A multivariate regression-cum-exponential estimator for population variance vector in two phase sampling

1 Introduction

Auxiliary information plays a vital role in illustrating conclusion about the parameters of population for the characteristics under study. It is used to enhance estimation of population parameters of the main variable under study. At the stage of manipulating as well as estimation, auxiliary information is used for improved estimation. Sometimes auxiliary information is known in prior of a survey and sometimes it is not known in advanced. There are many examples in survey sampling where auxiliary information is known in advance; number of banks in a city, number of employees, educational status, number of educated male and females in a city etc.

Graunt (1662) was the first who estimated the population of England using auxiliary information. Olkin (1958) suggested ratio estimator based on multi-auxiliary variables for multivariate case. John (1969) provided multivariate ratio and product type estimators for estimating the population means. Further comprehensive contribution of multivariate ratio and regression estimators using multi-auxiliary variables were taken up by Ahmad and Hanif (2010) for estimating population mean. Isaki (1983) proposed ratio and regression type estimators for estimating the population variance. Cebrian and Garcia (1997) worked on variance estimation by using auxiliary variables. Following Isaki (1983), Singh et al. (2009) proposed exponential estimator for estimating population variance and Abu-Dayyeh and Ahmed (2005) provided some multi-variate ratio and regression-type estimators in two-phase sampling and studied some properties of the proposed estimator through simulation study using real data. Kadilar and Cingi (2006) suggested the regression type estimator for estimating variance using known population variance of the auxiliary variable. Many other authors including Upadhyaya and Singh (2006), Ahmed et al. (2000), Yadav and Kadilar (2013), Singh and Solanki (2013), Ahmad et al. (2016) and Singh and Pal (2016) etc. have worked on variance estimation for using population variance of the auxiliary information. Asghar et al. (2014) provided some exponential-type estimator for variance using population means of multi-auxiliary auxiliary variables.

In fact, there is no significant work on variance estimation in the literature for estimating finite population variance under two-phase sampling using multivariate multi-auxiliary variables. Therefore to fulfill this gap we proposed a multivariate regression type exponential estimator for estimating the finite population variance using multi-auxiliary variables under two-phase sampling for full information case.

Let $S_{y_{(m\times m)}}^2$ be the population variance and its usual unbiased estimator is defined as, $$t_0=[\begin{array}{cccccc}\hfill t_{01}\hfill & \hfill t_{02}\hfill & \hfill \dots \hfill & \hfill t_{0j}\hfill & \hfill \dots \hfill & \hfill t_{0m}\hfill \end{array}]\text{,}$$ $$t_{0j}=s_{y_{j(2)}}^2\text{,}j=1\text{,}2\text{,}...\text{,}m$$

The variance of unbiased estimator is defined as,

We develop Isaki’s (1983) uni-variate regression estimator into a multivariate regression estimator under two- phase i.e. $$t_{\mathit{reg}}=[\begin{array}{cccccc}\hfill t_{\mathit{reg}1}\hfill & \hfill t_{\mathit{reg}2}\hfill & \hfill \dots \hfill & \hfill t_{\mathit{regj}}\hfill & \hfill \dots \hfill & \hfill t_{\mathit{regm}}\hfill \end{array}]\text{,}$$ where,

The variance covariance matrix of $t_{\mathit{reg}}$ is,

We modify Shabbir and Gupta’s (2015) estimator into a multivariate regression-type estimator for estimating a vector of population variance using populations mean of auxiliary information under two- phase as, $$t_{\mathit{rg}}=[\begin{array}{cccccc}\hfill t_{\mathit{rg}1}\hfill & \hfill t_{\mathit{rg}2}\hfill & \hfill \dots \hfill & \hfill t_{\mathit{rgj}}\hfill & \hfill \dots \hfill & \hfill t_{\mathit{rgm}}\hfill \end{array}]\text{,}$$ where,

The variance covariance matrix of $t_{\mathit{rg}}$ is,

We modify Asghar et al. (2014) into a multivariate exponential ratio type estimator for estimating for estimating a vector of population variances using multi-auxiliary variables as, $$t_a=[\begin{array}{cccccc}\hfill t_{a1}\hfill & \hfill t_{a2}\hfill & \hfill \dots \hfill & \hfill t_{\mathit{aj}}\hfill & \hfill \dots \hfill & \hfill t_{\mathit{am}}\hfill \end{array}]\text{,}$$ where,

The variance covariance matrix of $t_a$ is,

The presentation of the paper is as follows; Section 2 is based on some useful results for multivariate under two-phase sampling design. Section 3 is based on the derivation of our proposed estimator. Whereas in Sections 4 and 5 numerical study with real data and simulated study with simulated data are discussed respectively. Finally, conclusion and discussions are presented in Section 6.

2 Some useful results under two-phase sampling

We consider a finite population with U (<∞) identifiable units. Let $Y$ be the variable under study taking value $y_j$ where $j=1\text{,}2\text{,}\dots \text{,}m\text{.}$ Let ${\overline{X}}_1\text{,}{\overline{X}}_2\text{,}\dots \text{,}{\overline{X}}_n$ be the population means of auxiliary variables and $S_{\mathit{yj}(2)}^2$ be the population variance of study variable. Further let $S_x^2$ an $S_y^2$ be the population variances and $C_{x_i}^2$ , $C_y^2$ be the coefficient of variation and ${\rho }_{\mathit{yx}}$ denotes the correlation coefficient between study and auxiliary variables.

Under two-phase sampling design, $x_{(1)1}\text{,}{x}_{(1)2}\text{,}\dots \text{,}{x}_{(1)n}$ is observed by $n_1$ and $y_{(2)1}\text{,}{y}_{(2)2}\text{,}\dots \text{,}{y}_{(2)m}$ is observed by n₂.

In order to derive the expressions of mean square error, let ${\gamma }_2=\frac{1}{n_2}$ and ${\gamma }_1=\frac{1}{n_1}$ be the sampling fractions and $s_{y_{j_{\left(2\right)}}}^2=S_{y_{j_{(2})}}^2(1+{\epsilon }_{y_{j\left(2\right)}})\text{,}{s}_{x\left(1\right)k}^2=S_{x\left(1\right)k}^2(1+{\epsilon }_{x_{\left(1\right)k}})\text{,}{\overline{v}}_{\left(1\right)k}={\overline{V}}_k(1+{\overline{e}}_{v_{\left(1\right)k}})\text{,}$ where ${\epsilon }_{y_{j(2)}}\text{,}$ ${\epsilon }_{x_{(1)k}}$ and ${\overline{e}}_{x_{(1)k}}$ be the sampling errors and $E\left({\epsilon }_{y_{j\left(2\right)}}\right)=E\left({\epsilon }_{x_{\left(1\right)k}}\right)=E\left({\overline{e}}_{v_{\left(1\right)k}}\right)=0$ .

For multivariate estimators, we use following expectation results for the derivation of variance covariance matrix expressions, $$\begin{array}{c}\text{Let}\hfill \\ {\mathrm{\Delta }}_y=[\begin{array}{ccccc}\hfill {\epsilon }_{y_{(2)1}}\hfill & \hfill {\epsilon }_{y_{(2)2}}\hfill & \hfill .\hfill & \hfill .\hfill & \hfill {\epsilon }_{y_{(2)m}}\hfill \end{array}]\text{,}{\mathrm{\Delta }}_x=[\begin{array}{ccccc}\hfill {\epsilon }_{x_{(1)1}}\hfill & \hfill {\epsilon }_{x_{(x)2}}\hfill & \hfill .\hfill & \hfill .\hfill & \hfill {\epsilon }_{x_{(1)n}}\hfill \end{array}]\text{,}{\overline{D}}_v=[\begin{array}{ccccc}\hfill e_{v_{(1)1}}\hfill & \hfill e_{v_{(1)2}}\hfill & \hfill .\hfill & \hfill .\hfill & \hfill e_{v(1)n}\hfill \end{array}]\hfill \\ E_1{E}_{2/1}({\mathrm{\Delta }}_x^{\prime }{\mathrm{\Delta }}_x)={\gamma }_1{\displaystyle \sum _{x_{(n\times n)}}}\text{,}{E}_1{E}_{2/1}({\mathrm{\Delta }}_y^{\prime }{\mathrm{\Delta }}_y)={\gamma }_2{\displaystyle \sum _{y_{(m\times m)}}}\text{,}{E}_1{E}_{2/1}({\mathrm{\Delta }}_x^{\prime }{\mathrm{\Delta }}_y)={\gamma }_1{\displaystyle \sum _{{\mathit{xy}}_{(n\times m)}}}\text{,}{E}_1{E}_{2/1}({\overline{D}}_v^{\prime }{\overline{D}}_v)={\gamma }_1{\displaystyle \sum _{v_{(n\times n)}}}\hfill \\ E_1{E}_{2/1}({\mathrm{\Delta }}_y^{\prime }{\overline{D}}_v)={\gamma }_1{\displaystyle \sum _{{\mathit{yv}}_{(m\times n)}}}\text{,}{E}_1{E}_{2/1}({\overline{D}}_v^{\prime }{\mathrm{\Delta }}_y)={\gamma }_1{\displaystyle \sum _{{\mathit{vy}}_{(n\times m)}}}\hfill & \hfill \end{array}$$

3 Proposed generliazed regression-cum-exponential estimator

A multivariate regression-cum-exponential type estimator is proposed for full information case using multi-auxiliary variable in two-phase sampling design. We propose following multivariate estimator for population variance vector,

4 Numerical illustration

We conducted an empirical study to demonstrate the efficiency of the proposed class of Multivariate (trivariate and bivariate) estimator using one, two, and three auxiliary variables. Similarly a univariate version from the proposed multivariate estimator has also been presented for its numerical efficiency. This empirical study has been constructed using a real population data and the population detail is given in Appendix. We consider the three study variables (Y₁, Y₂, Y₃) for trivariate estimator, two study variables (Y₁, Y₂) for bivariate estimator and one study variable Y1 for univariate estimator and further in each case used three auxiliary variables, two auxiliaries and single auxiliary variable were used. Variances and co-variances are presented in Appendix and expressions in (23), (25) and (27) were used to calculate the MSE values respectively for the trivariate, bivariate and univariate estimators. In case of multivariate estimators traces were compared with. We have computed the percent relative efficiencies (PREs) of our regression-cum-exponential estimator $t_s$ with respect to $t_0$ following, $$\mathit{PREs}(t_{\ast }\text{,}{t}_0)=\frac{\mathit{Var}(t_0)}{\mathit{MSE}(t_{\ast })}\times 100$$

Table 1 demonstrates the relative efficiency of each estimator. It is observed that by increasing number of the auxiliary variables we get more efficient results in all three cases i.e. multivariate, bivariate and univariate.

5 Simulation study

However to assess the performance of our proposed estimators we have also computed the results by simulation study. We have taken a model for our regression-cum-exponential estimator as, $$\text{Observed Model}{Y}_{\mathit{ij}}=\sum X+{\epsilon }_i{\epsilon }_i\sim N(0\text{,}1)$$

The three study variables with three auxiliary variables under Model are distinct by the equation as, $$Y_1=0.2X_1+0.3X_2+0.9X_3+\epsilon \text{,}$$ $$Y_2=0.6X_1+0.5X_2+0.3X_3+\epsilon \text{,}$$ $$Y_3=0.5X_1+0.7X_2+0.4X_3+\epsilon \text{,}$$

On the first-phase we selected a sample of size $n_1=0.5N$ units by SRSWOR using R function. In second-phase, we again selected a sample of size $n_2=0.4n_1$ units from the samples selected at first phase by SRSWOR using R function when $N=3000$ . This procedure is repeated 5000 times to calculate the several values of $t$ . At last we have calculated the variance covariance matrices as, $$\text{Variance Vector}=S={[\begin{array}{ccc}{S}_{y1}^2\hfill & S_{y2}^2\hfill & S_{y3}^2\hfill \end{array}]}_{(1\times 3)}\text{and}{\displaystyle \sum _{(3\times 3)}}=\left[\begin{array}{ccc}\text{var}(t_1)\hfill & \text{cov}(t_1{t}_2)\hfill & \text{cov}(t_1{t}_3)\hfill \\ \text{cov}(t_2{t}_1)\hfill & \text{var}(t_2)\hfill & \text{cov}(t_2{t}_3)\hfill \\ \text{cov}(t_1{t}_3)\hfill & \text{cov}(t_2{t}_3)\hfill & \text{var}(t_3)\hfill \end{array}\right]$$ where $$\text{var}(t_i)=\frac{1}{q}{\displaystyle \sum _{i=1}^q}{(t_i-T)}^2\text{,}T=\frac{1}{q}{\displaystyle \sum _{i=1}^q}{t}_i\text{and}\text{cov}(t_i)=\frac{1}{q}{\displaystyle \sum _{i=1}^q}(t_i-T)(t_j-T)$$

6 Concluding remarks

From empirical study given in Table 1 it is concluded that our proposed estimator (trivariate/bivariate/univariate) ${\mathbf{t}}_s$ is more efficient than $t_0\text{,}{t}_a\text{,}{t}_{\mathit{rg}}$ estimators. From Table 1, it is also observe that by increasing the auxiliary information, our proposed estimator gives more efficient result. From results of the simulation study presented in Tables 2 and 3 it is concluded that proposed multivariate estimator are more efficient as they have less MSE values as well as values of the determinants are minimum than the determinant values of the existing estimators. Table 4 reveals that univariate version of the proposed estimator has minimum value of the MSE than the MSE’s of the existing estimator and therefore it is concluded that our proposed regression-cum-exponential estimator performs more efficiently. In simulation it is confirmed that proposed estimator is asymptotically normal and more consistent the existent estimators.