Improved generalized class of estimators in estimating the finite population mean using two auxiliary variables under two-stage sampling

1.   Introduction

In survey sampling, it is well known fact that suitable use of the auxiliary information may improves the precision of an estimator for the unknown population parameters. The auxiliary information can be used either at the design stage or at estimation stage to increase the accuracy of the population parameter estimators. Several authors presented modified different type of estimators for estimating the finite population mean including ^{[4,9,21,22,23,24,25,26,27]}.

The problem of estimation of finite population mean or total in two-stage sampling scheme using the auxiliary information has been well established. The two stage sampling scheme is an improvement over the cluster sampling, when it is not possible or easy to calculate all the units from the selected clusters. One of the main characteristic could be the budget, and it becomes too difficult to collect information from all the units within the selected clusters. To overcome this, one way is to select clusters, called first stage unit (fsus) and from the given population of interest, select a subsample from the selected clusters called the second stage units (ssu). This also benefits to increase the size of the first stage samples which consist of clusters, and assume to be heterogeneous groups. If there is no variation within clusters then might not be possible to collect information from all the units within selected clusters. In many situations, it is not possible to obtain the complete list of ultimate sampling units in large scale sample surveys, while a list of primary units of clusters may be available. In such situations, we select a random sample of first stage units or primary units using certain probability sampling schemes i.e simple random sampling (with or without replacement), systematic sampling and probability proportional to size (PPS), and then we can perform sub-sampling in selected clusters (first stage units). This approach is called two-stage sampling scheme.

Two-stage has a great varaity of applications, which go far beyond the immediate scop of sample survey. Whenever any process involves in chemical, physical, or biological tests that can be performed on a small amount of materail, it is likely to be drawn as a subsample from a larger amount that is itself a sample.

In large scale survey sampling, it is usual to adopt multistage sampling to estimate the population mean or total of the study variable $y$. ^[13] proposed a general class of estimators of a finite population mean using multi-auxiliary information under two stage sampling scheme. ^[1] proposed an alternative class of estimators in two stage sampling with two auxiliary variables. ^[10] proposed estimators for finite population mean under two-stage sampling using multivariate auxiliary information. ^[12] suggested a detailed note on ratio estimates in multi-stage sampling. ^[6] given some stratagies in two stage sampling using auxiliary information. ^[3] suggested a class of predictive estimaotrs in two stage sampling using auxiliary information. ^[8] gave a generalized method of estimation for two stage sampling using two auxiliary variables. ^[5] suggested chain ratio estimators in two stage sampling. For certain related work, we refer some latest articles, i.e., ^{[14,15,16,17,18,19,20]}.

In this article, we propose an improved generalized class of estimators using two auxiliary variables under two-stage sampling scheme. The biases and mean sqaure errors of the proposed generalized class of estimators are derived up to first order of approximation. Based on the numerical results, the proposed class of estimators are more efficient than their existing counterparts.

2.   Symbols and notation

Consider a finite population $U$ = $\{{U}_{1}, {U}_{2}, ..., {U}_{N}\}$ is divided into $N$ first-stage units (fsus) clusters in the population. Let N be the total number of first stage unit in population, n be the number of first stage units selected in the sample, ${M}_{i}$ be the number of second stage units (ssus) belongs to the ${i}^{th}$ first stage units (fsus), (i = 1, 2, …, N), and mi be the number of fsus selected from the ${i}^{th}$ fsu in the sample of n fsus, (i = 1, 2, …, n).

Let ${y}_{ij}$, ${x}_{ij}$ and ${z}_{ij}$ be values of the study variable $y$ and the auxiliary variables $\left(x\mathrm{a}\mathrm{n}\mathrm{d}z\right)$ respectively, for the ${j}^{th}$ ssus ${U}_{i} = (j = \mathrm{1, 2}, ..., {M}_{i})$, in the ${i}^{th}$ fsus. The population mean of the study variable $y$ and the auxiliary variables $(x, z)$ are given by:

where

Similarly for sample data:

where

In order to obtain the biases and mean sqaured errors, we consider the following relative error terms:

where,

3.   Existing estimators

In this section, we consider several estimators of the finite population mean under two-stage sampling that are available in the sampling literature, the properties of all estimators considered here are obtained up-to the first order of approximation.

(ⅰ) The usual mean estimator ${\overline {y}}^{\mathrm{*}} = {\overline {y}}_{0}^{\mathrm{*}}$ and its variance under two-stage sampling are given by:

and

(ⅱ) The usual ratio estimator under two-stage sampling, is given by:

where $\overline {X}$ is the known population mean of $x$.

The bias and $\mathrm{M}\mathrm{S}\mathrm{E}$ of ${\overline {y}}_{R}^{\mathrm{*}}$ to first order of approximation, are given by:

and

(ⅲ) ^[2] Exponential ratio type estimator under two-stage sampling, is given by:

The bias and $MSE$ of ${\overline {y}}_{E}^{\mathrm{*}}$ to first order of approximation, are given by:

and

(ⅳ) The traditional difference estimator under two-stage sampling is given by:

where $d$ is the constant.

The minimum variance of ${\overline {y}}_{D}^{\mathrm{*}}$, is given by:

where ${\rho }^{\mathrm{*}} = \frac{{V}_{yx}}{\sqrt{{V}_{y}}\sqrt{{V}_{x}}}$.

The optimum value of $d$ is ${d}_{opt} = \frac{\overline {Y}{V}_{yx}}{\overline {X}{V}_{x}}$.

(ⅴ) ^[7] Difference type estimator under two-stage sampling, is given by:

where ${d}_{0}$ and ${d}_{1}$ are constants.

The bias and minimum $\mathrm{M}\mathrm{S}\mathrm{E}$ of ${\overline {y}}_{Rao}^{\mathrm{*}}$ to first order of approximation, is given by:

and

The optimum values of ${d}_{0}$ and ${d}_{1}$ are:

${d}_{0opt} = \frac{{V}_{x}}{{V}_{x}{V}_{y}-{V}_{yx}^{2}+{V}_{x}}$ and ${d}_{1opt} = \frac{\overline {Y}{V}_{yx}}{\overline {X}({V}_{x}{V}_{y}-{V}_{yx}^{2}+{V}_{x})}$.

(ⅵ) The difference-in-ratio type estimator under two-stage sampling, is given by:

where ${d}_{2}$ and ${d}_{3}$ are constants.

The bias and minimum $\mathrm{M}\mathrm{S}\mathrm{E}$ of ${\overline {y}}_{DR}^{\mathrm{*}}$ to first order of approximation, are given by:

and

The optimum values of ${d}_{2}$ and ${d}_{3}$ are:

(ⅶ) The difference-in-exponential ratio type estimator under two-stage sampling, is given by:

where ${d}_{4}$ and ${d}_{5}$ are constants.

The bias and minimum $\mathrm{M}\mathrm{S}\mathrm{E}$ of ${\overline {y}}_{DE}^{\mathrm{*}}$ to first order of approximation, are given by:

and

The optimum values of ${d}_{4}$ and ${d}_{5}$ are:

(ⅷ) The difference-difference type estimator under two stage sampling, is given by:

where ${d}_{6}$ and ${d}_{7}$ are constants.

The minimum variance or M$SE$ of ${\overline {y}}_{DD}^{\mathrm{*}}$ to first order of approximation, is given by:

The optimum values of ${d}_{6}$ and ${d}_{7}$ are:

(ix) The difference-difference type estimator under two stage sampling, is given by:

where ${d}_{8}$, ${d}_{9}$ and ${d}_{10}$ are constants.

The bias and $\mathrm{M}\mathrm{S}\mathrm{E}$ of ${\overline {y}}_{DD\left(R\right)}^{\mathrm{*}}$ to first order of approximation is given by:

and

The optimum values of ${d}_{8}$, ${d}_{9}$ and ${d}_{10}$ are given by:

4.   Proposed estimator

The principal advantage of our proposed improved generalized class of estimators under two-stage sampling is that it is more flexible, efficient, than the existing estimators. The mean square errors based on two data sets are minimum and percentage relative efficiency is more than hundred as compared to the existing estimators considered here. We identified 11 estimators as members of the proposed class of estimators by substituting the different values of ${w}_{i}(i = \mathrm{1, 2}, 3)$, $\delta$ and $\gamma$. On the lines of ^[2,7], we propose the following generalized improved class of estimators under two stage sampling for estimation of finite population mean using two auxiliary varaible as given by:

where ${w}_{i}(i = \mathrm{1, 2}, 3)$ are constants, whose values are to be determined; $\delta$ and $\gamma$ are constants i.e., ($0\le \delta$, $\gamma \le 1$) and can be used to construct the different estimators.

Using (25), solving ${\overline {y}}_{G}^{\mathrm{*}}$ in terms of errors, we have

where

${\alpha }_{1} = \delta +2\gamma$ and ${\alpha }_{2} = \delta (\delta +2)+4\gamma (\delta +\gamma +1)$.

The bias and $MSE$ of ${\overline {y}}_{G}^{\mathrm{*}}$ are given by:

and

where

Solving (27), the minimum $MSE$ of ${\overline {y}}_{G}^{\mathrm{*}}$ to first order of approximation are given by:

where

and

The optimum values of ${w}_{i}(i = 1, \;2, \;3)$ are given by:

${w}_{1opt} = \frac{{\mathrm{\Omega }}_{3}}{2{\mathrm{\Omega }}_{1}}, {w}_{2opt} = \frac{\overline {Y}{\mathrm{\Omega }}_{4}}{2\overline {X}{\mathrm{\Omega }}_{1}},$ and ${w}_{3opt} = \frac{\overline {Y}{\mathrm{\Omega }}_{5}}{2\overline {Z}{\mathrm{\Omega }}_{1}}$,

where

From (28), we produce the following two estimators called ${\overline {y}}_{G1}^{\mathrm{*}}$ and ${\overline {y}}_{G2}^{\mathrm{*}}$. Put $(\delta = 0, \gamma = 1)$ and $(\delta = 1, \gamma = 0)$ in (25), we get the following two estimators respectively:

where ${w}_{i}(i = 4, \;5, \;6, \;7, \;8, \;9)$ are constants. Solving ${\overline {y}}_{G1}^{\mathrm{*}}$, in terms of errors, we have:

The bias and MSE of ${\overline {y}}_{G1}^{\mathrm{*}}$, to first order of approximation is given by:

By squaring and taking expectation of (29), we get the mean square error:

Differentiate (30) with respect to ${w}_{4}$, ${w}_{5}$ and ${w}_{6}$, we get the optimum values of ${w}_{4}$, ${w}_{5}$ and ${w}_{6}$ i.e.,

Substituting the optimum values of ${w}_{4}$, ${w}_{5}$ and ${w}_{6}$ in (30), we get the minimum mean square error of ${\overline {y}}_{G1}^{\mathrm{*}}$, given by:

Solving ${\overline {y}}_{G2}^{\mathrm{*}}$, in terms of errors, we have

or

The Bias and MSE ${\overline {y}}_{G2}^{\mathrm{*}}$, to first order of approximation, is given by:

Bias(${\overline {y}}_{G2}^{\mathrm{*}}$) = ${w}_{8}\overline {X}{V}_{x}^{2}-{w}_{9}\overline {Z}{V}_{xz}+{w}_{7}\overline {Y}{V}_{x}^{2}-\overline {Y}{V}_{x}^{2}.$

By squaring and taking expectation of (32), we get the mean square error:

Differentiate (33) with respect to ${w}_{7}$, ${w}_{8}$ and ${w}_{9}$, we get the optimum values of ${w}_{7}$, ${w}_{8}$ and ${w}_{9}$ i.e.,

Substituting the optimum values of ${w}_{7}$, ${w}_{8}$ and ${w}_{9}$ in (33), we get the minimum mean square error of ${\overline {y}}_{G2}^{\mathrm{*}}$, given by:

We can generate the considered and many more estimators from (25), by substituting the different values of ${w}_{i}(i = \mathrm{1, 2}, 3),$ $\delta$ and $\gamma$, given in Table 1.

5.   Numerical study

Population 1. [Source: ^[11], Model Assisted Survey Sampling]

There are 124 countries (second stage units) divided into 7 continents (first stage units) according to locations. Continent ${7}^{th}$ consists of only one country therefore, we placed ${7}^{th}$ continent in ${6}^{th}$ continent.

We considered:

$y$ = 1983 import (in millions U.S dollars),

$x$ = 1983 export (in millions U.S dollars),

$z$ = 1982 gross national product (in tens of millions of U.S dollars).

The data are divided into 6 clusters, having $N =$6, and $n =$3. Also ${\sum }_{i = 1}^{N}\;{M}_{i} =$124, $\overline {M} =$20.67. In Table 2, we show cluster sizes, and population means of the study variable $\left(y\right)$ and the auxiliary variables $(x, z)$. Tables 3 and 4 give some results.

Population 2. [Source: ^[11], Model Assisted Survey Sampling]

Similarly we considered the data as mentioned in Population 1,

$y$ = 1983 import (in millions U.S dollars),

$x$ = 1981 military expenditure (in tens of millions U.S dollars),

$z$ = 1980 population (in millions).

The data are divided into 6 clusters having $N = 6$, $n = 3$, ${\sum }_{i = 1}^{N}\;{M}_{i} = 124, \;\overline {M} = 20.67$.

In Table 5, we show cluster sizes, and means of the study variable $\left(y\right)$ and the auxiliary variables $(x, z)$. Tables 6 and 7 give some computation results.

The results based on Tables 2–7 are given in Tables 8 and 9 having biasses, mean square errors, and percentage relative efficiencies of the poposed and exisitng estimators w.r.t ${\overline {y}}_{0}^{\mathrm{*}}.$ Tables 8 and 9 show that the proposed estimators perform well as compared to the existing estimators considered here.

The following expression is used to obtain the Percent Relative Efficiency (PRE), i.e.,

where $i = 0, R, E, D, Rao, DR, DE, DD, DD\left(R\right), {G}_{1}, {G}_{2}.$

6.   Discussion

As mentioned above, we used two real data sets to obtain the biases, MSEs or variances and PREs of all estimators under two-stage sampling scheme when using two auxiliary variables. In Tables 2–4 and Tables 5–7, we present the summary statistic of both population. From Tables 8 and 9, we observed that the proposed class of estimators ${\overline {y}}_{G1}^{\mathrm{*}}$ and ${\overline {y}}_{G2}^{\mathrm{*}}$ are more precise than the existing estimators ${\overline {y}}_{0}^{\mathrm{*}}$, ${\overline {y}}_{R}^{\mathrm{*}}$, ${\overline {y}}_{E}^{\mathrm{*}}$, ${\overline {y}}_{D}^{\mathrm{*}}$, ${\overline {y}}_{Rao}^{\mathrm{*}}$, ${\overline {y}}_{DR}^{\mathrm{*}}$, ${\overline {y}}_{DE}^{\mathrm{*}}$, ${\overline {y}}_{DD}^{\mathrm{*}}$, ${\overline {y}}_{DD\left(R\right)}^{\mathrm{*}}$ in terms of MSEs and PREs. It is clear that the proposed improved generalized class of estimators, i.e., performs better than the estimators. As we increase the sample size the mean square error values decreases, and percentage relative efficiency give best results, which are the expected results.

7.   Conclusions

In this manuscript, we proposed a generalized class of estimators using two auxiliary variables under two-stage sampling for estimating the finite population mean. In addition, some well-known estimators of population mean like traditional unbiased estimator, usual ratio, exponential ratio type, traditional difference type, Rao difference type, difference-in- ratio type, difference-in-exponential ratio type, difference-in-difference, difference-difference ratio type estimator are created to be members of our suggested improved generalized class of estimators. Expression for the biases and mean squared error have been generated up to the first order of approximation. We identified 11 estimators as members of the proposed class of estimators by substituting the different values of ${w}_{i}(i = \mathrm{1, 2}, 3)$, $\delta$ and $\gamma$. Both generalized class of estimators ${\overline {y}}_{G1}^{\mathrm{*}}$ and ${\overline {y}}_{G2}^{\mathrm{*}}$ perform better as compared to all other considered estimators, although ${\overline {y}}_{G2}^{\mathrm{*}}$ is the best. In Population 2, the performance of ratio estimator $\left({\overline {y}}_{R}^{\mathrm{*}}\right)$ is weak. The gain in Population 1 is more as compared to Population 2.

Acknowledgments

The authors are thankful to the Editor-in-Chief and two anonymous referees for their careful reading of the paper and valuable comments which leads to a significant improvement in article.

Conflict of interest

The authors declare no conflict of interest.