, and the population mean square are defined using a conventional estimator. The expression for the mean square errors of the proposed class is derived up to the first order of approximation. To compare the effectiveness of the proposed class of estimators, an empirical study is conducted utilizing real and simulated data sets. Theoretical and empirical research demonstrates that the suggested generalized class of estimators outperforms other current estimators.


Introduction
e collection of data on sensitive topics such as induced abortions, drug misuse, and family income through personal interviews and surveys is a severe problem. Some questions, for example, are delicate: (a) On your 2009 tax return, how much did you underreport your income? (c) Have you undergone any abortions? (b) Have you molested any children? (e) Do you use illicit substances? One approach to get individuals to answer honestly is to use randomised response techniques. Horvitz et al. [1] and Greenberg et al. [2] have expanded Warner's [3] model to include numeric responses to the sensitive question rather than simple "yes" or "no" responses. e respondent chooses one of two questions using a randomization device: one is the sensitive question, and the other is unrelated. Direct true responses on the study variable might become di cult to obtain in survey sampling, especially when the variable is sensitive. When the study variable is sensitive in nature, many survey statisticians have estimated the population mean of sensitive variables, such as Eichhron and Hayre [4], Gupta et al. [5], Saha [6], and Diana and Perri [7]. Di erent ratio, regression, and exponential estimators for estimating population parameters of sensitive variables based on scrambled responses were reported by Sousa et al. [8], Koyuncu et al. [9], Kalucha et al. [10], and Gupta et al. [11,12]. Using a generalized quantitative optional randomised response model, Noor-Ul-Amin et al. [13] developed estimators based on generalized ratio and regression types, in which the nature of the auxiliary variable is nonsensitive. In the presence of measurement error, Khalil et al. [14] and Zahid and Shabbir [15] developed improved mean estimators of a sensitive research variable. Sanaullah et al. [16] introduced a new family of di erence-cum-exponential-type estimators of the nite population mean of the susceptible research variable by using a single nonsensitive secondary information. In the case of nonresponse using RRT under two-phase sampling, Sanaullah et al. [17] proposed a generalized family of estimators for the nite population mean of a susceptible research variable. Quantitative scrambled randomised response models are considerably improved by Saleem et al. [18] by assessing the performance of the population mean estimator. Waseem et al. [19] presented a generalized exponential type estimator for the mean of a sensitive variable using information from a nonsensitive auxiliary variable. Singh et al. [20] provided some alternative additive randomised response models by offering a blank card option for predicting the population mean of a quantitative sensitive variable. Vishwakarma and Singh [21] developed the estimation procedure to estimate the effect of measurement errors under additive scramble response for sensitive variables.
ere have been a few extreme values in numerous populations, and estimating the unknown population parameters without considering this knowledge is extremely delicate. Otherwise, the results will be understated or exaggerated, respectively. In order to resolve this problem, it is necessary to incorporate this knowledge into the estimation of population parameters. Several researchers, including Isaki [22], Bahl and Tuteja [23], Upadhyaya and Singh [24], Kadilar and Cingi [25], Dubey and Sharma [26], Singh and Vishwakarm [27], Shabbir and Gupta [28], and Yadav et al. [29], have proposed some broader classes of estimators for estimating finite population variance. Gupta et al. [30] and Daraz and Khan [31] presented several variance estimators for sensitive variables that make use of additional information in SRSWOR.
Motivated by Adichwal et al. [32], we suggest the general difference-cum exponential class of estimators for estimating general population parameters of the sensitive research variable.
Let Y represent the research variable, which contains sensitive elements that may not be accessible precisely as a result of the respondent's response. Let X be a nonsensitive secondary variable with a positive correlation to Y. Let S be an uncorrelated scrambled variate with a predetermined distribution. e variable S is used to scramble the sensitive variable Y. Each respondent is given the task of selecting a random number from the S distribution, say s, and adding it to the actual value of Y to assess Z. Let Z stand for the reported scrambled answer to Y, which was first proposed by Warner [3] and then developed by Pollock and Bek [33]. e general population parameter used in the study are discussed as follows: δ rs � μ rs /μ r/2 20 μ s/2 02 where r and s are nonnegative integers.
(2) e error terms used in the study are defined as follows: Under the condition that

Conventional Estimators
e population parameter of a sensitive research variable under discussion can be stated in its most general form as where a and b are suitably chosen scalars that take different values in order to get different parameters of a population. e parameters obtained after substituting a different combination of the scalars are given by For estimating the general population parameters discussed previously, the general traditional estimator z (a,b) is given by On expressing (6) in terms of errors, we have Squaring (7) and excluding terms with powers of three or more, we get We obtained the MSE of the general estimator by taking expectations on both the side of the abovementioned equation, as

Proposed Generalized Class of Estimators
Auxiliary information can be analyzed in a range of categories, including variables and attributes. It has been widely used by many researchers to develop numerous estimators for estimating population parameters in various sampling systems. Motivated by Adichwal et al. [32], we propose the general difference-cum exponential class of estimators for estimating population parameters of the sensitive research variable, given by where k 1 and k 2 are considered as scalar constant which takes the values (0, − 1, 1), and w, α, and β are suitably chosen constants. Expressing (11) in terms of the errors, we get Multiplying and neglecting the terms of higher-order containing errors in (12) which are greater than order two, we get Equivalently, Mathematical Problems in Engineering Squaring (14) on both the sides, and neglecting error terms of power greater than two, we get Equivalently, Taking expectations on both sides of the abovementioned equation, we get MSE of z d , given as where Differentiating (17) partially w.r.t P and Q, and equating to zero, we get After substituting the optimal values of P and Q, we get the minimum MSE(z d ) as Using known population mean X and substituting k 2 � 0, in the general difference-cum exponential class of estimators (11), we get the following general estimator for estimating population parameters of the sensitive research variable as where k 1 takes real values (-1, 0, 1) and w and α are suitably chosen constants. Following the abovementioned procedure, the MSE of z d1 is given by e minimum value of MSE (z d1 ) is obtained, after minimizing the above equation for (k 1 /α) opt � F 2 (a, b)/C x , as Table 1 illustrates the existing estimators derived from the family of z d1 estimators by selecting appropriate values for a, b, w, k 1 , α and applying them appropriately.
Using known population variance S 2 x and substituting k 1 � 0 in the general difference-cum exponential class of estimators (11), we get another general estimator for estimating population parameters of the sensitive research variable as where k 2 takes real values (-1, 0, 1) and w and β are suitably chosen constants. Following the abovementioned procedure, the MSE of z d2 is given by e minimum value of MSE (z d1 ) is obtained, after minimizing the above equation for (k 2 /β) opt � F 3 (a, b)/(δ 04 − 1), as Table 2 demonstrates the estimators derived from the family of z d2 estimators by selecting appropriate values for a, b, k 2 , α and applying them appropriately.

Efficiency Comparision of
Mathematical Problems in Engineering provided that (δ 04 − δ 2 03 − 1) ≥ 0 , the proposed class of estimator z d outperforms other competing estimators.

Estimtion of Population Mean of the Research Variable
Using known population parameters and substituting differentvaluesofa, b, w, k 1 , k 2 , α, β as (1,0 w, k 1 , k 2 , , α, β) in the general difference-cum exponential class of estimators (equation 4), we obtain a class of estimators for estimating the population mean of the sensitive research variable as e expression of MSE (z d ′ ) upto O(n-1) is given by where MSE(z d ′ ) is minimized for the optimal values of P and Q, given by On substituting the optimal values of P and Q in (30), the least value of MSE(z ′ ) takes the form, given by

Relative Efficiency
In the current section, we analyze the theoretical efficiency of the proposed class of estimators and the criteria is discussed under which the estimator performs better than other existing estimators.

Empirical Study
In a view to analysing the accuracy of the theoretical results, we use an empirical investigation to assess the effectiveness of the suggested class of estimators. e real population included in the study is Cochran ([34], page no. 325), where X denotes the number of rooms per block and Y denotes the number of person per block. e descriptive statistics of the real data population are given as Table 1: Several members of the z d1 family of estimator.

S.No.
Estimators Table 2: Several members of the z d2 family of estimator.
Mathematical Problems in Engineering Table 3: Several members of the z d ′ family of estimator.  gamma, Poisson, uniform, and log-normal with differing population parameters. Following the procedure adopted by Singh and Horn [35], we generated study and ancillary variables as Y � 10 + ������� � (1 − ρ 2 xy )Y * + ρ xy (S y /S x )X * and X � 5 + X * , where X * and Y * are independent variates of pertaining family of distributions. e sensitive variable Z is derived by scrambling the variable Y with S where S follows the corresponding parent distribution with predefined parameters. With various selections of correlation coefficient, i.e., 0.6, 0.7, 0.8, and 0.9, samples of different sizes, i.e., 40, 60, 80, and 100, are collected from each bivariate distribution.

Results and Discussion
e findings of the study for real data set are presented in Table 4 which describes the MSE and PRE of suggested class of estimators z d ′ along with the competing estimators, i.e., RRT sample mean (z 0 ), (z d1(3) ), Gupta et al. [12]'s estimator (z d1 (3) ) and difference estimator (z d1(4) ). Table 5-10 summarise Table 6 PRE Table 7 of known Table 8 and proposed  Table 9 estimators for various sample sizes, i.e., (40, 60, 80, and 100) and correlation coefficients, i.e., (0.6, 0.7, 0.8, and 0.9). Under simulation investigation, the proposed estimate z d ′ has the lowest MSE for all available regression coefficient