A generalized class of estimators for sensitive variable in the presence of measurement error and non-response under stratified random sampling

1 Introduction

In survey sampling, certain surveys cause some problems for the researchers due to the fact that the respondents are reluctant to discuss sensitive topics such as drug use, abortion, sexually transmitted diseases etc. When surveying on those topics, measurement error and non-response can occur since the respondent may choose, not to respond some specific questions, not to give the accurate answers, or not to take part in the survey. The problem of measurement error is usually ignored during the sensitive surveys and the assumption is made that the information obtained through surveys is free from error. Another important factor in surveys is non-response, which may arises due to refusal of respondents to give the information or not at home or lack of interest due to some sensitive issues. Usually measurement error and non-response are studied separately for the sensitive variable using the known auxiliary or additional information. In reality, when the variable of interest is sensitive, the respondents hesitate to provide the personal information, which give rise to measurement error. In most of the cases, the information is not obtained from all units in surveys, specially when the variable of interest is stigmatizing in nature. Many researchers studied the problem of non-response, including (Hansen and Hurwitz, 1946; Cochran, 1977; Rao, 1986; Khare and Srivastava, 2010; Andridge and Little, 2010; Singh et al., 2011; Khare et al., 2013; Shabbir and Khan, 2013; Shabbir et al., 2018 and Singh and Khalid, 2020). In survey sampling, when the variable under study contains social stigma, then the respondents are not comfortable to provide their personal information. Direct survey on sensitive question increases the relative bias. Warner (1965) introduced the randomized response technique (RRT), which reduces the possible bias and is used to obtain the true information while insuring the privacy of the respondents. For estimation of mean of a sensitive quantitative variable the Randomized Response model (RRM) is extended by Greenberg et al. (1971). Further work is done by Eichhorn and Hayre (1983); Gupta and Shabbir (2004), Kim and Warde (2004); Singh and Mathur (2005), Gjestvang and Singh (2006); Diana and Perri (2010), Gupta et al. (2010); Chaudhuri and Pal (2015), Gupta et al. (2016) and Bouza et al. (2018).

The researchers dealt with the problem of measurement error for estimating the population mean. For more details, see Cochran (1968); Fuller (1995); Shalabh (1997); Biemer et al. (2011); Shukla et al. (2012), etc. Recently few researchers studied the problem of measurement error and non-response together like Kumar et al. (2015); Singh and Sharma (2015); Azeem and Hanif (2017) and Kumar (2016). Zahid and Shabbir (2018); Khalil et al. (2018) and Zahid and Shabbir (2019) have discussed the problem of measurement error and non-response under stratified random sampling.

In practice, the researchers who have studied measurement error, have ignored the presence of non-response and randomized response at the same time. In this study, we have proposed a class of estimators for estimating the population mean of a sensitive variable in the presence of measurement error and non-response simultaneously under stratified random sampling. The efficiency of the suggested class of estimators over the existing estimators is shown through simulation study and real data sets.

Consider a finite population of N identifiable units which are partitioned into L homogeneous subgroups called strata, such that the $h^{\mathit{th}}$ strata consist of $N_h$ units, where $h=1,2,\dots ,L$ and ${\sum }_{h=1}^L{N}_h=N$ . It is assumed that N consists of two mutually exclusive groups called response and non-response groups. Let $N_{1h}$ and $N_{2h}$ are the responding and non-responding units in the $h^{\mathit{th}}$ stratum respectively. We select a sample of size $n_h$ from $N_h$ by using simple random sampling without replacement (SRSWOR) and assume that $n_{1h}$ units respond and $n_{2h}$ units do not respond. We select a sub-sample of size $k_h,\left(k_h=\frac{n_{2h}}{g_h},g_h>1\right)$ from $n_{2h}$ non-responding units in the $h^{\mathit{th}}$ stratum.

Let $(z_{\mathit{hi}}^{\ast },y_{\mathit{hi}}^{\ast },x_{\mathit{hi}}^{\ast },r_{x_{\mathit{hi}}}^{\ast })$ be the observed values and $(Z_{\mathit{hi}}^{\ast },Y_{\mathit{hi}}^{\ast },X_{\mathit{hi}}^{\ast },R_{x_{\mathit{hi}}}^{\ast })$ be the actual values of the $i^{\mathit{th}}(i=1,2,\dots ,n)$ sampled units in the $h^{\mathit{th}}$ stratum. Let $r_{x_{\mathit{hi}}}^{\ast }$ , and $R_{x_{\mathit{hi}}}^{\ast }$ be the corresponding ranks of $x_{\mathit{hi}}^{\ast }$ and $X_{\mathit{hi}}^{\ast }$ respectively, then the measurement errors be.

$Q_{\mathit{hi}}^{\ast }=z_{\mathit{hi}}^{\ast }-Z_{\mathit{hi}}^{\ast },V_{\mathit{hi}}^{\ast }=x_{\mathit{hi}}^{\ast }-X_{\mathit{hi}}^{\ast }$ and $T_{\mathit{hi}}=r_{x_{\mathit{hi}}}-R_{x_{\mathit{hi}}}$ .

Let $S_{\mathit{hZ}}^2,S_{\mathit{hX}}^2$ and $S_{{\mathit{hR}}_x}^2$ be the population variances for the responding units and $S_{\mathit{hZ}\left(2\right)}^2,S_{\mathit{hX}\left(2\right)}^2$ and $S_{{\mathit{hR}}_x\left(2\right)}^2$ be the population variances for non-responding units. Let $S_{\mathit{hQ}}^2,S_{\mathit{hV}}^2$ and $S_{\mathit{hT}}^2$ be the population variances associated with the measurement error for responding units. Let $S_{\mathit{hQ}\left(2\right)}^2,S_{\mathit{hV}\left(2\right)}^2$ and $S_{\mathit{hT}\left(2\right)}^2$ be the population variances associated with measurement error for the non-responding part of the population. Let ${\rho }_{\mathit{hZX}},{\rho }_{{\mathit{hZR}}_x},{\rho }_{{\mathit{hXR}}_x}$ be the coefficients of correlation, between their subscripts for respondents and ${\rho }_{\mathit{hZX}\left(2\right)},{\rho }_{{\mathit{hZR}}_x\left(2\right)},{\rho }_{{\mathit{hXR}}_x\left(2\right)}$ be the coefficients of correlation, between their subscripts for non-respondents in the population.

In Section 2, some existing estimators of the finite population mean are given. In Section 3, a generalized class of estimators is suggested for estimating the finite population mean by incorporating both measurement error and non-response information simultaneously. Numerical results and simulation study are presented in Section 4. Conclusion is given in Section 5.

2 Existing Estimators in Literature

In this section we consider the following existing estimators.

3 Proposed Generalized Class of Estimators

We suggest a generalized class of estimators for estimating the finite population mean for a sensitive variable, considering the problem of measurement error and non-response simultaneously in stratified random sampling. Measurement error and non-response are present on both the study variable and the auxiliary variable. The suggested estimator, is given by

${\delta }_{\mathit{hZ}}^{\ast }={\sum }_{i=1}^{n_h}(Y_{\mathit{hi}}^{\ast }-{\bar{Y}}_h)$ , ${\delta }_{\mathit{hU}}^{\ast }={\sum }_{i=1}^{n_{h_h}}{U}_{\mathit{hi}}^{\ast }$ ,

${\delta }_{\mathit{hX}}^{\ast }={\sum }_{i=1}^{n_h}(X_{\mathit{hi}}^{\ast }-{\bar{X}}_h)$ , ${\delta }_{\mathit{hV}}^{\ast }={\sum }_{i=1}^{n_{h_h}}{V}_{\mathit{hi}}^{\ast }$ ,

${\delta }_{R_{\mathit{hx}}}^{\ast }={\sum }_{i=1}^n(R_{x_{\mathit{hi}}}^{\ast }-{\bar{R}}_{x_h})$ , ${\delta }_T^{\ast }={\sum }_{i=1}^{n_h}{T}_{\mathit{hi}}^{\ast }$ .

Adding ${\delta }_{\mathit{hY}}^{\ast }$ and ${\delta }_{\mathit{hU}}^{\ast }$ , we get.

${\delta }_{\mathit{hZ}}^{\ast }+{\delta }_{\mathit{hU}}^{\ast }={\sum }_{i=1}^{n_h}(Z_{\mathit{hi}}^{\ast }-{\bar{Z}}_h)+{\sum }_{i=1}^{n_h}{U}_{\mathit{hi}}^{\ast }$ .

Dividing both sides by $n_h$ , and then simplifying, we get

Similarly, we can get

Further

$E{\left(\frac{{\delta }_{\mathit{hZ}}^{\ast }+{\delta }_{\mathit{hQ}}^{\ast }}{n_h}\right)}^2={\lambda }_{2h}(S_{\mathit{hZ}}^2+S_{\mathit{hQ}}^2)+{\theta }_h(S_{\mathit{hZ}\left(2\right)}^2+S_{\mathit{hQ}\left(2\right)}^2)=A_h^{*\prime }$ ,

$E{\left(\frac{{\delta }_{\mathit{hX}}^{\ast }+{\delta }_{\mathit{hV}}^{\ast }}{n_h}\right)}^2={\lambda }_{2h}(S_{\mathit{hX}}^2+S_{\mathit{hV}}^2)+{\theta }_h(S_{\mathit{hX}\left(2\right)}^2+S_{\mathit{hV}\left(2\right)}^2)=B_h^{*\prime }$ ,

$E{\left(\frac{{\delta }_{{\mathit{hR}}_x}^{\ast }+{\delta }_{\mathit{hT}}^{\ast }}{n_h}\right)}^2={\lambda }_{2h}(S_{{\mathit{hR}}_x}^2+S_{\mathit{hT}}^2)+{\theta }_h(S_{{\mathit{hR}}_x\left(2\right)}^2+S_{\mathit{hT}\left(2\right)}^2)=D_h^{*\prime }$ ,

$E\left(\frac{{\delta }_{\mathit{hZ}}^{\ast }+{\delta }_{\mathit{hQ}}^{\ast }}{n_h}\right)\left(\frac{{\delta }_{\mathit{hX}}^{\ast }+{\delta }_{\mathit{hV}}^{\ast }}{n}\right)={\lambda }_{2h}{\rho }_{\mathit{hZX}}{S}_{\mathit{hZ}}{S}_{\mathit{hX}}+{\theta }_h{\rho }_{\mathit{hZX}\left(2\right)}{S}_{\mathit{hZ}\left(2\right)}{S}_{\mathit{hX}\left(2\right)}=C_h^{*\prime }$ ,

$E\left(\frac{{\delta }_{\mathit{hZ}}^{\ast }+{\delta }_{\mathit{hQ}}^{\ast }}{n_h}\right)\left(\frac{{\delta }_{{\mathit{hR}}_x}^{\ast }+{\delta }_{\mathit{hT}}^{\ast }}{n}\right)={\lambda }_{2h}{\rho }_{{\mathit{hZR}}_x}{S}_{\mathit{hZ}}{S}_{{\mathit{hR}}_x}+{\theta }_h{\rho }_{{\mathit{hZR}}_x\left(2\right)}{S}_{\mathit{hZ}\left(2\right)}{S}_{{\mathit{hR}}_x\left(2\right)}=E_h^{*\prime }$ ,

$E\left(\frac{{\delta }_{\mathit{hX}}^{\ast }+{\delta }_{\mathit{hV}}^{\ast }}{n_h}\right)\left(\frac{{\delta }_{{\mathit{hR}}_x}^{\ast }+{\delta }_{\mathit{hT}}^{\ast }}{n_h}\right)={\lambda }_{2h}{\rho }_{{\mathit{hXR}}_x}{S}_{\mathit{hX}}{S}_{{\mathit{hR}}_x}+{\theta }_h{\rho }_{{\mathit{hXR}}_x\left(2\right)}{S}_{\mathit{hX}\left(2\right)}{S}_{{\mathit{hR}}_x\left(2\right)}=F_h^{*\prime }$ .

On simplifying, we get

$b^{*\prime }={\alpha }_3$ ,

$c^{*\prime }=\frac{1-{\alpha }_0}{2}$ ,

$d^{*\prime }={\alpha }_2+\frac{1-{\alpha }_0}{2}$ ,

$e^{*\prime }={\alpha }_1+\frac{1-{\alpha }_0}{2}$ , and.

$f^{*\prime }=\frac{{\alpha }_0^2-4{\alpha }_0+3}{8}+\frac{{\alpha }_1(2-{\alpha }_0+{\alpha }_1)}{2}$ .

$W_{\mathit{hZ}}=\frac{{\delta }_Z^{\ast }+{\delta }_Q^{\ast }}{n},W_{\mathit{hX}}=\frac{{\delta }_X^{\ast }+{\delta }_V^{\ast }}{n}$ and $W_{{\mathit{hR}}_x}=\frac{{\delta }_{R_x}^{\ast }+{\delta }_T^{\ast }}{n}$ .

Further simplifying, and ignoring error terms greater than two, we have

Using Eq. (25), the bias and MSE of ${\bar{y}}_{S\left(\mathit{GP}\right)}^{*\prime }$ to first order of approximation, is given by

$A_{h1}^{*\prime }={\bar{Z}}_h^2+A_h+e^{*\prime 2}{t}_h^2{R}_h^{\prime 2}{B}_h+4e^{*\prime }{t}_h{R}_h^{\prime }{C}_h+2f^{*\prime }{t}_h^2{R}_h^{\prime 2}{B}_h$ ,

$B_{h1}^{*\prime }=t_h^2{B}_h$ ,

$C_{h1}^{*\prime }=t_h{C}_h+t_h^2{R}_h^{\prime }{B}_h(e^{*\prime }+d^{*\prime })$ ,

$D_{h1}^{*\prime }={\bar{Z}}_h^2+e^{*\prime }{t}_h{R}_h^{\prime }{C}_h+f^{*\prime }{t}_h^2{R}_h^{\prime 2}{B}_h$ ,

$E_{h1}^{*\prime }=d^{*\prime }{t}_h^2{R}_h^{\prime }{B}_h$ ,

$F_{h1}^{*\prime }=t_h^2{D}_h$ ,

$G_{h1}^{*\prime }=c^{*\prime }{t}_h{R}_h^{\prime }{F}_h+e^{*\prime }{t}_h^2{R}_h^{\prime }{F}_h+t_h{E}_h+b^{*\prime }{t}_h^2{R}_{1h}^{\prime }{D}_h$ ,

$H_{h1}^{*\prime }=t_h^2{F}_h$ ,

$I_{h1}^{*\prime }=c^{*\prime }{t}_h{R}_h^{\prime }{F}_h+b^{*\prime }{t}_h^2{R}_{1h}^{\prime }{D}_h$ .

For finding the optimal values of $m_{1h},m_{2h}$ and $m_{3h}$ , we differentiate Eq. (27) with respect to $m_{1h},m_{2h}$ and $m_{3h}$ respectively. The optimal values, are given by.

$m_{1h\left(\mathit{opt}\right)}=\frac{B_{h1}^{*\prime }{D}_{h1}^{*\prime }{F}_{h1}^{*\prime }-C_{h1}^{*\prime }{E}_{h1}^{*\prime }{F}_{h1}^{*\prime }+E_{h1}^{*\prime }{G}_{h1}^{*\prime }{H}_{h1}^{*\prime }-D_{h1}^{*\prime }{H}_{h1}^{*2}-B_{h1}^{*\prime }{G}_{h1}^{*\prime }{I}_{h1}^{*\prime }+C_{h1}^{*\prime }{H}_{h1}^{*\prime }{I}_{h1}^{*\prime }}{A_{h1}^{*\prime }{B}_{h1}^{*\prime }{F}_{h1}^{*\prime }-C_{h1}^{*2}{F}_{h1}^{*\prime }+2C_{h1}^{*\prime }{G}_{h1}^{*\prime }{H}_{h1}^{*\prime }-A_{h1}^{*\prime }{H}_{h1}^{*2}}$ ,

$m_{2h\left(\mathit{opt}\right)}=\frac{A_{h1}^{*\prime }{E}_{h1}^{*\prime }{F}_{h1}^{*\prime }-C_{h1}^{*\prime }{D}_{h1}^{*\prime }{F}_{h1}^{*\prime }-E_{h1}^{*\prime }{G}_{h1}^{*2}+D_{h1}^{*\prime }{G}_{h1}^{*\prime }{H}_{h1}^{*\prime }+C_{h1}^{*\prime }{G}_{h1}^{*\prime }{I}_{h1}^{*\prime }-A_{h1}^{*\prime }{H}_{h1}^{*\prime }{I}_{h1}^{*\prime }}{A_{h1}^{*\prime }{B}_{h1}^{*\prime }{F}_{h1}^{*\prime }-C_{h1}^{*2}{F}_{h1}^{*\prime }+2C_{h1}^{*\prime }{G}_{h1}^{*\prime }{H}_{h1}^{*\prime }-A_{h1}^{*\prime }{H}_{h1}^{*2}}$ , and.

$m_{3\left(\mathit{opt}\right)}=\frac{C_{h1}^{*\prime }{E}_{h1}^{*\prime }{G}_{h1}^{*\prime }-B_{h1}^{*\prime }{D}_{h1}^{*\prime }{G}_{h1}^{*\prime }+C_{h1}^{*\prime }{D}_{h1}^{*\prime }{H}_{h1}^{*\prime }-A_{h1}^{*\prime }{E}_{h1}^{*\prime }{H}_{h1}^{*\prime }+A_{h1}^{*\prime }{B}_{h1}^{*\prime }{I}_{h1}^{*\prime }-C_{h1}^{*2}{I}_{h1}^{*\prime }}{A_{h1}^{*\prime }{B}_{h1}^{*\prime }{F}_{h1}^{*\prime }-C_{h1}^{*2}{F}_{h1}^{*\prime }+2C_{h1}^{*\prime }{G}_{h1}^{*\prime }{H}_{h1}^{*\prime }-A_{h1}^{*\prime }{H}_{h1}^{*2}}$ .