In sample survey, it is expected that the information would be collected from all the selected units in the sample, but practically, it is generally not possible because of non-response. Some of the units may not respond or may not be contacted during the survey period. This work focuses on domain estimation of population mean with sub-sampling the non-respondents. In this study, we consider calibration technique as a method of correcting non-response in domains of study by minimizing the chi-square distance function between the weight of the main estimator and the calibrated weight subject to the formulated constraint on the auxiliary variable. As a result, two estimators are proposed; these are the ratio estimator for domain mean and a ratio estimator for double sampling. Bias and Mean Square Error (MSE) for the proposed estimators are derived.

We have used an auxiliary variable to estimate the population mean assuming that the non-response is observed only on the study variable. The proposed estimators and the existing estimators where compared empirically in the domains with small sampling units and two populations where considered in terms of the MSE and Percentage Relative Efficiency (PRE). We considered two cases where non-responses are uniform in the two strata at approximately (30%) and a case where the non-response rates are different with 20% and 40% in strata 1 and 2 respectively. The proposed estimators are more efficient than the existing estimators.

Keywords: auxiliary variable, calibration, non-response, ratio estimator, sub-sampling, domain

It is obvious that society cannot run effectively on the basis of hunches or trial and error. Decisions based on data will provide better results than those based on intuitions or gut feelings. Statistics is a range of procedures for gathering, organizing, analyzing and presenting quantitative data. In the modern society, the need for statistical information seems endless. In particular, data are regularly collected to satisfy the need for information about specified sets of elements, called as finite population. Statistics helps us to turn data into information. One of the most important modes of data collection for satisfying such needs is sample survey, that is, a partial investigation of the finite population and on the basis of such partial information (sample information) one tries to inference about the finite population characteristics (parameters). Sample survey is less expensive than a complete enumeration, it is usually less time consuming, and may even be more accurate than the method of complete enumeration. The term sample is used for the set of units or portion of the aggregate of material which has been selected with the belief that it will be representative of the whole aggregate. The sampling theory deals with scientific and objective procedure of choosing an appropriate sampling design, i.e. selecting a sample from the population which is representative of the population as a whole and also provides suitable estimation procedure to estimate the population parameters. Most challenging about the sample representation of the population is the effect of non-response on the estimation of the population parameter. Different authors have suggested different techniques for a reliable and efficient estimator, among which is the calibration technique.

Calibration estimation in sample surveys has since its introduction by Deville JC, et al.¹ developed an established theory and method for estimation of finite population parameter. Calibration of weights is a technique that uses population data on auxiliary variables to improve estimates in sample surveys. If auxiliary data are available, some improvement in the precision of estimate may be achieved. Incorporation of auxiliary data in the estimation process is known as calibration. In stratified random sampling, calibration approach is used to obtain optimum strata weights for improving the precision of survey estimates of population parameters. Koyuncu N, et al.² defined some calibration estimators in stratified random sampling for population characteristics and Clement EP, et al.³ applied the concept of calibration estimators for domain totals in stratified random sampling. Clement EP, et al.⁴ combined some scalars with the mean of the auxiliary variable and proposed calibration alternative ratio estimator of mean in stratified sampling.

When a researcher is interested in obtaining information from a local or small area, it becomes challenging with small sample size in some of the areas of interest and even very difficult when non-response occurs. Several authors have made attempts to obtain reliable estimates in such areas of interests popularly called domains of study. Among them is Godwin A, et al.⁵ The author considers modifications of some of the procedures for global ratio estimation in single-phase sampling with sub-sampling the non-respondents proposed by Rao P⁶ to obtain an estimate of mean for a small domain that cuts across constituent strata of a population with unknown weights. The bias and mean-square error of each of the modified estimators were obtained for comparison However, the estimators were not subjected numerical test to validate the analytical claims most importantly in areas of small/zero sample sizes. Unlike,⁶ the population mean of the auxiliary variable adopted by Godwin A, et al.⁵ is assumed to be unknown before the start of the survey and hence double sampling was applied under stratified simple random sampling.

In a bid to improve on the efficiency of the estimators under non-response,^7,8 adopted the concept of calibration with a single constraint to estimate the population mean and the result was encouraging. Cochran WG⁹ showed that knowledge of, of domain j that is of interest reduces the variance of the estimator of domain mean in a single-phase simple random design. The reduction in variance is shown to be greater when the proportion of non-domain elements in the population is large and the study variable varies little among the domain elements. Ashutosh¹⁰ proposed estimators for domain mean utilizing stratified sampling with non-response. The proposed estimator was compared to a direct ratio estimator for domain mean utilizing stratified sampling with non-response. Clement EP, et al.⁴ stated that in the presence of powerful auxiliary variables, the calibration estimation meets the objective of reducing both non-response bias and the sampling error. Etebong P¹¹ develops a new approach to ratio estimation that produces a more efficient class of ratio estimators that do not depend on any optimality conditions for optimum performance using calibration weightings. Iseh MJ, et al.,¹² Iseh MJ, et al.,¹³ Iseh MJ, et al.¹⁴ considered the challenges of population mean estimation in small area that is characterized by small or no sample size and in the presence of unit non-response and presents a calibration estimator that produces reliable estimates under stratified random sampling from a class of synthetic estimators using calibration approach with alternative distance measure. To overcome the challenges of poor performance of the ratio estimator in small area occasioned with small/no sample size as a result of non-response, this work considers the calibration approach using the constraints of equal weights adjustment criteria, unbiased estimator of the population mean and variance of the auxiliary variable.

In this paper, based on the attempt by Godwin A, et al.⁵ who suggested the global ratio estimation in single-phase sampling with sub-sampling the non-respondents to obtain an estimate of mean for a small domain that cuts across constituent strata of a population with unknown weights, a new improved ratio estimator for population mean in stratified random sampling is suggested using the theory of calibration estimation with three constraints to achieve optimal precision and efficiency.

Some existing estimator and theoretical underpinnings

This section considers some existing ratio estimators for estimation of domain population mean and the theoretical underpinnings for the proposed ratio estimator. Though not much have been done in the area of domains of study in the presence of non-response probably due to the intricate nature of the estimation, this paper highlights some existing estimators as applicable to domain estimation which applied the concept of sub-sampling the non-respondents.

Some existing estimator

Study notations and definitions

$N=$ population size under study

$N_d=$ population size for the $d^{th}$  domain

$N_{dh}=$ population size of $h^{th}$ stratum in $d^{th}$ domain

$n_{dh}=$ sample size for the $d^{th}$ domain in the $h^{th}$  stratum

$n_{dh}=$ domain sample Size

$n_{1dh}=$ sample size for respondent units for the $d^{th}$ domain in the $h^{th}$ Stratum

$n_{2dh}=$ Sample size for nonrespondents units for the $d^{th}$ domain in the $h^{th}$ Stratum

$W_d{h}^{*}=$ The calibration weight

$W_{dh}=$ Stratum weight

$W_{dh1}$ = Response rate of the $d^{th}$ domain in the $h^{th}$ Stratum

$W_{dh2}=$ Non-response rate of the $d^{th}$ domain in the $h^{th}$ Stratum

${\lambda }_1,{\lambda }_2$ and ${\lambda }_3$  = the LaGrange multipliers

$X=$ Auxiliary variable

$Y=$ Study variable

${\overline{x}}_{dh}=$ Sample mean for the $d^{th}$ domain in the $h^{th}$ Stratum of the auxiliary variable

${\overline{y}}^{*}{}_{dh}=$ Unbiased estimator of the population mean for the $d^{th}$ domain in the $h^{th}$ Stratum of the study variable

${\overline{X}}_{dh}=$ Population mean for $d^{th}$ domain of the auxiliary variable in the $h^{th}$ Stratum

${\overline{Y}}_{dh}=$ Population mean for $d^{th}$ domain of the study variable in the $h^{th}$ Stratum

${\overline{X}}_d=$ Population mean for $d^{th}$ domain of the auxiliary variable

${\overline{Y}}_d=$ Population mean for $d^{th}$ domain of the study variable

$S_{ydh}^2=$ Mean square of the $d^{th}$ domain in the $h^{th}$ Stratum of the study variable

$S_{xdh}^2=$ Mean square of the $d^{th}$ domain in the Stratum of the $h^{th}$ auxilliary variable

$C_{xdh}=$ Coefficient of variation for the $d^{th}$ domain in the $h^{th}$ Stratum of the auxilliary variable

$C_{ydh}=$ Coefficient of variation for the $d^{th}$ domain in the $h^{th}$ Stratum of the study variable

$S_{ydh2}^2=$ Mean square of non-respondence of the $d^{th}$ domain in the $h^{th}$ Stratum of the study variable

$k_{dh}=$ Inverse sampling rate

$Q_{dh}=$ Tuning parameter

Udofia (2004) estimator

An alternative ratio estimator for domain mean was suggested by [5] is as follows:

$t_{2j}={\displaystyle \sum }_h^k{W}_h\frac{{\overline{y}}_h^{*}}{{\overline{x}}_h}{\overline{X}}_h$    (1)

With

$Bias\left(t_{2j}\right)={\displaystyle \sum }_{h=1}^k{W}_h\frac{1-f_h}{n_h{\overline{X}}_h}\left(R_h{S}_{x_h}^2-S_{x_h{y}_h}\right)$

and

$MSE\left(t_{2j}\right)={\displaystyle \sum }_{h=1}^k{W}_h^2\left[\frac{1-f_h}{n_h}\left(S_{yh}^2+R_h^2{S}_{x_h}^2-2R_h{S}_{x_h{y}_h}+\frac{W_{2h}\left(k-1\right)}{n_h}{S}_{2y_h}^2\right)\right]$    (2)

where

$R_h=\frac{{\overline{Y}}_h}{{\overline{X}}_h},f_h=\left(\frac{1}{n_h}-\frac{1}{N_h}\right)$

Pal and Singh HP estimator

Pal and Singh¹⁵ proposed a class of ratio-cum-ratio-type exponential estimators for population mean with sub sampling the non-respondents. The estimator and the mean square error is given as:

$t_{ps1}=\alpha {\overline{y}}^{*}\left(\frac{\overline{X}}{\overline{x}}\right)+\left(1-\alpha \right){\overline{y}}^{*}\exp \left(\frac{\overline{X}-\overline{x}}{\overline{X}+\overline{x}}\right)$

And

$MSE\left(t_{ps1}\right)={\overline{Y}}^2\left(\lambda C_y^2\left(1-{\rho }_{xy}^2\right)+\frac{W_2\left(Z-1\right)}{n}{C}_{y\left(2\right)}^2\right)$    (3)

Where 

$W_2=\frac{n_2}{n},\lambda =\frac{1-f}{n},f=\frac{n}{N}$  and α is a constant

Ashutosh estimator

Ashutosh¹⁰ proposed a direct ratio generalized estimator for domain mean through stratified sampling with non-response as;

$T_{DG.st.\beta .d}={\overline{y}}_{st.d}{\left[\frac{{\overline{x}}_{st.d}}{{\overline{X}}_{st.d}}\right]}^{\beta }$

Where β is a chosen constant of $d^{th}$ domain mean of x and the value of y respondents can be written as;

$\begin{array}{l}{\overline{y}}_{st.d}={\displaystyle \sum }_{h=1}^H{W}_{h.d}{\overline{y}}_{h.d}\\ {\overline{x}}_{st.d}={\displaystyle \sum }_{h=1}^H{W}_{h.d}{\overline{x}}_{h.d}\end{array}$

Members of the proposed estimators  $T_{DG.st.\beta .d}^{*}$

$T_{DG.st.\beta .d}^{*}={\overline{y}}_{st.a}^{*}$  if $\beta =0$

$T_{DG.st.-1.a}^{*}=\frac{{\overline{y}}_{st.a}^{*}}{{\overline{x}}_{st.a}^{*}}{\overline{X}}_{h.a}$  if $\beta =-1$

$T_{DG.st\mathrm{.1.}a}^{*}={\overline{y}}_{st.a}^{*}\frac{{\overline{x}}_{st.a}^{*}}{{\overline{x}}_{st.a}^{*}}$  if $\beta =1$

$T_{DG.st\mathrm{.2.}a}^{*}={\overline{y}}_{st.a}^{*}{\left[\frac{{\overline{x}}_{st.a}^{*}}{{\overline{x}}_{st.a}^{*}}\right]}^2$  if $\beta =2$

Bias and Mean Square Error of $T_{DG.st.-1.a }^{*}$ is given as;

$Bias\left(T_{DG.st.-1.a}^{*}\right)={\displaystyle \sum }_{h=1}^H{W}_{h.a}{\overline{Y}}_{h.a}\left[\frac{N_{h.a}-n_{h.a}}{N_{h.a}{n}_{h.a}}{C}_{Xh.a}^2+\frac{\left(g_{h.a}-1\right)W_{2h.a}}{n_{h.a}}{C}_{2Yh.a}^2\right]-{\overline{Y}}_a$

$MSE\left(T_{DG.st.-1a}^{*}\right)={\displaystyle \sum }_{h=1}^H{W}_{h.a}^2{\overline{Y}}_{h.a}^2\left[\begin{array}{l}\frac{N_{h.a}-n_{h.a}}{N_{h.a}{n}_{ha}}\left(C_{Yh.a}^2+C_{Xh.a}^2-2C_{YXha}\right)+\\ \frac{\left(g_{h.a}-1\right)W_{2h.a}}{n_{h.a}}\left(C_{2Yh.a}^2+C_{2Xh.a}^2-2C_{2YXha}\right)\end{array}\right]$    (4)

Sampling design in single phase

Let $\pi =\left\{U_1,U_2,\mathrm{...},U_N\right\}$ denote a finite population, the elements of which fall into L known strata with $N_{dh}$  elements the $h^{th}$ stratum, $h=1,2,\mathrm{...},L,{{\displaystyle \sum }}^{\text{}}h{N}_{dh}=N_d$ . It is assumed that π can also be partitioned according to the distribution of variable Z into exhaustive set of D sub-populations or domains of study that is denoted by $\left\{A_d^{*};d=1,2,\mathrm{...},D\right\}$ . Each stratum consist of a substratum of $N_{1dh}$ respondents and a substratum of $N_{2dh}$  non-respondents, $N_{1dh}+N_{2dh}=N_{dh}$ for all h. Let $A_{dh}^{*}$  denote the part of domain $d\left(A_d^{*}\right)$ in stratum h and $N_{dhj}$ the unknown number of elements in $A_{dh}^{*}$ . Let $y_{dhj}$  denote the value of characteristic Y for element i in $A_{dh}^{*}$ .

Proposed estimator

Calibration has been proven to be an estimation technique to smoothen an existing estimator for a better precision and an improved efficiency. For household survey and other economic data that requires knowledge of the supplementary information, a new ratio estimator is suggested to enhance efficiency in domains of study even in the presence of non-response.  Motivated by [5] in an Alternative Ratio Estimator for domain mean, we proposed the following estimator:

$t_{cal}^{*}={\displaystyle \sum }_{h=1}^L{W}_{dh}^{*}\frac{{\overline{y}}_{dh}^{*}}{{\overline{x}}_{dh}}{\overline{X}}_{dh}$    (5)

(5) can be written as

$t_{cal}^{*}={\displaystyle \sum }_{h=1}^L{W}_{dh}^{*}{\overline{y}}_{dhr}$    (6)

where

${\overline{y}}_{dhr}=r_{dh}{\overline{X}}_{dh}$

and

$r_{dh}=\frac{{\overline{y}}_{dh}^{*}}{{\overline{x}}_{dh}},{\overline{X}}_{dh}$  is assume to be known and $W_{dh}^{*}$ is the calibration weight aimed at adjusting the existing weight in [5] estimators using a chi-square distance measure.

$\phi =\frac{{{\displaystyle \sum }}_{h=1}^L{\left(W_{dh}^{*}-W_{dh}\right)}^2}{Q_{dh}{W}_{dh}}$

Subject to the following constraints

$\begin{array}{l}{\displaystyle \sum }_{h=1}^L{W}_{dh}^{*}=1\\ {\displaystyle \sum }_{h=1}^L{W}_{dh}^{*}{\overline{x}}_{dh}={\displaystyle \sum }_{h=1}^L{W}_{dh}{\overline{X}}_{dh}\\ {\displaystyle \sum }_{h=1}^L{W}_{dh}^{*}{s}_{dh}^2={\displaystyle \sum }_{h=}^L{W}_{dh}{S}_{dh}^2\end{array}$

Thus the optimization problem is given by:

$\begin{array}{l}\phi =\frac{{{\displaystyle \sum }}_{h=1}^L{\left(W_{dh}^{*}-W_{d_h}\right)}^2}{Q_{dh}{W}_{dh}}-2{\lambda }_1\left({\displaystyle \sum }_{h=1}^L{W}_{dh}^{*}-1\right)-2{\lambda }_2\left({\displaystyle \sum }_{h=1}^L{W}_{dh}^{*}{\overline{x}}_{dh}-{\displaystyle \sum }_{h=1}^L{W}_{dh}{\overline{X}}_{dh}\right)\\ -2{\lambda }_3\left({\displaystyle \sum }_{h=1}^L{W}_{dh}^{*}{s}_{dh}^2-{\displaystyle \sum }_{h=}^L{W}_{dh}{S}_{dh}^2\right)\end{array}$

where ${\lambda }_1,{\lambda }_2$  and ${\lambda }_3$  are the Lagrange multipliers such that

$\frac{\partial \phi }{\partial W_{dh}^{*}}=\frac{2\left(W_{dh}^{*}-W_{d_h}\right)}{Q_{dh}{W}_{dh}}-2{\lambda }_1-2{\lambda }_2{\overline{x}}_{dh}-2{\lambda }_3{s}_{dh}^2=0$

$\Rightarrow W_{dh}^{*}=W_{dh}+Q_{dh}{W}_{dh}\left({\lambda }_1+{\lambda }_2{\overline{x}}_{dh}+{\lambda }_3{s}_{dh}^2\right)$

Substituting $W_{dh}^{*}$  in Eq. 6 gives

$\begin{array}{l}{\widehat{\overline{t}}}_{cal}={\displaystyle \sum }_{h=1}^L{W}_{dh}{\overline{y}}_{dhr}+{\beta }_{1(dh}\left(1-{\displaystyle \sum }_{h=1}^L{W}_{dh}\right)+{\beta }_{2\left(dh\right)}\left({\displaystyle \sum }_{h=1}^L{W}_{dh}\left({\overline{X}}_{dh}-{\overline{x}}_{dh}\right)\right)\\ +{\beta }_{3\left(dh\right)}\left({\displaystyle \sum }_{h=1}{W}_{dh}\left(S_{dh}^2-s_{dh}^2\right)\right)\end{array}$    (7)

Where

${\beta }_{1\left(d_h\right)}=\left[\begin{array}{l}\left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}{\overline{y}}_{dhr}\right)\left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}{\overline{x}}_{dh}^2\right)\left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}{s}_{dh}^4\right)-\\ \left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}{\overline{y}}_{dhr}\right){\left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}{\overline{x}}_{dh}{s}_{dh}^2\right)}^2-\\ \left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}{\overline{x}}_{dh}\right)\left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}{\overline{x}}_{dh}{\overline{y}}_{dhr}\right)\left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}{s}_{dh}^4\right)+\\ \left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}{\overline{x}}_{dh}\right)\left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}{s}_{dh}^2{\overline{y}}_{dhr}\right)\left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}{\overline{x}}_{dh}{s}_{dh}^2\right)\\ \left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}{s}_{dh}^2\right)\left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}{\overline{x}}_{dh}{\overline{y}}_{dhr}\right)\left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}{\overline{x}}_{dh}{s}_{dh}^2\right)-\\ \left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}{s}_{dh}^2\right)\left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}{s}_{dh}^2{\overline{y}}_{dhr}\right)\left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}{\overline{x}}_{dh}^2\right)\raisebox{1ex}{$$}\!\left/ \!\raisebox{-1ex}{$$}\right.\\ \left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}\right)\left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}{\overline{x}}_{dh}\right)\left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}{\overline{x}}_{dh}{s}_{dh}^2\right)-\\ \left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}\right){\left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}{\overline{x}}_{dh}{s}_{dh}^2\right)}^2-\\ {\left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}{\overline{x}}_{dh}\right)}^2\left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}{s}_{dh}^4\right)+\left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}{\overline{x}}_{dh}\right)\\ \left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}{s}_{dh}^2\right)\left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}{\overline{x}}_{dh}{s}_{dh}^2\right)+\\ \left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}{s}_{dh}^2\right)\left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}{\overline{x}}_{dh}\right)\left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}{\overline{x}}_{dh}{s}_{dh}^2\right)-\\ {\left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}{s}_{dh}^2\right)}^2\left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}{\overline{x}}_{dh}^2\right)\end{array}\right]$

${\beta }_{2\left(d_h\right)}=\left[\begin{array}{l}\left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}\right)\left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}{\overline{x}}_{dh}{\overline{y}}_{dhr}\right)\left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}{s}_{dh}^4\right)-\\ \left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}\right)\left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}{s}_{dh}^2{\overline{y}}_{dhr}\right)\left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}{\overline{x}}_{dh}{s}_{dh}^2\right)-\\ \left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}{\overline{y}}_{dhr}\right)\left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}{\overline{x}}_{dh}^2\right)\left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}{s}_{dh}^4\right)+\\ \left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}{\overline{y}}_{dhr}\right){\left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}{\overline{x}}_{dh}{s}_{dh}^2\right)}^2+\\ \left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}{s}_{dh}^2\right)\left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}{\overline{x}}_{dh}\right)\left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}{s}_{dh}^2{\overline{y}}_{dhr}\right)-\\ \left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}{s}_{dh}^2\right)\left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}{s}_{dh}^2\right)\left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}{s}_{dh}^2{\overline{y}}_{dhr}\right)\raisebox{1ex}{$$}\!\left/ \!\raisebox{-1ex}{$$}\right.\\ \left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}\right)\left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}{\overline{x}}_{dh}\right)\left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}{\overline{x}}_{dh}{s}_{dh}^2\right)-\\ \left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}\right){\left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}{\overline{x}}_{dh}{s}_{dh}^2\right)}^2-\\ {\left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}{\overline{x}}_{dh}\right)}^2\left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}{s}_{dh}^4\right)+\left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}{\overline{x}}_{dh}\right)\\ \left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}{s}_{dh}^2\right)\left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}{\overline{x}}_{dh}{s}_{dh}^2\right)+\\ \left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}{s}_{dh}^2\right)\left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}{\overline{x}}_{dh}\right)\left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}{\overline{x}}_{dh}{s}_{dh}^2\right)-\\ {\left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}{s}_{dh}^2\right)}^2\left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}{\overline{x}}_{dh}^2\right)\end{array}\right]$

${\beta }_{3d_h)}=\left[\begin{array}{l}\left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}\right)\left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}{\overline{x}}_{dh}^2\right)\left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}{s}_{dh}^2{\overline{y}}_{dhr}\right)-\\ \left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}\right)\left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}{\overline{x}}_{dh}{s}_{dh}^2\right)\left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}{\overline{x}}_{dh}{\overline{y}}_{dhr}\right)-\\ \left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}{\overline{x}}_{dh}\right)\left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}{\overline{x}}_{dh}\right)\left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}{s}_{dh}^2{\overline{y}}_{dhr}\right)+\\ \left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}{\overline{x}}_{dh}\right)\left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}{s}_{dh}^2\right)\left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}{\overline{x}}_{dh}{\overline{y}}_{dhr}\right)+\\ \left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}{\overline{y}}_{dhr}\right)\left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}{\overline{x}}_{dh}\right)\left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}{\overline{x}}_{dh}{s}_{dh}^2\right)-\\ \left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}{\overline{y}}_{dhr}\right)\left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}{s}_{dh}^2\right)\left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}{\overline{x}}_{dh}^2\right)\raisebox{1ex}{$$}\!\left/ \!\raisebox{-1ex}{$$}\right.\\ \left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}\right)\left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}{\overline{x}}_{dh}\right)\left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}{\overline{x}}_{dh}{s}_{dh}^2\right)-\\ \left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}\right){\left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}{\overline{x}}_{dh}{s}_{dh}^2\right)}^2-\\ {\left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}{\overline{x}}_{dh}\right)}^2\left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}{s}_{dh}^4\right)+\left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}{\overline{x}}_{dh}\right)\\ \left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}{s}_{dh}^2\right)\left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}{\overline{x}}_{dh}{s}_{dh}^2\right)+\\ \left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}{s}_{dh}^2\right)\left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}{\overline{x}}_{dh}\right)\left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}{\overline{x}}_{dh}{s}_{dh}^2\right)-\\ {\left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}{s}_{dh}^2\right)}^2\left({\displaystyle \sum }_{h=1}^L{Q}_{dh}{W}_{dh}{\overline{x}}_{dh}^2\right)\end{array}\right]$

Bias and variance of the proposed estimator

From the proposed estimator above

Let

$\begin{array}{l}{e}_0=\frac{\left({\overline{y}}_{dh}^{*}-{\overline{Y}}_{dh}\right)}{{\overline{Y}}_{dh}}\\ \Rightarrow W_{dh}^{*}=W_{dh}+Q_{dh}{W}_{dh}\left({\lambda }_1+{\lambda }_2{\overline{x}}_{dh}+{\lambda }_3{s}_{dh}^2\right)\end{array}$ ,

$e_1=\frac{\left({\overline{x}}_{dh}-{\overline{X}}_{dh}\right)}{{\overline{X}}_{dh}}$ ,

$e_2=\frac{\left(s_{xdh}^2-S_{xdh}^2\right)}{S_{xdh}^2}$

Where

${\overline{y}}_{dh}^{*}=\frac{{{\displaystyle \sum }}_{h=1}^L{y}_{dh}^{*}}{n_{dh}}$ , ${\overline{x}}_{dh}=\frac{{{\displaystyle \sum }}_{h=1}^L{x}_{dh}}{n_{dh}}$ , ${\overline{X}}_{dh}=\frac{{{\displaystyle \sum }}_{h=1}^L{X}_{dh}}{N_{dh}}$ , $s_{dh}^2=\frac{{{\displaystyle \sum }}_{h=1}^L{\left({\overline{x}}_{dh}-{\overline{X}}_{dh}\right)}^2}{n_{dh}-1}$  and $S_{dh}^2=\frac{{{\displaystyle \sum }}_{h=1}^L{\left({\overline{X}}_{dh}-\overline{X}\right)}^2}{N_{dh}-1}$

Also,

$\begin{array}{l}{\overline{y}}_{dh}^{*}={\overline{Y}}_{dh}\left(1+e_0\right)\\ {\overline{x}}_{dh}={\overline{X}}_{dh}\left(1+e_1\right)\\ s_{xdh}^2=S_{xdh}^2\left(1+e_2\right)\end{array}$

Let

$\begin{array}{l}E\left[e_0^2\right]=\frac{Var\left({\overline{y}}_{dh}^{*}\right)}{{\overline{Y}}_{dh}^2}=\left(\frac{1}{n_{dh}}-\frac{1}{N_{dh}}\right)C_{y_{dh}}^2+\frac{\left(K_{dh}-1\right)}{n_{dh2}}{W}_{dh2}{C}_{ydh2}^2\\ =\left(\frac{1}{n_{dh}}-\frac{1}{N_{dh}}\right)\frac{S_{ydh}^2}{{\overline{Y}}_{dh}^2}+\frac{\left(K_{dh}-1\right)}{n_{dh}{\overline{Y}}_{dh}^2}{W}_{dh2}{S}_{ydh2}^2\\ E\left[e_1^2\right]=\frac{Var\left({\overline{x}}_{dh}\right)}{{\overline{X}}_{dh}^2}=\left(\frac{1}{n_{dh}}-\frac{1}{N_{dh}}\right)C_{xdh}^2=\left(\frac{1}{n_{dh}}-\frac{1}{N_{dh}}\right)\frac{S_{xdh}^2}{{\overline{X}}_{dh}^2}\\ E\left[e_2^2\right]=\frac{Var\left(s_{xdh}^2\right)}{S_{xd}^2}=\left(\frac{1}{n_{dh}}-\frac{1}{N_{dh}}\right)\frac{S_{xdh}^4}{S_{xdh}^2}=\left(\frac{1}{n_{dh}}-\frac{1}{N_{dh}}\right)S_{xdh}^2\\ E\left[e_0{e}_1\right]=\frac{COV\left({\overline{x}}_{dh,}{\overline{y}}_{dh}^{*}\right)}{{\overline{X}}_{dh}{\overline{Y}}_{dh}}=\frac{1}{{\overline{X}}_{dh}{\overline{Y}}_{dh}}\left[C\left(E\left[{\overline{x}}_{dh}\right],E\left[{\overline{y}}_{dh}^{*}\right]\right)\right]\\ =\left(\frac{1}{n_{dh}}-\frac{1}{N_{dh}}\right){\rho }_{xy}{C}_{ydh}{C}_{xdh}\\ =\frac{1}{{\overline{X}}_{dh}{\overline{Y}}_{dh}}\left(\frac{1}{n_{dh}}-\frac{1}{N_{dh}}\right){\rho }_{xy}{S}_{xdh}{S}_{ydh}\\ \therefore E\left[e_0\right]=E\left[e_1\right]=E\left[e_2\right]=0\\ E\left[e_1{e}_2\right]=\left(\frac{1}{n_{dh}}-\frac{1}{N_{dh}}\right)C_{xdh}{\lambda }_{03}=\left(\frac{1}{n_{dh}}-\frac{1}{N_{dh}}\right)\frac{S_{xdh}}{{\overline{X}}_{dh}}{\lambda }_{03}\end{array}$

where

${\lambda }_{rs}=\frac{{\mu }_{rs}}{{\mu }_{20}^{r/2}{\mu }_{02}^{s/2}}$

And

$\begin{array}{l}{\mu }_{rs}=\frac{1}{N_{dh}-1}{\displaystyle \sum }_{i=1}^N{\left(Y_{dhi}-{\overline{Y}}_{dh}\right)}^r{\left(X_{dhi}-{\overline{X}}_{dh}\right)}^s\\ {\mu }_{20}=S_{ydh}^2\\ {\mu }_{02}=S_{xdh}^2\end{array}$

Hence

${\lambda }_{03}=\frac{{\mu }_{03}}{{\mu }_{20}^{0/2}{\mu }_{02}^{3/2}}$

$E\left[e_0{e}_2\right]=\left(\frac{1}{n_{dh}}-\frac{1}{N_{dh}}\right)C_{ydh}{\lambda }_{12}=\left(\frac{1}{n_{dh}}-\frac{1}{N_{dh}}\right)\frac{S_{ydh}}{{\overline{Y}}_{dh}}{\lambda }_{12}$

Where ${\lambda }_{12}=\frac{{\mu }_{12}}{{\mu }_{20}^{1/2}{\mu }_{02}}$

${\overline{y}}_{dhr}=\frac{{\overline{y}}_{dh}^{*}}{{\overline{x}}_{dh}}{\overline{X}}_{dh}$

$={\overline{Y}}_{dh}\left(1+e_0-e_1+e_1^2-e_0{e}_1\right)$

To obtain the bias

$\begin{array}{l}B\left(t_{cal}^{*}\right)=E\left[t_{cal}^{*}-{\overline{Y}}_d\right]\\ =E\left[\begin{array}{l}{\displaystyle \sum }_{h=1}^L{W}_{dh}\left[{\overline{Y}}_{dh}\left(1+e_0-e_1+e_1^2-e_0{e}_1\right)\right]\\ -{\beta }_{2\left(dh\right)}{\displaystyle \sum }_{h=1}^L{W}_{dh}{\overline{X}}_{dh}{e}_1-{\beta }_{3\left(dh\right)}{\displaystyle \sum }_{h=1}^L{W}_{dh}{S}_{xdh}^2{e}_2-{\overline{Y}}_d\end{array}\right]\\ =E\left[\begin{array}{l}{\displaystyle \sum }_{h=1}^L{W}_{dh}\left[{\overline{Y}}_{dh}\left(e_0-e_1+e_1^2-e_0{e}_1\right)\right]\\ -{\beta }_{2\left(dh\right)}{\displaystyle \sum }_{h=1}^L{W}_{dh}{\overline{X}}_{dh}{e}_1-{\beta }_{3\left(dh\right)}{\displaystyle \sum }_{h=1}^L{W}_{dh}{S}_{xdh}^2{e}_2\end{array}\right]\\ ={\displaystyle \sum }_{h=1}^L{W}_{dh}{\overline{Y}}_{dh}\left[\left(E\left(e_1^2\right)-E\left(e_0{e}_1\right)\right)\right]\\ B\left(t_{cal}^{*}\right)={\displaystyle \sum }_{h=1}^L{W}_{dh}{\overline{Y}}_{dh}\left[\left(\frac{1}{n_{dh}}-\frac{1}{N_{dh}}\right)\frac{S_{xdh}^2}{{\overline{X}}_{dh}^2}\right]-\\ {\displaystyle \sum }_{h=1}^L{W}_{dh}{\overline{Y}}_{dh}\left[\frac{1}{{\overline{X}}_{dh}{\overline{Y}}_{dh}}\left(\frac{1}{n_{dh}}-\frac{1}{N_{dh}}\right){\rho }_{xy}{S}_{xdh}{S}_{ydh}\right]\end{array}$    (8)