A Generalized Class of Exponential-type Estimators for Estimating the Population Mean in Post Stratified Sampling

Abstract

This paper addresses the problem of estimating the population mean using auxiliary information under post stratified sampling. We have suggested a generalized class of estimators for population mean in post stratified sampling. Expressions of bias and mean squared error of suggested class of estimators up to the first order of approximation have been obtained. We have obtained the optimum values of scalars under which the class of estimators attained its minimum mean squared error. Efficiency conditions are obtained under which the suggested class of estimators is better than the usual unbiased estimator, combined ratio estimator and difference estimator. We have also studied the properties of some members of the proposed class of estimators. We have further compared the minimum mean squared error of the suggested class of estimators with that of the members. An empirical study is carried out through a natural population data. Empirical study gives insight on the magnitude of the efficiency of the members of the developed class of estimators.

Keywords

auxiliary variable study variable population mean bias mean squared error post stratification

1. Introduction

It is known fact that in case of heterogeneous population stratified sampling technique is used for selecting a good representative sample of the population. In stratified random sampling the entire heterogeneous population is divided into homogeneous groups, usually termed as strata. Then the units are sampled at random from each stratum. Here the size of the sample is predefined. Use of stratified random sampling requires the size of the strata as well as sampling frame for each stratum. When stratified sampling could not apply for the estimation purpose because of unknown frames of each stratum then the use of post stratification is well recognized. In stratified setup, strata sizes are manageable but list of stratum units are often hard to get. Moreover stratum frames may incomplete or overlapping or several population units may fall under multiple strata while classification. Under these circumstances post stratification is a useful sampling design. The post stratification is as precise as the stratified sampling with proportional allocation subject to the condition of a large sample. Jagers et al. (1985) advocated that the post stratification with respect to relevant criteria may improve estimation strategy subsequently over the sample mean or ratio estimator.

1.1 Notations and Definitions

Let us consider a finite population $U = (U_{1}, U_{2}, \dots U_{N})$ of N units, which can be uniquely partitioned into L strata of size $N_{1}, N_{2}, \dots N_{L}$ such that $\sum_{h = 1}^{L} N_{h} = N$ . Let $(y, x)$ be the (study, auxiliary) variables respectively. The strata weights $W_{h} = N_{h} / N$ , (h = 1, 2,…, L) are assumed known. Let $(y_{h i}, x_{h i}); (i = 1, 2, \dots, N_{h})$ , denote the values of variables $(y, x)$ respectively, for $i^{t h}$ unit in $h^{t h}$ stratum and $({\bar{Y}}_{h}$ , ${\bar{X}}_{h})$ denote strata means. A simple random sample of size n is drawn without replacement from the population which results into the configuration $\underline{n} = (n_{1}, n_{2}, \dots, n_{L}), n_{h}$ denoting the number of units in the sample falling in stratum h, $\sum_{h = 1}^{L} n_{h} = n$ . Assume that n is large enough so that the probability of n_h being zero is small [i.e., $Pr (n_{h} = 0)$ =0]. Now let us define

\bar{Y} = \frac{1}{N} \sum_{h = 1}^{L} \sum_{i = 1}^{N_{h}} y_{h i} = \sum_{h = 1}^{L} W_{h} {\bar{Y}}_{h}, \bar{X} = \frac{1}{N} \sum_{h = 1}^{L} \sum_{i = 1}^{N_{h}} x_{h i} = \sum_{h = 1}^{L} W_{h} {\bar{X}}_{h},

{\bar{Y}}_{h} = \frac{1}{N_{h}} \sum_{i = 1}^{N_{h}} y_{h i}, {\bar{X}}_{h} = \frac{1}{N_{h}} \sum_{i = 1}^{N_{h}} x_{h i}, {\bar{Z}}_{h} = \frac{1}{N_{h}} \sum_{i = 1}^{N_{h}} z_{h i},

{\bar{y}}_{p s} = \sum_{h = 1}^{L} W_{h} {\bar{y}}_{h}, {\bar{x}}_{p s} = \sum_{h = 1}^{L} W_{h} {\bar{x}}_{h}, {\bar{y}}_{h} = \frac{1}{n_{h}} \sum_{i = 1}^{n_{h}} y_{h i}, {\bar{x}}_{h} = \frac{1}{n_{h}} \sum_{i = 1}^{n_{h}} x_{h i} .

f_{h} = n_{h} / N_{h}

C_{y h}^{2} = S_{y h}^{2} / {\bar{Y}}^{2}

C_{x h}^{2} = S_{x h}^{2} / {\bar{X}}^{2}

S_{y x h} = ρ_{y x h} S_{y h} S_{x h}

;

R_{1} = \bar{Y} / \bar{X}

R_{2} = \bar{Y} / \bar{Z}

β = (\sum_{h = 1}^{L} ξ_{h} b_{h} / \sum_{h = 1}^{L} ξ_{h}) = (\sum_{h = 1}^{L} W_{h} S_{x y h} / \sum_{h = 1}^{L} W_{h} S_{x h}^{2})

is a weighted mean of strata regression coefficient of y on x; with

ξ_{h} = W_{h} S_{x h}^{2}

and

b_{h} = S_{x y h} / S_{x h}^{2}

, where

S_{y x h} = \frac{1}{(N_{h} - 1)} \sum_{i = 1}^{N_{h}} (y_{h i} - {\bar{Y}}_{h}) (x_{h i} - {\bar{X}}_{h})

S_{y h}^{2} = \frac{1}{(N_{h} - 1)} \sum_{i = 1}^{N_{h}} {(y_{h i} - {\bar{Y}}_{h})}^{2},

S_{x h}^{2} = \frac{1}{(N_{h} - 1)} \sum_{i = 1}^{N_{h}} {(x_{h i} - {\bar{X}}_{h})}^{2} .

For given configuration of sample

\underline{n} = (n_{1}, n_{2}, \dots, n_{L})

we have

V a r ({\bar{y}}_{p s} | \underline{n}) = E ({({\bar{y}}_{p s} - \bar{Y})}^{2} | \underline{n}) = \sum_{h = 1}^{L} W_{h}^{2} (\frac{1 - f_{h}}{n_{h}}) S_{y h}^{2},

V a r ({\bar{x}}_{p s} | \underline{n}) = E ({({\bar{x}}_{p s} - \bar{X})}^{2} | \underline{n}) = \sum_{h = 1}^{L} W_{h}^{2} (\frac{1 - f_{h}}{n_{h}}) S_{x h}^{2},

C o v ({\bar{y}}_{p s}, {\bar{x}}_{p s}) | \underline{n} = E (({\bar{y}}_{p s} - \bar{Y}) ({\bar{x}}_{p s} - \bar{X})) | \underline{n} = \sum_{h = 1}^{L} W_{h}^{2} (\frac{1 - f_{h}}{n_{h}}) S_{y x h} .

Using the results from Stephan (1945) for $E (n_{h}^{- 1})$ ; we get the following results to the first degree of approximation (fda):

V a r ({\bar{y}}_{p s}) = λ \sum_{h = 1}^{L} W_{h} S_{y h}^{2}, V a r ({\bar{x}}_{p s}) = λ \sum_{h = 1}^{L} W_{h} S_{x h}^{2}, C o v ({\bar{y}}_{p s}, {\bar{x}}_{p s}) = λ \sum_{h = 1}^{L} W_{h} S_{y x h},

where

λ = (1 - f) / n

and

f = n / N

(sampling fraction).

For detailed study on conditional and unconditional post-stratification, the reader is referred to Holt and Smith (1979) and Espejo and Pineda (1997).

2. Review of Some Existing Estimators of Population Mean Using Auxiliary Information in Post Stratified Random Sampling

2.1 Combined Ratio and Product Estimators in Post Stratified Random Sampling

The usual unbiased estimator ${\bar{y}}_{p s}$ can be improved through the knowledge of population mean $\bar{X}$ of the auxiliary variable x. This led Ige and Tripathi (1989) to suggest combined ratio and product estimators based on post stratification and using auxiliary variable x for population mean $\bar{Y}$ of y, respectively, as

{\hat{\bar{Y}}}_{R (p s) C} = {\bar{y}}_{p s} (\bar{X} / {\bar{x}}_{p s}) and {\hat{\bar{Y}}}_{P (p s) C} = {\bar{y}}_{p s} ({\bar{x}}_{p s} / \bar{X}) .

It can be easily shown that ${\hat{\bar{Y}}}_{R (p s) C}$ and ${\hat{\bar{Y}}}_{P (p s) C}$ are more efficient than ${\bar{y}}_{p s}$ if $\frac{β}{R_{1}} > \frac{1}{2}$ and $\frac{β}{R_{1}} < - \frac{1}{2}$ respectively, hold good.

Tuteja et al. (1995) have further extended the study of Ige and Tripathi (1989) based on post stratification and auxiliary information.

2.2 Separate Ratio and Product Estimators in Post Stratified Random Sampling

If ${\bar{X}}_{h} (h = 1, 2, \dots, L)$ , the h^th stratum mean of the auxiliary variable x in the population is known, then the separate ratio and product estimators (based on post stratification and using auxiliary variable x) for $\bar{Y}$ of y, are respectively, defined by

{\hat{\bar{Y}}}_{R (p s) S} = \sum_{h = 1}^{L} W_{h} {\bar{y}}_{h} ({\bar{X}}_{h} / {\bar{x}}_{h}) and {\hat{\bar{Y}}}_{P (p s) S} = \sum_{h = 1}^{L} W_{h} {\bar{y}}_{h} ({\bar{x}}_{h} / {\bar{X}}_{h}) .

It can be easily seen that the separate ratio estimator ${\hat{\bar{Y}}}_{R (p s) S}$ and the separate product estimator ${\hat{\bar{Y}}}_{P (p s) S}$ are more efficient than ${\bar{y}}_{p s}$ if the conditions $\frac{β_{h}}{R_{1 h}} > \frac{1}{2}, \forall h = 1, 2, \dots, L;$ and $\frac{β_{h}}{R_{1 h}} < - \frac{1}{2},$

$\forall h = 1, 2, \dots, L;$ hold good respectively, where $β_{h} = S_{y x h} / S_{x h}^{2}$ and $R_{1 h} = {\bar{Y}}_{h} / {\bar{X}}_{h} .$

2.3 Combined Ratio-type and Product-Type Exponential Estimators in Post Stratified Sampling

Motivated by Bahl & Tuteja (1991), Chouhan (2012) and Tailor et al. (2017) considered the combined ratio-type and product-type exponential estimators for $\bar{Y}$ of y in post stratified sampling respectively as

{\hat{\bar{Y}}}_{R (p s) C e} = {\bar{y}}_{p s} e x p (\bar{X} - {\bar{x}}_{p s}) / \bar{X} + {\bar{x}}_{p s} and {\hat{\bar{Y}}}_{P (p s) C e} = {\bar{y}}_{p s} e x p ({\bar{x}}_{p s} - \bar{X}) / (\bar{X} + {\bar{x}}_{p s}) .

It can be proved that the combined ratio-type exponential estimator ${\hat{\bar{Y}}}_{R (p s) C e}$ and combined product-type exponential estimator ${\hat{\bar{Y}}}_{P (p s) C e}$ are more efficient than ${\bar{y}}_{p s}$ if the conditions $\frac{β}{R_{1}} > \frac{1}{4}$ and $\frac{β}{R_{1}} < - \frac{1}{4}$ are respectively satisfied.

2.4 Separate Ratio-Type and Product-Type Exponential Estimators in Post Stratified Sampling

Chouhan (2012) suggested the following ratio-type and product-type exponential estimators for $\bar{Y}$ of y respectively as

{\hat{\bar{Y}}}_{R (p s) S e} = \sum_{h = 1}^{L} W_{h} {\bar{y}}_{h} e x p ({\bar{X}}_{h} - {\bar{x}}_{h}) / ({\bar{X}}_{h} + {\bar{x}}_{h}) and {\hat{\bar{Y}}}_{P (p s) S e} = \sum_{h = 1}^{L} W_{h} {\bar{y}}_{h} e x p ({\bar{x}}_{h} - {\bar{X}}_{h}) / ({\bar{X}}_{h} + {\bar{x}}_{h}) .

It can be observed that the separate ratio-type exponential ${\hat{\bar{Y}}}_{R (p s) S e}$ and product-type exponential ${\hat{\bar{Y}}}_{P (p s) S e}$ estimators are better than ${\bar{y}}_{p s}$ as long as the conditions $\frac{β_{h}}{R_{1 h}} > \frac{1}{4}, \forall h = 1, 2, \dots L;$ and $\frac{β_{h}}{R_{1 h}} < - \frac{1}{4}, \forall h = 1, 2, \dots L;$ are satisfied, respectively.

2.5 Combined Dual to Ratio and Product Estimators in Post Stratified Sampling

Motivated by Srivenkataramana (1980) and Bandyopadhyaya (1980), one can define the combined dual to ratio and product estimators in post stratified sampling for $\bar{Y}$ _{of y} respectively as

{\hat{\bar{Y}}}_{d R (p s) C} = {\bar{y}}_{p s} \frac{{\bar{x}}_{p s} *}{\bar{X}} and {\hat{\bar{Y}}}_{d P (p s) C} = {\bar{y}}_{p s} \frac{\bar{X}}{{\bar{x}}_{p s} *}, where {\bar{x}}_{p s} * = \frac{N \bar{X} - n {\bar{x}}_{p s}}{(N - n)} = (1 + π) \bar{X} - π {\bar{x}}_{p s} with π = \frac{n}{(N - n)} .

It can be looked upon that the estimators

{\hat{\bar{Y}}}_{d R (p s) C}

and

{\hat{\bar{Y}}}_{d P (p s) C}

are more efficient than

{\bar{y}}_{p s}

if the conditions

\frac{β}{R_{1}} > \frac{π}{2}

and

\frac{β}{R_{1}} < - \frac{π}{2}

are respectively hold true.

Further it is easy to see that ${\hat{\bar{Y}}}_{d R (p s) C}$ and ${\hat{\bar{Y}}}_{d P (p s) C}$ are better than ${\hat{\bar{Y}}}_{P (p s) C}$ and ${\hat{\bar{Y}}}_{P (p s) C}$ respectively if the conditions $\frac{β}{R_{1}} < \frac{1 + π}{2}, n < \frac{N}{2}$ and $\frac{β}{R_{1}} > - \frac{(1 + π)}{2}, n < \frac{N}{2}$ hold good.

Finally, it can be observed that ${\hat{\bar{Y}}}_{d R (p s) C}$ is more efficient than ${\bar{y}}_{p s}$ and ${\hat{\bar{Y}}}_{R (p s) C}$ if $\frac{π}{2} < \frac{β}{R_{1}} < \frac{1 + π}{2}, n < \frac{N}{2} .$

Further it can be seen that ${\hat{\bar{Y}}}_{d P (p s) C}$ is more efficient than the unbiased estimator ${\bar{y}}_{p s}$ and ${\hat{\bar{Y}}}_{P (p s) C}$ if $- (\frac{1 + π}{2}) < \frac{β}{R} < - \frac{π}{2} .$

2.6 Dual to Separate Ratio and Product Estimators in Post Stratified Sampling

Following the same procedure as adopted by Srivenkataramana (1980) and Bandyopadhyaya (1980), one can define the dual to separate ratio and product estimators in post stratified sampling for $\bar{Y}$ of y respectively as

{\hat{\bar{Y}}}_{d R (p s) S} = \sum_{h = 1}^{L} W_{h} {\bar{y}}_{h} \frac{{\bar{x}}_{h} *}{{\bar{X}}_{h}} and {\hat{\bar{Y}}}_{d P (p s) S} = \sum_{h = 1}^{L} W_{h} {\bar{y}}_{h} \frac{{\bar{X}}_{h}}{{\bar{x}}_{h} *},

where

{\bar{x}}_{h} * = \frac{N_{h} {\bar{X}}_{h} - n_{h} {\bar{x}}_{h}}{(N_{h} - n_{h})} = (1 + π_{h}) {\bar{X}}_{h} - π_{h} {\bar{x}}_{h}

with

π_{h} = \frac{n_{h}}{N_{h} - n_{h}} .

It is easy to see that under proportional allocation ${\hat{\bar{Y}}}_{d R (p s) S}$ is more efficient than ${\bar{y}}_{p s}$ and ${\hat{\bar{Y}}}_{R (p s) S}$ if $\frac{π_{h}}{2} < \frac{β_{h}}{R_{1 h}} < \frac{(1 + π_{h})}{2},$ $\forall h = 1, 2, \dots L$ .

It can be further proved that under proportional allocation ${\hat{\bar{Y}}}_{d P (p s) S}$ is more efficient than ${\bar{y}}_{p s}$ and ${\hat{\bar{Y}}}_{P (p s) S}$ if $- \frac{(1 + π_{h})}{2} < \frac{β_{h}}{R_{1 h}} < - \frac{π_{h}}{2}, \forall h = 1, 2, \dots, L .$

2.7 Dual to Combined Ratio-Type and Product-Type Exponential Estimators of the Population Mean $\bar{Y}$ in Post Stratified Sampling

Following the same procedure as adopted by Srivenkataramana (1980) and Bandyopadhyaya (1980), one can define a dual to combined ratio-type exponential and product-type exponential estimators for $\bar{Y}$ of y_, respectively as

{\hat{\bar{Y}}}_{d R (p s) C e} = {\bar{y}}_{p s} e x p {\frac{{\bar{x}}_{p s} * - \bar{X}}{{\bar{x}}_{p s} * + \bar{X}}} and {\hat{\bar{Y}}}_{d P (p s) C e} = {\bar{y}}_{p s} e x p {\frac{\bar{X} - {\bar{x}}_{p s} *}{{\bar{x}}_{p s} * + \bar{X}}},

The estimator ${\hat{\bar{Y}}}_{d R (p s) C e}$ is more efficient than ${\bar{y}}_{p s}$ and ${\hat{\bar{Y}}}_{R (p s) C e}$ if the condition $\frac{π}{4} < \frac{β}{R_{1}} < \frac{1 + π}{4}$ hold true.

Further it can be shown that the ${\hat{\bar{Y}}}_{d P (p s) C e}$ is more efficient than ${\bar{y}}_{p s}$ and ${\hat{\bar{Y}}}_{P (p s) C e}$ if the conditions $- (\frac{1 - π}{4}) < \frac{β}{R_{1}} < - \frac{π}{4}$ hold good.

2.8 Dual to Separate Ratio-Type and Product-Type Exponential Estimators for Population Mean $\bar{Y}$ in Post Stratified Sampling

Motivated by Srivenkataramana (1980) and Bandyopadhyaya (1980), Lone and Tailor (2014) suggested the dual to separate ratio type exponential estimator for population mean $\bar{Y}$ as

{\hat{\bar{Y}}}_{d R (p s) S e} = \sum_{h = 1}^{L} W_{h} {\bar{y}}_{h} e x p (\frac{{\bar{x}}_{h} * - {\bar{X}}_{h}}{{\bar{x}}_{h} * + {\bar{X}}_{h}}) .

In a similar fashion one can also define the dual to separate product-type exponential estimator for $\bar{Y}$ of y as

{\hat{\bar{Y}}}_{d P (p s) S e} = \sum_{h = 1}^{L} W_{h} {\bar{y}}_{h} e x p (\frac{{\bar{X}}_{h} - {\bar{x}}_{h} *}{{\bar{x}}_{h} * + {\bar{X}}_{h}}) .

It can be easily looked upon that ${\hat{\bar{Y}}}_{d R (p s) S e}$ is more efficient than ${\bar{y}}_{p s}$ and ${\hat{\bar{Y}}}_{R (p s) S}$ as long as the condition $\frac{π_{h}}{4} < \frac{β_{h}}{R_{1 h}} < \frac{(1 - \frac{1}{4} π_{h}^{2})}{(2 - π_{h})}, \forall h = 1, 2, \dots, L$ is satisfied.

The estimator ${\hat{\bar{Y}}}_{d R (p s) S e}$ is more efficient than ${\hat{\bar{Y}}}_{R (p s) S e}$ and ${\hat{\bar{Y}}}_{d R (p s) S}$ if the conditions $\frac{β_{h}}{R_{1 h}} < \frac{(1 + π_{h})}{4}, \forall h = 1, 2, \dots, L;$ and $\frac{β_{h}}{R_{1 h}} < \frac{3}{4} π_{h}, \forall h = 1, 2, \dots, L$ are respectively hold good. Thus the condition $\frac{β_{h}}{R_{1 h}} < \frac{3}{4} π_{h}, \forall h = 1, 2, \dots, L,$ is sufficient for the estimator ${\hat{\bar{Y}}}_{d R (p s) S e}$ to be more efficient than the estimators ${\hat{\bar{Y}}}_{R (p s) S e}$ and ${\hat{\bar{Y}}}_{d R (p s) S}$ as $\frac{n_{h}}{N_{h}} < \frac{1}{3}$ is tipically true in sample survey situations.

The estimator ${\hat{\bar{Y}}}_{d P (p s) S e}$ is better than ${\bar{y}}_{p s}$ if $\frac{β_{h}}{R_{1 h}} < - \frac{π_{h}}{4}, \forall h = 1, 2, \dots, L;$ while it is more efficient than ${\hat{\bar{Y}}}_{P (p s) S}$ if $\frac{β_{h}}{R_{1 h}} < - (1 - \frac{1}{4} π_{h}^{2}) / (2 - π_{h}), \forall h = 1, 2, \dots, L .$

Further, the estimator ${\hat{\bar{Y}}}_{d P (p s) S e}$ is more efficient than ${\hat{\bar{Y}}}_{P (p s) S e}$ if $\frac{β_{h}}{R_{1 h}} > - \frac{(1 + π_{h})}{4}, \forall h = 1, 2, \dots, L;$ while it is better than ${\hat{\bar{Y}}}_{d P (p s) S}$ if $\frac{β_{h}}{R_{1 h}} > - \frac{3}{4} π_{h},$ $\forall h = 1, 2, \dots, L$ .

Lone and Tailor (2014) have also obtained the conditions under which ${\hat{\bar{Y}}}_{d P (p s) S e}$ is more efficient than ${\bar{y}}_{p s}$ , ${\hat{\bar{Y}}}_{R (p s) S}$ , ${\hat{\bar{Y}}}_{R (p s) S e}$ and ${\hat{\bar{Y}}}_{d R (p s) S}$ .

2.9 A Family of Combined Estimators in Post Stratified Sampling

Following Khoshnevisan et al. (2007) and Koyuncu and Kadilar (2009), Onyeka (2012) developed a general family of estimators of $\bar{Y}$ of y in post stratified sampling scheme as

{\hat{\bar{Y}}}_{R (p s) C} * = {\bar{y}}_{p s} {(\frac{a \bar{X} + b}{δ (a {\bar{x}}_{p s} + b) + (1 - δ) (a \bar{X} + b)})}^{g},

where

a (\neq 0), b

are either constants or functions of known population parameters of the auxiliary variable x, such as standard deviation

σ_{x}

, coefficient of variation

C_{x}

, skewness

β_{1} (x)

, kurtosis

β_{2} (x)

and correlation coefficient

ρ_{y x}

etc., and

(δ, g)

are constants.

The family of estimators ${\hat{\bar{Y}}}_{R (p s) C} *$ is capable of generating a large number of combined estimators of $\bar{Y}$ of y by making appropriate choices of the values $δ, g, a$ and b.

It is identified due to Onyeka (2012) for g = 0, ${\hat{\bar{Y}}}_{R (p s) C} *$ reduces to the ${\bar{y}}_{p s}$ and for $(δ, g) = (1, 1) and (a, b) = (1, 0), (1, C_{x}), (1, β_{2} (x)), (β_{2} (x), C_{x}),$ $(C_{x}, β_{2} (x)), (1, ρ_{y x})$ , ${\hat{\bar{Y}}}_{R (p s) C} *$ boils down to ${\hat{\bar{Y}}}_{R (p s) C} = {\bar{y}}_{p s} (\bar{X} / {\bar{x}}_{p s}),$ Sisodia and Dwivedi's (1981) ratio-type estimator ${\hat{\bar{Y}}}_{R (p s) S D} = {\bar{y}}_{p s} (\bar{X} + C_{x}) / ({\bar{x}}_{p s} + C_{x})$ , Singh et al.'s (2004) ratio-type estimator ${\hat{\bar{Y}}}_{R (p s) S K 1} = {\bar{y}}_{p s} (\bar{X} + β_{2} (x)) / ({\bar{x}}_{p s} + β_{2} (x))$ , Upadhyaya and Singh's (1999) ratio-type estimators ${\hat{\bar{Y}}}_{R (p s) U S 1} = {\bar{y}}_{p s} (\bar{X} β_{2} (x) + C_{x}) / ({\bar{x}}_{p s} β_{2} (x) + C_{x})$ and ${\hat{\bar{Y}}}_{R (p s) U S 2} = {\bar{y}}_{p s} (\bar{X} C_{x} + β_{2} (x)) / ({\bar{x}}_{p s} C_{x} + β_{2} (x))$ and Singh and Tailor's (2003) ratio-type estimator ${\hat{\bar{Y}}}_{R (p s) S T 1} = {\bar{y}}_{p s} (\bar{X} + ρ_{y x}) / ({\bar{x}}_{p s} + ρ_{y x})$ respectively. Further for $(δ, g) = (1, - 1) and (a, b) = (1, 0), (1, C_{x}),$ $(β_{2} (x), C_{x}), (C_{x}, β_{2} (x)), (1, σ_{x})$ , $(β_{1} (x), σ_{x})$ , $(β_{2} (x), σ_{x}),$ $(1, ρ_{y x}), (1, β_{2} (x))$ , ${\hat{\bar{Y}}}_{R (p s) C} *$ respectively reduces to ${\bar{y}}_{P (p s)} = {\bar{y}}_{p s} ({\bar{x}}_{p s} / \bar{X})$ , Pandey and Dubey's (1988) product-type estimator ${\hat{\bar{Y}}}_{P (p s) P D} = {\bar{y}}_{p s} ({\bar{x}}_{p s} + C_{x}) / (\bar{X} + C_{x})$ , Upadhyaya and Singh's (1999) product-type estimator ${\hat{\bar{Y}}}_{P (p s) U S 3} = {\bar{y}}_{p s} ({\bar{x}}_{p s} β_{2} (x) + C_{x}) / (\bar{X} β_{2} (x) + C_{x})$ , ${\hat{\bar{Y}}}_{P (p s) U S 4} = {\bar{y}}_{p s} ({\bar{x}}_{p s} C_{x} + β_{2} (x)) / (\bar{X} C_{x} + β_{2} (x))$ , Singh (2003) product-type estimator ${\hat{\bar{Y}}}_{P (p s) S 1} = {\bar{y}}_{p s} ({\bar{x}}_{p s} + σ_{x}) / (\bar{X} + σ_{x})$ , ${\hat{\bar{Y}}}_{P (p s) S 2} = {\bar{y}}_{p s} ({\bar{x}}_{p s} β_{1} (x) + σ_{x}) / (\bar{X} β_{1} (x) + σ_{x})$ and ${\hat{\bar{Y}}}_{P (p s) S 3} = {\bar{y}}_{p s} ({\bar{x}}_{p s} β_{2} (x) + σ_{x}) / (\bar{X} β_{2} (x) + σ_{x})$ , Singh and Tailor's (2003) product-type estimator ${\hat{\bar{Y}}}_{P (p s) S T 2} = {\bar{y}}_{p s} ({\bar{x}}_{p s} + ρ_{y x}) / (\bar{X} + ρ_{y x})$ and Singh and Kakran's (1993) product type estimator ${\hat{\bar{Y}}}_{P (p s) S K 2} = {\bar{y}}_{p s} ({\bar{x}}_{p s} + β_{2} (x)) / (\bar{X} + β_{2} (x))$ .

Onyeka (2012) obtained MSE of the family of estimators ${\hat{\bar{Y}}}_{R (p s) C} *$ to the fda as

M S E ({\hat{\bar{Y}}}_{R (p s) C} *) = λ \sum_{h = 1}^{L} W_{h} [S_{y h}^{2} + δ^{2} θ^{2} g^{2} R^{2} S_{x h}^{2} - 2 δ θ g R S_{y x h}],

where

θ = a \bar{X} / (a \bar{X} + b)

The MSEs of the estimators ${\bar{y}}_{p s}$ , ${\hat{\bar{Y}}}_{R (p s) C}$ , ${\hat{\bar{Y}}}_{R (p s) S D}$ , ${\hat{\bar{Y}}}_{R (p s) S K 1}$ , ${\hat{\bar{Y}}}_{R (p s) U S 1}$ , ${\hat{\bar{Y}}}_{R (p s) U S 2}$ , ${\hat{\bar{Y}}}_{R (p s) S T 1}$ , ${\bar{y}}_{P (p s)}$ , ${\hat{\bar{Y}}}_{P (p s) P D}$ , ${\hat{\bar{Y}}}_{P (p s) U S 3}$ , ${\hat{\bar{Y}}}_{P (p s) U S 4}$ , ${\hat{\bar{Y}}}_{P (p s) S 1}$ , ${\hat{\bar{Y}}}_{P (p s) S 2}$ , ${\hat{\bar{Y}}}_{P (p s) S 3}$ , ${\hat{\bar{Y}}}_{P (p s) S T 2}$ , ${\hat{\bar{Y}}}_{P (p s) S K 2}$ can be easily obtained just by putting the suitable values of the scalars $(a, b, δ, g)$ .

Onyeka (2012) obtained the optimum value $δ_{o p t} = β / (θ g R)$ of $δ$ and the resulting minimum MSE of ${\hat{\bar{Y}}}_{R (p s) C} *$ as

M S E_{min} ({\hat{\bar{Y}}}_{R (p s) C} *) = λ (1 - ρ_{0}^{2}) \sum_{h = 1}^{L} W_{h} S_{y h}^{2},

where

β = \sum_{h = 1}^{L} W_{h} S_{y x h} / \sum_{h = 1}^{L} W_{h} S_{x h}^{2}

and

ρ_{0} = \sum_{h = 1}^{L} W_{h} S_{y x h} / \sqrt{(\sum_{h = 1}^{L} W_{h} S_{y h}^{2}) (\sum_{h = 1}^{L} W_{h} S_{x h}^{2})}

Onyeka (2012) has shown that the family of estimators ${\hat{\bar{Y}}}_{R (p s) C} *$ is more efficient than ${\bar{y}}_{p s}$ and ${\hat{\bar{Y}}}_{R (p s) C}$ at the optimum condition. He has also remarked that none of the special cases of the family of estimators ${\hat{\bar{Y}}}_{R (p s) C} *$ is more efficient than the optimum estimator.

2.10 Separate Ratio-Type Estimators in Post Stratified Sampling When the Coefficient of Variation $(C_{x h})$ and Kurtosis $(β_{2 h} (x))$ of the $h^{t h} (h = 1, 2, \dots, L)$ Stratum in the Population are known

Motivated by Sisodia and Dwivedi (1981) and Upadhyaya and Singh (1999), Bacanli & Aksu (2012) suggested the following separate ratio-type estimators in post stratified sampling for the population mean $\bar{Y}$ of the study variable y as

{\hat{\bar{Y}}}_{R (p s) S 1} = \sum_{h = 1}^{L} {\bar{y}}_{h} W_{h} \frac{({\bar{X}}_{h} + C_{x h})}{({\bar{x}}_{h} + C_{x h})}, {\hat{\bar{Y}}}_{R (p s) S 2} = \sum_{h = 1}^{L} {\bar{y}}_{h} W_{h} \frac{({\bar{X}}_{h} + β_{2 h} (x))}{({\bar{x}}_{h} + β_{2 h} (x))},

{\hat{\bar{Y}}}_{R (p s) S 3} = \sum_{h = 1}^{L} {\bar{y}}_{h} W_{h} \frac{({\bar{X}}_{h} β_{2 h} (x) + C_{x h})}{({\bar{x}}_{h} β_{2 h} (x) + C_{x h})}, {\hat{\bar{Y}}}_{R (p s) S 4} = \sum_{h = 1}^{L} {\bar{y}}_{h} W_{h} \frac{({\bar{X}}_{h} C_{x h} + β_{2 h} (x))}{({\bar{x}}_{h} C_{x h} + β_{2 h} (x))} .

Bacanli and Aksu (2012) obtained the MSEs of the above estimators and computed the efficiency of each estimator in post stratification with respect to the corresponding estimator in stratified random sampling using the data from 2000 population and housing census of Turkey which was carried out by the Turkish Statistical Institute. They have shown that the estimators in post stratified sampling yield the highest efficiency i.e., the estimators in post stratified sampling have smaller MSEs than the separate ratio estimator in stratified random sampling.

2.11 Dual to Ratio Estimators of Population Mean in Post Stratified Sampling Using Known Value of Some Population Parameters

Motivated by Srivenkataramana (1980), Bandyopadhyaya (1980) and Onyeka (2012), Onyeka (2013) proposed a class of dual to ratio estimators of $\bar{Y}$ of y, in post stratified sampling using known population parameters of an auxiliary character x, as

{\hat{\bar{Y}}}_{d R (p s) C} * = {\bar{y}}_{p s} {(\frac{δ (a {\bar{x}}_{p s} * + b) + (1 - δ) (a \bar{X} + b)}{a \bar{X} + b})}^{g},

where

{\bar{x}}_{p s} *

is a transformed sample mean of the auxiliary variable x

(i . e . {\bar{x}}_{p s} * = (N \bar{X} - n {\bar{x}}_{p s}) / N - n)

, based on the transformed variable

x_{h i} * = (N \bar{X} - n x_{h i}) / (N - n)

and satisfying the relationship

\bar{X} = f {\bar{x}}_{p s} + (1 - f) {\bar{x}}_{p s} *

and

(a, b, δ, g)

are same as defined earlier for

{\hat{\bar{Y}}}_{R (p s) C}

The MSE of the proposed estimator ${\hat{\bar{Y}}}_{d R (p s) C} *$ to the fda is given by

M S E ({\hat{\bar{Y}}}_{d R (p s) C} *) = λ \sum_{h = 1}^{L} W_{h} [S_{y h}^{2} + π^{2} δ^{2} θ^{2} g^{2} R^{2} S_{x h}^{2} - 2 π δ θ g R S_{y x h}],

where

θ = a \bar{X} / (a \bar{X} + b)

and

π = n / (N - n)

Onyeka (2013) obtained the optimum value $δ_{o p t} = β / (π θ g R)$ of $δ$ and thus the resulting minimum MSE of $({\hat{\bar{Y}}}_{d R (p s) C} *)$ is given by

M S E_{min} ({\hat{\bar{Y}}}_{d R (p s) C} *) = λ (1 - ρ_{0}^{2}) \sum_{h = 1}^{L} W_{h} S_{y h}^{2},

where

β

and

ρ_{0}

are same as defined earlier.

It is observed that the minimum MSE of ${\hat{\bar{Y}}}_{d R (p s) C} *$ , is same as the approximate variance of the usual post stratified regression estimator ${\bar{y}}_{p s R E G} = {\bar{y}}_{p s} - {\hat{β}}_{0} ({\bar{x}}_{p s} - \bar{X})$ , indicating that the efficiency of the proposed class of estimators, ${\hat{\bar{Y}}}_{d R (p s) C} *$ , just like the class of estimators ${\hat{\bar{Y}}}_{R (p s) C} *$ , proposed by Onyeka (2012), may not be improved beyond the efficiency of the customary regression-type estimator ${\bar{y}}_{p s R E G}$ in post stratified sampling.

Onyeka (2013) have also obtained the conditions under which the estimators ${\hat{\bar{Y}}}_{d R (p s) C} *$ dominates the estimators ${\bar{y}}_{p s}$ , ${\hat{\bar{Y}}}_{R (p s) C}$ , ${\hat{\bar{Y}}}_{P (p s) C}$ and ${\hat{\bar{Y}}}_{d R (p s) C}$ .

2.12 Improved Separate Ratio and Product-Type Exponential Estimators in Case of Post Stratification

Motivated by Upadhyaya et al. (2011), Lone and Tailor (2015) proposed a separate ratio-type exponential ${\hat{\bar{Y}}}_{P (p s) S e} *$ and separate product-type exponential ${\hat{\bar{Y}}}_{P (p s) S e} *$ estimators for $\bar{Y}$ of y in the case of post stratification as

{\hat{\bar{Y}}}_{R (p s) S e} * = \sum_{h = 1}^{L} W_{h} {\bar{y}}_{h} e x p (\frac{{\bar{X}}_{h} - {\bar{x}}_{h}}{{\bar{X}}_{h} + (a_{h} - 1) {\bar{x}}_{h}}) and {\hat{\bar{Y}}}_{P (p s) S e} * = \sum_{h = 1}^{L} W_{h} {\bar{y}}_{h} e x p (\frac{{\bar{z}}_{h} - {\bar{Z}}_{h}}{{\bar{Z}}_{h} + (b_{h} - 1) {\bar{z}}_{h}}),

where

(a_{h}, b_{h}) \geq 0

Lone and Tailor (2015) have obtained the minimum MSE of ${\hat{\bar{Y}}}_{R (p s) S e} *$ and ${\hat{\bar{Y}}}_{P (p s) S e} *$ respectively as

M S E_{min} ({\hat{\bar{Y}}}_{R (p s) S e} *) = λ \sum_{h = 1}^{L} W_{h} S_{y h}^{2} (1 - ρ_{h}^{2}),

M S E_{min} ({\hat{\bar{Y}}}_{P (p s) S e} *) = λ \sum_{h = 1}^{L} W_{h} S_{y h}^{2} (1 - ρ_{h}^{* 2}) .

for the optimum values of

a_{h} = \frac{R_{1 h}}{β_{h}} = a_{h 0}, \forall h = 1, 2, \dots, L;

and

b_{h} = - (\frac{R_{2 h}}{β_{h} *}) = b_{h 0}, \forall h = 1, 2, \dots, L;

respectively, where

ρ_{h} = \frac{S_{y x h}}{S_{y h} S_{x h}}

ρ_{h} * = \frac{S_{y z h}}{S_{y h} S_{z h}}

R_{2 h} = \frac{{\bar{Y}}_{h}}{{\bar{Z}}_{h}}

β_{h} * = \frac{S_{y z h}}{S_{z h}^{2}}

and

(R_{1 h}, β_{h})

are same as defined earlier.

Lone and Tailor (2015) have shown that the estimators ${\hat{\bar{Y}}}_{R (p s) S e}^{* (o)}$ and ${\hat{\bar{Y}}}_{P (p s) S e}^{* (o)}$ ${\hat{\bar{Y}}}_{P (p s) S e} *$ , say, based on ‘estimated optimum’ values have the same MSEs to the fda as of the minimum MSEs of the estimators ${\hat{\bar{Y}}}_{R (p s) S e} *$ and ${\hat{\bar{Y}}}_{P (p s) S e} *$ respectively.

They have also obtained the conditions under which the suggested estimators ${\hat{\bar{Y}}}_{R (p s) S e} *$ and ${\hat{\bar{Y}}}_{P (p s) S e} *$ is more efficient than ${\bar{y}}_{p s}$ , ${\hat{\bar{Y}}}_{R (p s) S}$ and ${\hat{\bar{Y}}}_{P (p s) S}$ .

2.13 Classes of Combined and Separate Estimators of Population Mean $\bar{Y}$ in Post Stratified Sampling

On the lines of Srivastava (1971), Singh and Vishwakarma (2010) suggested a class of combined estimators for $\bar{Y}$ of y as

{\hat{\bar{Y}}}_{H C} = H ({\bar{y}}_{p s}, {\bar{x}}_{p s}),

where

H ({\bar{y}}_{p s}, {\bar{x}}_{p s})

is a function of

({\bar{y}}_{p s}, {\bar{x}}_{p s})

such that

H ({\bar{y}}_{p s}, {\bar{x}}_{p s}) = \bar{Y}

and satisfies certain regularity conditions as given in Srivastava (1971).

For the optimum value $H_{2} (\bar{Y}, \bar{X}) = - β$ of $H_{2} (\bar{Y}, \bar{X})$ , the minimum MSE of the class of ${\hat{\bar{Y}}}_{H C}$ is given by

M S E_{min} ({\hat{\bar{Y}}}_{H C}) = λ (1 - ρ_{0}^{2}) \sum_{h = 1}^{L} W_{h} S_{y h}^{2},

where

ρ_{0} = \sum_{h = 1}^{L} W_{h} ρ_{h} S_{y h} S_{x h} / \sqrt{(\sum_{h = 1}^{L} ξ_{h}) (\sum_{h = 1}^{L} W_{h} S_{y h}^{2})}

and

H_{2} (\bar{Y}, \bar{X})

denotes the first order partial derivative of the function

H ({\bar{y}}_{p s}, {\bar{x}}_{p s})

with respect to

{\bar{x}}_{p s}

about the point

(\bar{Y}, \bar{X})

Further on the lines of Srivastava (1971), Singh and Vishwakarma (2010) suggested a class of separate estimators for $\bar{Y}$ of y as

{\hat{\bar{Y}}}_{H S} = \sum_{h = 1}^{L} W_{h} F_{h} ({\bar{y}}_{h}, {\bar{x}}_{h}),

where

F_{h} ({\bar{y}}_{h}, {\bar{x}}_{h})

is a function of

({\bar{y}}_{h}, {\bar{x}}_{h})

such that

F_{h} ({\bar{y}}_{h}, {\bar{x}}_{h}) = {\bar{Y}}_{h}

and satisfies the certain regularity conditions as given in Srivastava (1971).

The minimum MSE of ${\hat{\bar{Y}}}_{H S}$ for the optimum value $F_{h 2} ({\bar{Y}}_{h}, {\bar{X}}_{h}) = - b_{h}$ is given by

M S E_{min} ({\hat{\bar{Y}}}_{H S}) = λ \sum_{h = 1}^{L} W_{h} S_{y h}^{2} (1 - ρ_{h}^{2}) .

where

F_{h 2} ({\bar{Y}}_{h}, {\bar{X}}_{h})

is the first order partial derivative of the function

F_{h} ({\bar{y}}_{h}, {\bar{x}}_{h})

with respect to

{\bar{x}}_{h}

about the point

({\bar{Y}}_{h}, {\bar{X}}_{h})

Singh and Vishwakarma (2010) have shown that the separate class of estimators ${\hat{\bar{Y}}}_{H S}$ is superior to ${\hat{\bar{Y}}}_{H C}$ . They have further demonstrated that unless the regression coefficient is the same from stratum to stratum, ${\hat{\bar{Y}}}_{H S}$ is better than ${\hat{\bar{Y}}}_{H C}$ at optimum condition.

Vishwakarma (2017) suggested an alternative to conventional ratio estimator for $\bar{Y}$ as

t_{G} = {\bar{y}}_{p s} (\frac{A - {\bar{x}}_{p s}}{A - \bar{X}}),

where A being a constant.

The estimator $t_{G}$ is a member of ${\hat{\bar{Y}}}_{H C}$ due to Singh and Vishwakarma (2010). The properties of the estimator $t_{G}$ can be studied easily from ${\hat{\bar{Y}}}_{H C}$ .

2.14 Classes of Separate Estimators of Population Mean $\bar{Y}$ in Post Stratified Sampling

Vishwakarma and Singh (2017) proposed a class of separate estimators for $\bar{Y}$ of y as

{\bar{y}}_{P S} * = \sum_{h = 1}^{L} W_{h α} * {\bar{y}}_{h},

where

W_{h α} * = a [α w_{h} + (1 - α) W_{h}]

w_{h} = \frac{n_{h}}{n}, W_{h} = \frac{N_{h}}{N}

and

(a, α)

being suitable chosen constants.

We note that for $(a, α) = (1, 1), (1, 0)$ and $(1, α)$ the estimator ${\bar{y}}_{P S} *$ reduces to the estimators $\bar{y}$ , ${\bar{y}}_{p s}$ and Agarwal and Panda (1993) estimator $\bar{y}_{p s}^{'}$ respectively.

Vishwakarma and Singh (2017) have obtained the bias and MSE of the class of estimators ${\bar{y}}_{p s} *$ . Assuming ‘a’ is a scalar such that $0 < a \leq 1$ , the minimum MSE of ${\bar{y}}_{p s} *$ is obtained for optimum value of ‘ $α$ ’. They have obtained the conditions under which the estimator ${\bar{y}}_{p s} *$ (at optimum condition) is better than the sample mean $\bar{y} = \sum_{h = 1}^{L} w_{h} {\bar{y}}_{h}$ , ${\bar{y}}_{p s}$ (unbiased estimator in post stratified sampling) and the estimator ${\bar{y}}_{p s}^{'} = \sum_{h = 1}^{L} w_{h α} {\bar{y}}_{h}$ with $w_{h α} = {α \frac{n_{h}}{n} + (1 - α) \frac{N_{h}}{N}} = α w_{h} + (1 - α) W_{h}$ due to Agarwal and Panda (1993).

3. Ratio-Cum-Product-Type Estimators for Population Mean in Post Stratified Sampling

3.1 Ratio-Cum-Product-Type Estimators of Finite Population Mean in Post Stratified Sampling

Consider a finite population U of size N which is divided into L strata of size $N_{1}$ , $N_{2}$ ,…, $N_{L}$ such that $\sum_{h = 1}^{L} N_{h} = N$ . Let (y,x) be the (study, auxiliary) variables respectively correlated positively with y and z be the another auxiliary variable, negatively correlated with y. Let $y_{h i}$ be the observation on i^th unit in h^th stratum for y and $x_{h i}$ and $z_{h i}$ be the observations on i^th unit of the h^th stratum for auxiliary variables x and z respectively, then $({\bar{X}}_{h}, {\bar{Y}}_{h}, {\bar{Z}}_{h})$ and $(\bar{X}, \bar{Y}, \bar{Z})$ are the h^th stratum mean and population mean of the variables (x,y,z) respectively.

Following Singh (1967), Tailor et al. (2016) suggested a ratio-cum-product-type estimator for $\bar{Y}$ of y as

{\hat{\bar{Y}}}_{R P (p s) C} = {\bar{y}}_{p s} (\frac{\bar{X}}{{\bar{x}}_{p s}}) (\frac{{\bar{z}}_{p s}}{\bar{Z}}),

where

{\bar{y}}_{p s}, {\bar{x}}_{p s},

and

{\bar{z}}_{p s}

are the unbiased estimators of population means

\bar{Y}

\bar{X}

and

\bar{Z}

respectively.

Tailor et al. (2016) have obtained the conditions under which the estimator ${\hat{\bar{Y}}}_{R P (p s) C}$ is more efficient than ${\bar{y}}_{p s}$ , ${\hat{\bar{Y}}}_{R (p s) C}$ and ${\hat{\bar{Y}}}_{P (p s) C}$ .

3.2 A Class of Combined Ratio-Cum-Product Type Estimators

Motivated by Singh (1967) and Chouhan (2012), Lone and Tailor (2016) suggested a class of ratio-cum-product-type estimators for $\bar{Y}$ of y as

t = {\bar{y}}_{p s} {\sum_{h = 1}^{L} W_{h} (a_{h} {\bar{X}}_{h} + b_{h}) / \sum_{h = 1}^{L} W_{h} (a_{h} {\bar{x}}_{h} + b_{h})} {\sum_{h = 1}^{L} W_{h} (c_{h} {\bar{z}}_{h} + d_{h}) / \sum_{h = 1}^{L} W_{h} (c_{h} {\bar{Z}}_{h} + d_{h})},

where

(a_{h}, b_{h}, c_{h}

and

d_{h})

are the functions of the population parameters of the auxiliary variables.

Further Lone and Tailor (2016) have suggested the following product-type estimators for $\bar{Y}$ of y as

{\hat{\bar{Y}}}_{P (p s) C_{1}} = {\bar{y}}_{p s} {\sum_{h = 1}^{L} W_{h} ({\bar{z}}_{h} + C_{z h}) / \sum_{h = 1}^{L} W_{h} ({\bar{Z}}_{h} + C_{z h})},

{\hat{\bar{Y}}}_{P (p s) C_{2}} = {\bar{y}}_{p s} {\sum_{h = 1}^{L} W_{h} ({\bar{z}}_{h} + β_{2 h} (z)) / \sum_{h = 1}^{L} W_{h} ({\bar{Z}}_{h} + β_{2 h} (z))},

{\hat{\bar{Y}}}_{P (p s) C_{3}} = {\bar{y}}_{p s} {\sum_{h = 1}^{L} W_{h} ({\bar{z}}_{h} + ρ_{y z h}) / \sum_{h = 1}^{L} W_{h} ({\bar{Z}}_{h} + ρ_{y z h})} .

Lone and Tailor (2016) have obtained the conditions under which the class of combined ratio-cum-product-type estimators t is more efficient than the estimator

{\bar{y}}_{p s}

, Ige and Tripathi (1989) ratio and product estimators, Chouhan (2012) estimators, Tailor et al. (2016) estimator

{\hat{\bar{Y}}}_{R P (p s) C}

and the estimators

{\hat{\bar{Y}}}_{P (p s) C 1}

{\hat{\bar{Y}}}_{P (p s) C 2}

and

{\hat{\bar{Y}}}_{P (p s) C 3}

3.3 Ratio and Ratio-Type Exponential Estimator

Tailor and Mehta (2019) suggested a ratio and ratio-type exponential estimator for $\bar{Y}$ as

{\hat{\bar{Y}}}_{R Re (p s)} = {\bar{y}}_{p s} e x p (\bar{X} - {\bar{x}}_{p s} / \bar{X} + {\bar{x}}_{p s}) e x p (\bar{Z} - {\bar{z}}_{p s} / \bar{Z} - {\bar{z}}_{p s}) .

Tailor and Mehta (2019) obtained the bias and MSE of ${\hat{\bar{Y}}}_{R Re (p s)}$ to the fda and derived the conditions under which the estimator ${\hat{\bar{Y}}}_{R Re (p s)}$ is more efficient than ${\bar{y}}_{p s}$ , ${\hat{\bar{Y}}}_{R (p s) C}$ and ${\hat{\bar{Y}}}_{R (p s) C e}$ due to Tailor et al. (2017).

3.4 A Two Parameter Ratio-Product-Ratio Estimator in Post Stratified Sampling

For estimating $\bar{Y}$ of y in case of post stratification, following the procedure adopted by Chami et al. (2012), Singh and Nigam (2022) proposed a two parameter ratio-product-ratio estimator:

{\bar{y}}_{p s (η, δ)} = {\bar{y}}_{p s} [η {\frac{(1 - δ) {\bar{x}}_{p s} + δ \bar{X}}{δ {\bar{x}}_{p s} + (1 - δ) \bar{X}}} + (1 - η) {\frac{δ {\bar{x}}_{p s} + (1 - δ) \bar{X}}{(1 - δ) {\bar{x}}_{p s} + δ \bar{X}}}],

where

(η, δ)

are real constants. For

(η, δ) = (\frac{1}{2}, \frac{1}{2})

, ((1,0) and (0,1) and (0,0) and (1,1)), the proposed estimator

{\bar{y}}_{p s (η, δ)}

respectively reduces to

{\bar{y}}_{p s}

{\hat{\bar{Y}}}_{P (p s) C} = {\bar{y}}_{p s} {\bar{x}}_{p s} / \bar{X}

and

{\hat{\bar{Y}}}_{R (p s) C} = {\bar{y}}_{p s} \bar{X} / {\bar{x}}_{p s}

Singh and Nigam (2022) obtained the conditions under which the estimator ${\bar{y}}_{p s (η, δ)}$ has smaller MSE than that of ${\bar{y}}_{p s}$ , ${\hat{\bar{Y}}}_{R (p s) C}$ , ${\hat{\bar{Y}}}_{P (p s) C}$ and some other existing estimators.

Singh and Nigam (2022) envisaged the separate ratio-product-ratio estimator $\bar{Y}$ of y as

{\bar{y}}_{t s} = \sum_{h = 1}^{L} W_{h} {\bar{y}}_{h} [η_{h} {\frac{(1 - δ_{h}) {\bar{x}}_{h} + δ_{h} {\bar{X}}_{h}}{δ_{h} {\bar{x}}_{h} + (1 - δ_{h}) {\bar{X}}_{h}}} + (1 - η_{h}) {\frac{δ_{h} {\bar{x}}_{h} + (1 - δ_{h}) {\bar{X}}_{h}}{(1 - δ_{h}) {\bar{x}}_{h} + δ_{h} {\bar{X}}_{h}}}]

where

(η_{h}, δ_{h})

are real constants.

Further they have compared the estimator ${\bar{y}}_{t s}$ with different estimators such as ${\bar{y}}_{p s}$ , ${\hat{\bar{Y}}}_{R (p s) S} = \sum_{h = 1}^{L} W_{h} {\bar{y}}_{h} ({\bar{X}}_{h} / {\bar{x}}_{h})$ , separate product estimator ${\hat{\bar{Y}}}_{P (p s) S} = \sum_{h = 1}^{L} W_{h} {\bar{y}}_{h} ({\bar{x}}_{h} / {\bar{X}}_{h})$ and with other estimators. The conditions are obtained under which ${\bar{y}}_{t s}$ is more efficient than the estimators ${\bar{y}}_{p s}$ , ${\hat{\bar{Y}}}_{R (p s) S}$ , ${\hat{\bar{Y}}}_{P (p s) S}$ and other estimators.

3.5 A Ratio-Type Exponential Estimator for Population Mean Under Post Stratification

Rather Ul Islam et al. (2022) considered a ratio-type exponential estimator for $\bar{Y}$ of y as

{\hat{\bar{Y}}}_{R K} = {\bar{y}}_{p s} [k + (1 - k) e x p (\bar{X} - {\bar{x}}_{p s} / \bar{X} + {\bar{x}}_{p s})],

where k is a suitably chosen constant.

To the first degree of approximation, the correct MSE expression of ${\hat{\bar{Y}}}_{R K}$ is given by

M S E ({\hat{\bar{Y}}}_{R K}) = λ \sum_{h = 1}^{L} W_{h} (S_{y h}^{2} + \frac{1}{4} {(1 - k)}^{2} R_{1}^{2} S_{x h}^{2} - (1 - k) R_{1} S_{y x h})

The optimum value of k that minimizes the MSE of ${\hat{\bar{Y}}}_{R K}$ :

k_{o p t} = = 1 + 2 (β / R_{1}),

where

(β, R_{1})

are same as defined earlier.

Thus the correct minimum MSE of ${\hat{\bar{Y}}}_{R K}$ is given by

M S E_{min} ({\hat{\bar{Y}}}_{R K}) = λ (1 - ρ_{0}^{2}) \sum_{h = 1}^{L} W_{h} S_{y h}^{2},

where

ρ_{0}

is same as defined earlier.

It can be easily shown that the estimator ${\hat{\bar{Y}}}_{R K}$ is better than ${\bar{y}}_{p s}$ , ${\hat{\bar{Y}}}_{P (p s) C}$ and ${\hat{\bar{Y}}}_{R (p s) C}$ .

3.6 Ratio and Ratio Estimators for Population Mean in Demonstration of Post-Stratification

Let (x,z) be the auxiliary variates positively correlated with the study variable y. Assuming that the population means $({\bar{X}}_{h}, {\bar{Z}}_{h})$ of (x,z) of $h^{t h}$ stratum, h = 1,2,…,L; are known; Mehta and Tailor (2024) proposed the ratio and ratio estimator for $\bar{Y}$ of y under post stratified sampling as

{\hat{\bar{Y}}}_{C R P S} = {\bar{y}}_{p s} {\sum_{h = 1}^{L} W_{h} {\bar{X}}_{h} / \sum_{h = 1}^{L} W_{h} {\bar{x}}_{h}} {\sum_{h = 1}^{L} W_{h} {\bar{Z}}_{h} / \sum_{h = 1}^{L} W_{h} {\bar{z}}_{h}}

The properties of the estimator ${\hat{\bar{Y}}}_{C R P S}$ are studied by them up to the fda. Conditions are obtained under which the suggested estimator ${\hat{\bar{Y}}}_{C R P S}$ is better than ${\bar{y}}_{p s}$ and the combined ratio estimator ${\hat{\bar{Y}}}_{R (p s) C}$ . They have shown empirically that the estimator ${\hat{\bar{Y}}}_{C R P S}$ is more efficient than ${\bar{y}}_{p s}$ and ${\hat{\bar{Y}}}_{R (p s) C}$ with considerable gain efficiency.

4. Estimation of Population Ratio Using Auxiliary Information

4.1 Estimation of Population Ratio in Post Stratified Sampling Using variable Transformation

Onyeka et al. (2015a) proposed six combined-type estimators of the population ratio $R_{1} = \bar{Y} / \bar{X}$ in post stratified sampling as

{\hat{R}}_{1 C} = \frac{{\bar{y}}_{p s}}{{\bar{x}}_{p s} - b ({\bar{x}}_{p s} * - \bar{X})}, {\hat{R}}_{2 C} = \frac{{\bar{y}}_{p s}}{(\frac{{\bar{x}}_{p s}}{{\bar{x}}_{p s} *} \bar{X})} = \frac{{\bar{y}}_{p s} {\bar{x}}_{p s} *}{{\bar{x}}_{p s} \bar{X}}, {\hat{R}}_{3 C} = \frac{{\bar{y}}_{p s}}{(\frac{{\bar{x}}_{p s} {\bar{x}}_{p s} *}{\bar{X}})} = \frac{{\bar{y}}_{p s} \bar{X}}{{\bar{x}}_{p s} {\bar{x}}_{p s} *}, {\hat{R}}_{4 C} = \frac{{\bar{y}}_{p s}}{{\bar{x}}_{p s} *},

{\hat{R}}_{5 C} = \frac{{\bar{y}}_{p s}}{{\bar{x}}_{p s} * - b ({\bar{x}}_{p s} - \bar{X})}, {\hat{R}}_{6 C} = \frac{{\bar{y}}_{p s}}{(\frac{{\bar{x}}_{p s} *}{{\bar{x}}_{p s}} \bar{X})} = \frac{{\bar{y}}_{p s} {\bar{x}}_{p s}}{{\bar{x}}_{p s} * \bar{X}},

where

{\bar{x}}_{p s} * = (1 + π) \bar{X} - π {\bar{x}}_{p s}

with

π = \frac{n}{n - n}

and b is a suitable chosen constant.

Onyeka et al. (2015a) have obtained the efficiency conditions under conditional and unconditional arguments.

Onyeka et al. (2015a) has conducted the empirical study to examine the merits of their suggested estimators. He has shown that empirical results confirm the theoretical results in the context of estimators envisaged by him.

4.2 Separate-Type Estimators for Estimating Population Ratio in Post Stratified Sampling Using Variable Transformation

Onyeka et al. (2015b) proposed six separate-type estimators of the population ratio $R_{1} = \bar{Y} / \bar{X}$ in post stratified sampling as

{\hat{R}}_{1 S} = \sum_{h = 1}^{L} W_{h} \frac{{\bar{y}}_{h}}{{\bar{x}}_{h} - b ({\bar{x}}_{h}^{* *} - {\bar{X}}_{h})}, {\hat{R}}_{2 S} = \sum_{h = 1}^{L} W_{h} \frac{{\bar{y}}_{h}}{(\frac{{\bar{x}}_{h}}{{\bar{x}}_{h}^{* *}} {\bar{X}}_{h})} = \sum_{h = 1}^{L} W_{h} \frac{{\bar{y}}_{h} {\bar{x}}_{h}^{* *}}{{\bar{x}}_{h} {\bar{X}}_{h}},

{\hat{R}}_{3 S} = \sum_{h = 1}^{L} W_{h} \frac{{\bar{y}}_{h}}{(\frac{{\bar{x}}_{h} {\bar{x}}_{h}^{* *}}{{\bar{X}}_{h}})} = \sum_{h = 1}^{L} W_{h} \frac{{\bar{y}}_{h} {\bar{X}}_{h}}{{\bar{x}}_{h} {\bar{x}}_{h}^{* *}}, {\hat{R}}_{4 S} = \sum_{h = 1}^{L} W_{h} \frac{{\bar{y}}_{h}}{{\bar{x}}_{h}^{* *}}, {\hat{R}}_{5 S} = \sum_{h = 1}^{L} W_{h} \frac{{\bar{y}}_{h}}{{\bar{x}}_{h}^{* *} - b ({\bar{x}}_{h} - {\bar{X}}_{h})},

{\hat{R}}_{6 S} = \sum_{h = 1}^{L} W_{h} \frac{{\bar{y}}_{h}}{(\frac{{\bar{x}}_{h}^{* *}}{{\bar{x}}_{h}} {\bar{X}}_{h})} = \sum_{h = 1}^{L} W_{h} \frac{{\bar{y}}_{h} {\bar{x}}_{h}}{{\bar{x}}_{h}^{* *} {\bar{X}}_{h}},

where

{\bar{x}}_{h}^{* *} = (1 + π) \bar{X} - π {\bar{x}}_{h}

with

π = n / (N - n)

Onyeka et al. (2015b) have also derived the efficiency conditions under conditional and unconditional arguments.

Onyeka et al. (2015b) has conducted an empirical study to examine the merits of their suggested estimators.

4.3 Estimation of Ratio of Two Population Means Using Auxiliary Information in Case of Post Stratification

Let $(y_{0}, y_{1})$ and $(x_{1}, x_{2})$ be the study and auxiliary variables respectively. Let $({\bar{Y}}_{0}, {\bar{Y}}_{1})$ and $({\bar{X}}_{1}, {\bar{X}}_{2})$ be the population means of the study variables $(y_{0}, y_{1})$ and the auxiliary variables $(x_{1}, x_{2})$ respectively. It is desired to estimate the ratio $(R_{01} = {\bar{Y}}_{0} / {\bar{Y}}_{1})$ of two population means using information on two auxiliary variables $(x_{1}, x_{2})$ . Let $({\bar{y}}_{0 p s}, {\bar{y}}_{1 p s})$ and $({\bar{x}}_{1 p s}, {\bar{x}}_{2 p s})$ be the unbiased estimators of $({\bar{Y}}_{0}, {\bar{Y}}_{1})$ and $({\bar{X}}_{1}, {\bar{X}}_{2})$ respectively based on post stratified sampling. It is assumed that the population means $({\bar{X}}_{1}, {\bar{X}}_{2})$ of the auxiliary variables $(x_{1}, x_{2})$ are known in advance.

The conventional estimator of the population ratio $R_{01}$ in post stratified sampling is given by

{\hat{R}}_{01 (p s)} = \frac{{\bar{y}}_{0 p s}}{{\bar{y}}_{1 p s}} .

Tailor and Mehta (2018) suggested the following estimators for the population ratio $R_{01}$ as

{\hat{R}}_{R p s} = {\hat{R}}_{01 (p s)} ({\bar{X}}_{1} / {\bar{x}}_{1 p s}), {\hat{R}}_{P p s} = {\hat{R}}_{01 (p s)} ({\bar{x}}_{2 p s} / {\bar{X}}_{2}) .

Tailor and Mehta (2018) have derived the conditions under which the estimators ${\hat{R}}_{R p s}$ and ${\hat{R}}_{P p s}$ dominate over the conventional estimator ${\hat{R}}_{01 (p s)}$ .

5. Proposed Generalized Class of Estimators for Population Mean in Post Stratified Sampling

We suggest the following class of estimators

\hat{T} = [k_{1} {\bar{y}}_{p s} + k_{2} (\bar{X} - {\bar{x}}_{p s})] [d {(\frac{ϕ \bar{X} + η}{ϕ {\bar{x}}_{p s} + η})}^{α} + (1 - d) e x p {\frac{δ ϕ (\bar{X} - {\bar{x}}_{p s})}{ϕ (\bar{X} + {\bar{x}}_{p s}) + 2 η}}],

(1)

where

(ϕ, η)

are either real numbers or function of known parameters of the auxiliary variable x such as

C_{x}, β_{1} (x)

(coefficient of skewness),

β_{2} (x)

(coefficient of kurtosis),

S_{x} (or σ_{x})

(standard deviation of x), correlation coefficient

ρ_{y x}

and

Δ = (β_{2} (x) - β_{1} (x) - 1)

. Scalars

(α, δ)

take values (−1, 0, 1), d is suitably chosen scalar and

(k_{1}, k_{2})

are suitably chosen constants such that the sum of the constants

(k_{1}, k_{2})

need not be unity i.e.,

(k_{1} + k_{2} \neq 1)

. A large number of estimators for

\bar{Y}

can be generated for different values of

(d, ϕ, η, α, δ)

In order to derive the bias of T up to fda, we set

e_{0} = \frac{{\bar{y}}_{p s} - \bar{Y}}{\bar{Y}}, e_{1} = \frac{{\bar{x}}_{p s} - \bar{X}}{\bar{X}} .

Thus, we have ${\bar{y}}_{p s} = \bar{Y} (1 + e_{0})$ and $\bar{x} = \bar{X} (1 + e_{1})$ , and the relative estimators are given by

\hat{y} = \frac{\bar{y}}{\bar{Y}} = (1 + e_{0}), \hat{x} = \frac{\bar{x}}{\bar{X}} = (1 + e_{1}) .

Thus, the expectation value of e_i's is

$E (e_{i}) = 0$ for i = 1,2 .and the relative variances are

E (e_{0}^{2}) = λ \sum_{h = 1}^{L} W_{h} C_{y h}^{2}, E (e_{1}^{2}) = λ \sum_{h = 1}^{L} W_{h} C_{x h}^{2},

where

λ = (\frac{1 - f}{n})

and

f = \frac{n}{N}

is the sampling fraction. Also, we have

E (e_{0} e_{1}) = λ \sum_{h = 1}^{L} W_{h} ρ_{y x h} C_{y h} C_{x h} .

Now expressing (1) in terms of e_i's, we have

\hat{T} = [k_{1} \bar{Y} (1 + e_{0}) - k_{2} \bar{X} e_{1}] [d {(1 + υ e_{1})}^{- α} + (1 - d) e x p {- \frac{δ υ}{2} e_{1} {(1 + \frac{υ}{2} e_{1})}^{- 1}}]

(2)

Expanding the RHS of (2), multiplying out, neglecting terms of e's having power greater than two and subtracting $\bar{Y}$ from both sides we have

(\hat{T} - \bar{Y}) = \bar{Y} [k_{1} 1 + e_{0} - υ θ_{0} (e_{1} + e_{0} e_{1}) + θ_{1} υ^{2} e_{1}^{2} + k_{2} (\frac{1}{R}) θ_{0} υ e_{1}^{2} - e_{1} - 1],

(3)

where

υ = \frac{ϕ \bar{X}}{(ϕ \bar{X} + η)}, θ_{0} = {α d + \frac{(1 - d) δ}{2}} and θ_{1} = \frac{1}{2} {d α (α + 1) + \frac{δ (2 + δ) (1 - d)}{4}} .

Bias of the suggested estimator $\hat{T}$ up to fda is

B (\hat{T}) = \bar{Y} [E_{4} k_{1} + E_{5} k_{2} - 1],

(4)

where

E_{4} = [1 + λ υ \sum_{h = 1}^{L} W_{h} C_{x h}^{2} (θ_{1} υ - θ_{0} C_{h})], E_{5} = [\frac{θ_{0} υ λ}{R} \sum_{h = 1}^{L} W_{h} C_{x h}^{2}], C_{h} = ρ_{y x h} \frac{C_{y h}}{C_{x h}} .

Squaring both sides of (3) and neglecting terms of e's having power greater than two, we have

\begin{aligned} (\hat{T} - \bar{Y})^{2} = {\bar{Y}}^{2} [\begin{array}{l} 1 + k_{1}^{2} 1 + 2 e_{0} - 2 υ θ_{0} e_{1} + e_{0}^{2} + υ^{2} (θ_{0}^{2} + 2 θ_{1}) e_{1}^{2} - 4 υ θ_{0} e_{0} e_{1} + k_{2}^{2} (\frac{1}{R^{2}}) e_{1}^{2} \\ + 2 \frac{k_{1} k_{2}}{R} 2 θ_{0} υ e_{1}^{2} - e_{0} e_{1} - e_{1} \\ - 2 k_{1} 1 + e_{0} - υ θ_{0} e_{1} - υ θ_{0} e_{0} e_{1} + θ_{1} υ^{2} e_{1}^{2} - 2 k_{2} \frac{1}{R} (θ_{0} υ e_{1}^{2} - e_{1}) \end{array}], \end{aligned}

(5)

Taking expectation of both sides of (5), we get the MSE of $\hat{T}$ to the fda as

M S E (\hat{T}) = {\bar{Y}}^{2} [1 + E_{1} k_{1}^{2} + E_{2} k_{2}^{2} + 2 E_{3} k_{1} k_{2} - 2 E_{4} k_{1} - 2 E_{5} k_{2}],

(6)

where

\begin{aligned} E_{1} & = [1 + λ \sum_{h = 1}^{L} W_{h} C_{y h}^{2} + υ C_{x h}^{2} (υ (θ_{0}^{2} + 2 θ_{1}) - 4 θ_{0} C_{h})], \end{aligned}

\begin{aligned} E_{2} & = [\frac{λ}{R^{2}} \sum_{h = 1}^{L} W_{h} C_{x h}^{2}], E_{3} = [\frac{λ}{R} \sum_{h = 1}^{L} W_{h} C_{x h}^{2} (2 θ_{0} υ - C_{h})], \end{aligned}

E_{4}

and

E_{5}

are same as defined earlier .

The MSE ( $\hat{T}$ ) at (6) is minimum when

k_{1} = \frac{(E_{2} E_{4} - E_{3} E_{5})}{(E_{1} E_{2} - E_{3}^{2})} = k_{1 (o p t)} say, k_{2} = \frac{(E_{1} E_{5} - E_{3} E_{4})}{(E_{1} E_{2} - E_{3}^{2})} = k_{2 (o p t)} (say) .

Thus the resulting minimum MSE of the proposed estimator $\hat{T}$ is given by

M S E_{min} (\hat{T}) = {\bar{Y}}^{2} [1 - \frac{(E_{2} E_{4}^{2} - 2 E_{3} E_{4} E_{5} + E_{1} E_{5}^{2})}{(E_{1} E_{2} - E_{3}^{2})}] .

(7)

which holds good if

0 < \frac{(E_{2} E_{4}^{2} - 2 E_{3} E_{4} E_{5} + E_{1} E_{5}^{2})}{(E_{1} E_{2} - E_{3}^{2})} < 1 and (E_{1} E_{2} - E_{3}^{2}) > 0.

Now we consider the following cases:

Case I: For $k_{1} = 1$ and $α = δ = 0$ , the proposed estimator $\hat{T}$ boils down to the difference estimator

{\hat{T}}_{d} = [{\bar{y}}_{p s} + k_{2} (\bar{X} - {\bar{x}}_{p s})] .

(8)

The MSE of the estimator ${\hat{T}}_{d}$ to the fda is given as

M S E ({\hat{T}}_{d}) = {\bar{Y}}^{2} [E_{1 (0)} + E_{2 (0)} k_{2}^{2} - 2 E_{3 (0)} k_{2}],

(9)

where

E_{1 (0)} = [λ \sum_{h = 1}^{L} W_{h} C_{y h}^{2}], E_{2 (0)} = [\frac{λ}{R^{2}} \sum_{h = 1}^{L} W_{h} C_{x h}^{2}], E_{3 (0)} = [\frac{λ}{R} \sum_{h = 1}^{L} W_{h} ρ_{y x h} C_{y h} C_{x h}] .

Minimizing the MSE of t₀ with respect to $k_{2}$ yields the optimum value of $k_{2}$ as

k_{2} = \frac{E_{3 (0)}}{E_{2 (0)}} = k_{2 (o p t)} .

(10)

which yields the minimum MSE of

{\hat{T}}_{d}

is given by

M S E_{min} ({\hat{T}}_{d}) = {\bar{Y}}^{2} [E_{1 (0)} - \frac{E_{3 (0)}^{2}}{E_{2 (0)}}] = λ (1 - ρ_{0}^{2}) \sum_{h = 1}^{L} W_{h} S_{y h}^{2},

(11)

where

ρ_{0} = \sum_{h = 1}^{L} W_{h} β_{h} S_{x h}^{2} / \sqrt{(\sum_{h = 1}^{L} W_{h} S_{y h}^{2}) (\sum_{h = 1}^{L} W_{h} S_{x h}^{2})}, β_{h} = \frac{S_{y x h}}{S_{x h}^{2}} = \frac{ρ_{y x h} S_{y h}}{S_{x h}} .

Case II: For $α = δ = 0$ , the envisaged estimator $\hat{T}$ reduces to the estimator

{\hat{T}}_{1} = [k_{1} {\bar{y}}_{p s} + k_{2} (\bar{X} - {\bar{x}}_{p s})] .

(12)

The bias and MSE of the estimator ${\hat{T}}_{1}$ to the fda are respectively given as

B ({\hat{T}}_{1}) = \bar{Y} [k_{1} - 1],

(13)

M S E ({\hat{T}}_{1}) = {\bar{Y}}^{2} [1 + E_{1 (1)} k_{1}^{2} + E_{2 (1)} k_{2}^{2} + 2 E_{3 (1)} k_{1} k_{2} - 2 E_{4 (1)} k_{1}],

(14)

where

E_{1 (1)} = [1 + λ \sum_{h = 1}^{L} W_{h} C_{y h}^{2}], E_{2 (1)} = [\frac{λ}{R^{2}} \sum_{h = 1}^{L} W_{h} C_{x h}^{2}], E_{3 (1)} = [- \frac{λ}{R} \sum_{h = 1}^{L} W_{h} ρ_{y x h} C_{y h} C_{x h}], E_{4 (1)} = 1.

The MSE ( ${\hat{T}}_{1}$ ) is minimum when

\begin{array}{l} k_{1} = \frac{E_{2 (1)}}{(E_{1 (1)} E_{2 (1)} - E_{3 (1)}^{2})} = {[1 + λ (1 - ρ_{0}^{2}) \sum_{h = 1}^{L} W_{h} C_{y h}^{2}]}^{- 1} = k_{1 (1)}^{(o p t)}, \\ k_{2} = \frac{- E_{3 (1)}}{(E_{1 (1)} E_{2 (1)} - E_{3 (1)}^{2})} = [R (\sum_{h = 1}^{L} W_{h} C_{h} C_{x h}^{2}) / \sum_{h = 1}^{L} W_{h} C_{x h}^{2} {1 + λ (1 - ρ_{0}^{2}) \sum_{h = 1}^{L} W_{h} C_{y h}^{2}}] = k_{2 (1)}^{(o p t)} \end{array}},

(15)

Thus the minimum MSE of ${\hat{T}}_{1}$ is given by

M S E_{min} ({\hat{T}}_{1}) = {\bar{Y}}^{2} [1 - \frac{E_{2 (1)}}{(E_{1 (1)} E_{2 (1)} - E_{3 (1)}^{2})}] = \frac{λ (1 - ρ_{0}^{2}) \sum_{h = 1}^{L} W_{h} S_{y h}^{2}}{[1 + λ (1 - ρ_{0}^{2}) \sum_{h = 1}^{L} W_{h} C_{y h}^{2}]} .

(16)

Case III: For d = 1, the proposed estimator $T_{1}$ turns out to the class of estimators for $\bar{Y}$ as

{\hat{T}}_{2} = [k_{1} {\bar{y}}_{p s} + k_{2} (\bar{X} - {\bar{x}}_{p s})] {(\frac{ϕ \bar{X} + η}{ϕ {\bar{x}}_{p s} + η})}^{α}

(17)

The bias and MSE of the estimator ${\hat{T}}_{2}$ to the fda are respectively given as

B ({\hat{T}}_{2}) = \bar{Y} [k_{1} E_{4 (2)} + k_{2} E_{5 (2)} - 1],

(18)

M S E ({\hat{T}}_{2}) = {\bar{Y}}^{2} [1 + E_{1 (2)} k_{1}^{2} + E_{2 (2)} k_{2}^{2} + 2 E_{3 (2)} k_{1} k_{2} - 2 E_{4 (2)} k_{1} - 2 E_{5 (2)} k_{2}] .

(19)

where

E_{1 (2)} = [1 + λ \sum_{h = 1}^{L} W_{h} C_{y h}^{2} + α υ C_{x h}^{2} (υ (2 α + 1) - 4 C_{h})], E_{2 (2)} = [\frac{λ}{R^{2}} \sum_{h = 1}^{L} W_{h} C_{x h}^{2}],

E_{3 (2)} = [\frac{λ}{R} \sum_{h = 1}^{L} W_{h} C_{x h}^{2} (2 α υ - C_{h})], E_{4 (2)} = [1 + λ α υ \sum_{h = 1}^{L} W_{h} C_{x h}^{2} (\frac{(α + 1)}{2} υ - C_{h})],

E_{5 (2)} = [\frac{α υ λ}{R} \sum_{h = 1}^{L} W_{h} C_{x h}^{2}] .

The MSE of ${\hat{T}}_{2}$ at (19) is minimized for

k_{1} = \frac{(E_{2 (2)} E_{4 (2)} - E_{3 (2)} E_{5 (2)})}{(E_{1 (2)} E_{2 (2)} - E_{3 (2)}^{2})} = k_{1 (2)}^{(o p t)}, k_{2} = \frac{(E_{1 (2)} E_{5 (2)} - E_{3 (2)} E_{4 (2)})}{(E_{1 (2)} E_{2 (2)} - E_{3 (2)}^{2})} = k_{2 (2)}^{(o p t)},

(20)

Thus the minimum MSE of ${\hat{T}}_{2}$ is given by

M S E_{min} ({\hat{T}}_{2}) = {\bar{Y}}^{2} [1 - \frac{(E_{2 (2)} E_{4 (2)}^{2} - 2 E_{3 (2)} E_{4 (2)} E_{5 (2)} + E_{1 (2)} E_{5 (2)}^{2})}{(E_{1 (2)} E_{2 (2)} - E_{3 (2)}^{2})}] > 0

(21)

if $0 < \frac{(E_{2 (2)} E_{4 (2)}^{2} - 2 E_{3 (2)} E_{4 (2)} E_{5 (2)} + E_{1 (2)} E_{5 (2)}^{2})}{(E_{1 (2)} E_{2 (2)} - E_{3 (2)}^{2})} < 1 and (E_{1 (2)} E_{2 (2)} - E_{3 (2)}^{2}) > 0.$

Case IV: For d = 0, the suggested estimator $\hat{T}$ turns out to the estimator for $\bar{Y}$ of y as

{\hat{T}}_{3} = [k_{1} {\bar{y}}_{p s} + k_{2} (\bar{X} - {\bar{x}}_{p s})] e x p {\frac{δ ϕ (\bar{X} - {\bar{x}}_{p s})}{ϕ (\bar{X} + {\bar{x}}_{p s}) + 2 η}} .

(22)

The bias and MSE of the estimator ${\hat{T}}_{3}$ to the fda are respectively given as

B ({\hat{T}}_{3}) = \bar{Y} [k_{1} E_{4 (3)} + k_{2} E_{5 (3)} - 1],

(23)

M S E ({\hat{T}}_{3}) = {\bar{Y}}^{2} [1 + E_{1 (3)} k_{1}^{2} + E_{2 (3)} k_{2}^{2} + 2 E_{3 (3)} k_{1} k_{2} - 2 E_{4 (3)} k_{1} - 2 E_{5 (3)} k_{2}],

(24)

where

E_{1 (3)} = [1 + λ \sum_{h = 1}^{L} W_{h} {C_{y h}^{2} + \frac{δ υ}{2} C_{x h}^{2} (υ (δ + 1) - 4 C_{h})}], E_{2 (3)} = [\frac{λ}{R^{2}} \sum_{h = 1}^{L} W_{h} C_{x h}^{2}],

E_{3 (3)} = [\frac{λ}{R} \sum_{h = 1}^{L} W_{h} C_{x h}^{2} (δ υ - C_{h})], E_{4 (3)} = [1 + \frac{δ υ}{2} λ \sum_{h = 1}^{L} W_{h} C_{x h}^{2} (\frac{(2 + δ)}{4} υ - C_{h})],

E_{5 (3)} = [\frac{δ υ}{2 R} λ \sum_{h = 1}^{L} W_{h} C_{x h}^{2}] .

The MSE of ${\hat{T}}_{3}$ is minimum when

k_{1} = \frac{(E_{2 (3)} E_{4 (3)} - E_{3 (3)} E_{5 (3)})}{(E_{1 (3)} E_{2 (3)} - E_{3 (3)}^{2})} = k_{1 (3)}^{(o p t)}, k_{2} = \frac{(E_{1 (3)} E_{5 (3)} - E_{3 (3)} E_{4 (3)})}{(E_{1 (3)} E_{2 (3)} - E_{3 (3)}^{2})} = k_{2 (3)}^{(o p t)},

(25)

Thus the minimum MSE of ${\hat{T}}_{3}$ is given by

M S E_{min} ({\hat{T}}_{3}) = {\bar{Y}}^{2} [1 - \frac{(E_{2 (3)} E_{4 (3)}^{2} - 2 E_{3 (3)} E_{4 (3)} E_{5 (3)} + E_{1 (3)} E_{5 (3)}^{2})}{(E_{1 (3)} E_{2 (3)} - E_{3 (3)}^{2})}]

(26)

which is positive if

0 < \frac{(E_{2 (3)} E_{4 (3)}^{2} - 2 E_{3 (3)} E_{4 (3)} E_{5 (3)} + E_{1 (3)} E_{5 (3)}^{2})}{(E_{1 (3)} E_{2 (3)} - E_{3 (3)}^{2})} < 1

and

(E_{1 (3)} E_{2 (3)} - E_{3 (3)}^{2}) > 0

5.2 Efficiency Comparison

The Var/MSE of unbiased estimator ${\bar{y}}_{p s}$ and combine ratio-type estimator ${\hat{\bar{Y}}}_{R (p s) C}$ are given by

V a r ({\bar{y}}_{p s}) = \sum_{h = 1}^{L} W_{h}^{2} (\frac{1 - f_{h}}{n_{h}}) S_{y h}^{2},

(27)

M S E ({\hat{\bar{Y}}}_{R (p s) C}) = λ \sum_{h = 1}^{L} W_{h} [S_{y h}^{2} + R_{1} (R_{1} - 2 β) S_{x h}^{2}] .

(28)

From (27), (28), (7) and (11) it is observed that (i)

$M S E_{min} (\hat{T}) < M S E ({\bar{y}}_{p s})$ if

(1 - E_{1 (0)}) < E_{(T)} .

(29)

(ii)

$M S E_{min} (\hat{T}) < M S E ({\hat{\bar{Y}}}_{R (p s) C})$ if

(1 - E_{(T)}) < [E_{1 (0)} + λ \sum_{h = 1}^{L} W_{h} C_{x h}^{2} (1 - 2 C_{h})] .

(30)

(iii)

$M S E_{min} (\hat{T}) < M S E_{min} ({\hat{T}}_{d})$ if

(1 - E_{(T)}) < (1 - ρ_{0}^{2}) E_{1 (0)},

(31)

where

E_{1 (0)} = [λ \sum_{h = 1}^{L} W_{h} C_{y h}^{2}] and E_{(T)} = \frac{(E_{2} E_{4}^{2} - 2 E_{3} E_{4} E_{5} + E_{1} E_{5}^{2})}{(E_{1} E_{2} - E_{3}^{2})} .

Thus the proposed class of estimators $\hat{T}$ is better than ${\bar{y}}_{p s}$ , ${\hat{\bar{Y}}}_{R (p s) C}$ and ${\hat{T}}_{d}$ as long as the conditions given in (29), (30) and (31) respectively hold good.

Further from (7), (16), (21) and (26) we have that

$M S E_{min} (\hat{T}) < M S E_{min} ({\hat{T}}_{1})$ if

E_{(T)} > E_{(T_{1})},

(32)

where

E_{(T_{1})} = \frac{E_{2 (1)}}{(E_{1 (1)} E_{2 (1)} - E_{3 (1)}^{2})} .

$M S E_{min} (\hat{T}) < M S E_{min} ({\hat{T}}_{2})$ if

E_{(T)} > E_{(T_{2})},

(33)

where

E_{(T_{2})} = \frac{(E_{2 (2)} E_{4 (2)}^{2} - 2 E_{3 (2)} E_{4 (2)} E_{5 (2)} + E_{1 (2)} E_{5 (2)}^{2})}{(E_{1 (2)} E_{2 (2)} - E_{3 (2)}^{2})} .

(i)

$M S E_{min} (\hat{T}) < M S E_{min} ({\hat{T}}_{3})$ if

E_{(T)} > E_{(T_{3})},

(34)

where

E_{(T_{3})} = \frac{(E_{2 (3)} E_{4 (3)}^{2} - 2 E_{3 (3)} E_{4 (3)} E_{5 (3)} + E_{1 (3)} E_{5 (3)}^{2})}{(E_{1 (3)} E_{2 (3)} - E_{3 (3)}^{2})} .

Thus the suggested class of estimators $\hat{T}$ is superior to the estimators ${\hat{T}}_{1}$ , ${\hat{T}}_{2}$ and ${\hat{T}}_{3}$ as long as the conditions (32), (33) and (34) are satisfied respectively. Remark 5.1

One may also define the following separate class of exponential-type estimators for population mean $\bar{Y}$ based on post stratified sampling as

{\hat{T}}_{s} = \sum_{h = 1}^{L} W_{h} [k_{1 h} {\bar{y}}_{h} + k_{2 h} ({\bar{X}}_{h} - {\bar{x}}_{h})] [d_{h} {(\frac{ϕ_{h} {\bar{X}}_{h} + η_{h}}{ϕ_{h} {\bar{x}}_{h} + η_{h}})}^{α_{h}} + (1 - d_{h}) e x p {\frac{δ_{h} ϕ_{h} ({\bar{X}}_{h} - {\bar{x}}_{h})}{ϕ_{h} ({\bar{X}}_{h} + {\bar{x}}_{h}) + 2 η_{h}}}]

\ref{disp-formula101-15741699251336038}

where

(ϕ_{h}, η_{h})

are either real numbers or function of known parameters of the auxiliary variable x of the

h^{t h}

stratum such as

C_{x h}, β_{1 h} (x),

β_{2 h} (x)

S_{x h} (or σ_{x h})

ρ_{y x h}

and

Δ_{h} = (β_{2 h} (x) - β_{1 h} (x) - 1)

. Scalars

(α_{h}, δ_{h})

take values (−1, 0, 1),

d_{h}

being suitably chosen scalar and

(k_{1 h}, k_{2 h})

are suitably chosen constants such that the sum of

(k_{1 h}, k_{2 h})

need not be unity i.e.,

(k_{1 h} + k_{2 h} \neq 1)

. A large number of estimators for

\bar{Y}

can be generated for different values from the suggested class of estimators

{\hat{T}}_{s}

for various suitable values of

(d_{h}, ϕ_{h}, η_{h}, α_{h}, δ_{h})

5.3 Numerical Illustration

From the suggested class of estimators $\hat{T}$ we can generate several numbers of estimators for different combinations of constants $(α, δ, ϕ, η)$ . Some members of the suggested class of estimators are given in Table 1 below.

Table 1.
Some Members of the Proposed Class of Estimators $\hat{T}$ .

Estimators $(α, δ, ϕ, η)$

${\hat{T}}^{(1)} = [k_{1} {\bar{y}}_{p s} + k_{2} (\bar{X} - {\bar{x}}_{p s})] [d (\frac{\bar{X}}{{\bar{x}}_{p s}}) + (1 - d) e x p {\frac{(\bar{X} - {\bar{x}}_{p s})}{(\bar{X} + {\bar{x}}_{p s})}}]$ (1,1,1,0)

${\hat{T}}^{(2)} = [k_{1} {\bar{y}}_{p s} + k_{2} (\bar{X} - {\bar{x}}_{p s})] [d (\frac{{\bar{x}}_{p s}}{\bar{X}}) + (1 - d) e x p {\frac{({\bar{x}}_{p s} - \bar{X})}{(\bar{X} + {\bar{x}}_{p s})}}]$ (−1,-1,1,0)

${\hat{T}}^{(3)} = [k_{1} {\bar{y}}_{p s} + k_{2} (\bar{X} - {\bar{x}}_{p s})] [d (\frac{{\bar{x}}_{p s}}{\bar{X}}) + (1 - d) e x p {\frac{(\bar{X} - {\bar{x}}_{p s})}{(\bar{X} + {\bar{x}}_{p s})}}]$ (-1,1,1,0)

${\hat{T}}^{(4)} = [k_{1} {\bar{y}}_{p s} + k_{2} (\bar{X} - {\bar{x}}_{p s})] [d (\frac{\bar{X}}{{\bar{x}}_{p s}}) + (1 - d) e x p {\frac{({\bar{x}}_{p s} - \bar{X})}{(\bar{X} + {\bar{x}}_{p s})}}]$ (1,-1,1,0)

${\hat{T}}^{(5)} = [k_{1} {\bar{y}}_{p s} + k_{2} (\bar{X} - {\bar{x}}_{p s})] [d + (1 - d) e x p {\frac{(\bar{X} - {\bar{x}}_{p s})}{(\bar{X} + {\bar{x}}_{p s})}}]$ (0,1,1,0)

${\hat{T}}^{(6)} = [k_{1} {\bar{y}}_{p s} + k_{2} (\bar{X} - {\bar{x}}_{p s})] [d + (1 - d) e x p {\frac{({\bar{x}}_{p s} - \bar{X})}{(\bar{X} + {\bar{x}}_{p s})}}]$ (0,−1,1,0)

Estimators	$(α, δ, ϕ, η)$
${\hat{T}}^{(1)} = [k_{1} {\bar{y}}_{p s} + k_{2} (\bar{X} - {\bar{x}}_{p s})] [d (\frac{\bar{X}}{{\bar{x}}_{p s}}) + (1 - d) e x p {\frac{(\bar{X} - {\bar{x}}_{p s})}{(\bar{X} + {\bar{x}}_{p s})}}]$	(1,1,1,0)
${\hat{T}}^{(2)} = [k_{1} {\bar{y}}_{p s} + k_{2} (\bar{X} - {\bar{x}}_{p s})] [d (\frac{{\bar{x}}_{p s}}{\bar{X}}) + (1 - d) e x p {\frac{({\bar{x}}_{p s} - \bar{X})}{(\bar{X} + {\bar{x}}_{p s})}}]$	(−1,-1,1,0)
${\hat{T}}^{(3)} = [k_{1} {\bar{y}}_{p s} + k_{2} (\bar{X} - {\bar{x}}_{p s})] [d (\frac{{\bar{x}}_{p s}}{\bar{X}}) + (1 - d) e x p {\frac{(\bar{X} - {\bar{x}}_{p s})}{(\bar{X} + {\bar{x}}_{p s})}}]$	(-1,1,1,0)
${\hat{T}}^{(4)} = [k_{1} {\bar{y}}_{p s} + k_{2} (\bar{X} - {\bar{x}}_{p s})] [d (\frac{\bar{X}}{{\bar{x}}_{p s}}) + (1 - d) e x p {\frac{({\bar{x}}_{p s} - \bar{X})}{(\bar{X} + {\bar{x}}_{p s})}}]$	(1,-1,1,0)
${\hat{T}}^{(5)} = [k_{1} {\bar{y}}_{p s} + k_{2} (\bar{X} - {\bar{x}}_{p s})] [d + (1 - d) e x p {\frac{(\bar{X} - {\bar{x}}_{p s})}{(\bar{X} + {\bar{x}}_{p s})}}]$	(0,1,1,0)
${\hat{T}}^{(6)} = [k_{1} {\bar{y}}_{p s} + k_{2} (\bar{X} - {\bar{x}}_{p s})] [d + (1 - d) e x p {\frac{({\bar{x}}_{p s} - \bar{X})}{(\bar{X} + {\bar{x}}_{p s})}}]$	(0,−1,1,0)

For numerical illustration we consider a data set previously discussed by Onyeka (2012) such as

\begin{matrix} N = 96, N_{1} = 72, N_{2} = 24, n = 20, n_{1} = 8, n_{2} = 12, \bar{Y} = 2.44, {\bar{Y}}_{1} = 2.44, \end{matrix}

\begin{matrix} {\bar{Y}}_{2} = 2.46, \bar{X} = 68.13, {\bar{X}}_{1} = 68.11, {\bar{X}}_{2} = 68.17, \end{matrix}

\begin{matrix} C_{y} = 0.23, C_{y 1} = 0.24, C_{y 2} = 0.20, C_{x} = 0.10, C_{x 1} = 0.11, \end{matrix}

\begin{matrix} C_{x 2} = 0.20, ρ_{y x} = 0.82, ρ_{y x 1} = 0.80, ρ_{y x 2} = 0.90. \end{matrix}

Table 2 shows the percent relative efficiencies (PREs) of the estimators

{\bar{y}}_{p s}

{\hat{\bar{Y}}}_{R (p s) C}

{\hat{T}}_{d}

{\hat{T}}_{1}

{\hat{T}}_{2} (for α = - 1, 1)

and

{\hat{T}}_{3} (for δ = - 1, 1)

relative to

{\bar{y}}_{p s}

. It is observed from Table 2 that the estimator

{\hat{T}}_{3} (for δ = 1)

is better than

{\bar{y}}_{p s}

and

{\hat{\bar{Y}}}_{R (p s) C}

with considerable gain in efficiency, while it is better than

{\hat{T}}_{d}

{\hat{T}}_{1}

{\hat{T}}_{2} (for α = - 1, 1)

and

{\hat{T}}_{3} (for δ = - 1, 1)

with very marginal gain in efficiency.

Table 2.

PREs of Different Estimators ${\bar{y}}_{p s}$ , ${\hat{\bar{Y}}}_{R (p s) C}$ , ${\hat{T}}_{d}$ , ${\hat{T}}_{1}$ , ${\hat{T}}_{2} for (α = - 1, 1)$ , ${\hat{T}}_{3} for (δ = - 1, 1)$ with Respect to ${\bar{y}}_{p s}$ .

Estimators	${\bar{y}}_{p s}$	${\hat{\bar{Y}}}_{R (p s) C}$	${\hat{T}}_{d}$	${\hat{T}}_{1}$	${\hat{T}}_{2 (α = - 1)}$	${\hat{T}}_{2 (α = 1)}$	${\hat{T}}_{3 (δ = - 1)}$	${\hat{T}}_{3 (δ = 1)}$
$P R E (., {\bar{y}}_{p s})$	100.0	216.9	302.9	303.1	303.0	303.1	303.1	303.2

Table 3 presents the PREs of different members of the suggested class of estimators $\hat{T}$ with respect to ${\bar{y}}_{p s}$ as given in Table 1. It is observed from Table 3 that

The estimator ${\hat{T}}^{(1)}$ has higher PRE than ${\bar{y}}_{p s}$ , ${\hat{\bar{Y}}}_{R (p s) C}$ , ${\hat{T}}_{d}$ , ${\hat{T}}_{1}$ , ${\hat{T}}_{2} (for α = - 1, 1)$ and ${\hat{T}}_{3} (for δ = - 1, 1)$ for some selected values of d in the range [−13,13].

Table 3.

PRE of Different Members of the Proposed Class of Estimators $\hat{T}$ with Respect to ${\bar{y}}_{p s}$ .

* $P R E (T^{(1)}, {\bar{y}}_{p s})$
$d$	−13	−11	−9	−7	−5	5	7	9	11	13
PRE	633.6	408.5	340.5	314.5	305.2	304.28	311.1	331.7	358.2	546.7

* $P R E (T^{(2)}, {\bar{y}}_{p s})$
$d$	−14	−12	−10	−8	−6	4	5	8	10	12
PRE	648.6	412.5	342.2	315.3	305.6	304.5	307.1	333.0	387.9	554.0

* $P R E (T^{(3)}, {\bar{y}}_{p s})$
$d$	−4	−3	−2	2	3
PRE	407.6	324.1	305.1	305.1	324.1

* $P R E (T^{(4)}, {\bar{y}}_{p s})$
$d$	−4	−3	−2	−1	3	4	5
PRE	768.1	382.8	318.2	304.1	314.0	364.9	629.4

* $P R E (T^{(5)}, {\bar{y}}_{p s})$
$d$	−13	−11	−9	−7	−5	5	7	9	11	12	13
PRE	637.8	409.3	340.7	314.6	305.3	304.9	311.2	331.6	384.8	440.4	544.1

* $P R E (T^{(6)}, {\bar{y}}_{p s})$
$d$	−12	−10	−8	−6	−4	6	8	10	12	14
PRE	639.9	410.1	341.1	314.8	305.4	304.4	311.4	332.1	385.1	548.2

*Stands for the proposed estimators ${\hat{T}}^{(j)}; j = 1 to 6$ are better than the estimators ${\bar{y}}_{p s}$ , ${\hat{\bar{Y}}}_{R (p s) C}$ , ${\hat{T}}_{d}$ , ${\hat{T}}_{1}$ , ${\hat{T}}_{2} (for α = - 1, 1)$ and ${\hat{T}}_{3} (for δ = - 1, 1) .$

The highest PRE (633.6%) is observed at d = −13 followed by PRE (546.7%) at d = 13.

The estimator ${\hat{T}}^{(2)}$ is more efficient than ${\bar{y}}_{p s}$ , ${\hat{\bar{Y}}}_{R (p s) C}$ , ${\hat{T}}_{d}$ , ${\hat{T}}_{1}$ , ${\hat{T}}_{2} (for α = - 1, 1)$ and ${\hat{T}}_{3} (for δ = - 1, 1)$ for some selected values of d in the range [−14,12].

The largest PRE (648.6%) is observed at d = −14 followed by PRE (554.0%) at d = 12.

The estimator ${\hat{T}}^{(3)}$ is more adequate than ${\bar{y}}_{p s}$ , ${\hat{\bar{Y}}}_{R (p s) C}$ , ${\hat{T}}_{d}$ , ${\hat{T}}_{1}$ , ${\hat{T}}_{2} (for α = - 1, 1)$ and ${\hat{T}}_{3} (for δ = - 1, 1)$ for some appropriately chosen values of d in the range [−4,4].

The largest PRE (407.6%) is observed at $| d | = 4$ followed by PRE (324.1%) at $| d | = 3$ .

The estimator ${\hat{T}}^{(4)}$ is better than ${\bar{y}}_{p s}$ , ${\hat{\bar{Y}}}_{R (p s) C}$ , ${\hat{T}}_{d}$ , ${\hat{T}}_{1}$ , ${\hat{T}}_{2} (for α = - 1, 1)$ and ${\hat{T}}_{3} (for δ = - 1, 1)$ in the range [−4,5] for selected values of d.

The maximum PRE (768.1%) is observed at d = −4 followed by PRE (629.4%) at d = 5.

The estimator ${\hat{T}}^{(5)}$ is better than ${\bar{y}}_{p s}$ , ${\hat{\bar{Y}}}_{R (p s) C}$ , ${\hat{T}}_{d}$ , ${\hat{T}}_{1}$ , ${\hat{T}}_{2} (for α = - 1, 1)$ and ${\hat{T}}_{3} (for δ = - 1, 1)$ for selected values of d in the range [−13,13].

The largest PRE (637.8%) is looked upon at d = −13 followed by PRE (544.1%) at d = 13.

The estimator ${\hat{T}}^{(6)}$ is better than the estimator ${\bar{y}}_{p s}$ , ${\hat{\bar{Y}}}_{R (p s) C}$ , ${\hat{T}}_{d}$ , ${\hat{T}}_{1}$ , ${\hat{T}}_{2} (for α = - 1, 1)$ and ${\hat{T}}_{3} (for δ = - 1, 1)$ for suitable values of d lies in the range of [−12,14].

The highest PRE (639.9%) is seen at d = −12 followed by PRE (548.2%) at d = 14.

It is further observed that the estimator ${\hat{T}}^{(4)}$ has largest PRE (768.1%) at d = −4 among the estimators ${\bar{y}}_{p s}$ , ${\hat{\bar{Y}}}_{R (p s) C}$ , ${\hat{T}}_{d}$ , ${\hat{T}}^{(1)}$ , ${\hat{T}}^{(2)}$ , ${\hat{T}}^{(3)}$ , ${\hat{T}}^{(4)}$ , ${\hat{T}}^{(5)}$ and ${\hat{T}}^{(6)}$ .

From the above discussion finally we conclude that there is enough scope of selecting the constant

d in each member of the proposed class of estimators ${\hat{T}}^{(j)}; j = 1 to 6$ for obtaining estimators better

than the estimators ${\bar{y}}_{p s}$ , ${\hat{\bar{Y}}}_{R (p s) C}$ , ${\hat{T}}_{d}$ , ${\hat{T}}_{1}$ , ${\hat{T}}_{2} (for α = - 1, 1)$ and ${\hat{T}}_{3} (for δ = - 1, 1)$ .

Thus we recommend the studies included in this paper for theoretician as well practitioners for their use in the context of upliftment of the literature and practice.

6. Conclusion

In this article we have presented review of some existing estimators related to the estimation of population mean/ratio of two means using auxiliary information under post stratified sampling. The review presented in the paper is very useful to the researchers engaged in this area. We have further proposed a class of combined exponential-type estimators for estimating the population mean $\bar{Y}$ of the study variable y using auxiliary information under post stratified sampling. In addition to ratio, product and difference-type estimators it includes a large number of estimators of $\bar{Y}$ of y. Expressions for bias and MSE of the suggested class of estimators have been derived up to first order of approximation. Asymptotic optimum estimator (AOE) is investigated along with its MSE formula. Theoretical comparison of the suggested class of estimators is also presented with other well-known estimators. Results of the proposed class of estimators have substantial gain in efficiency over usual unbiased estimator ${\bar{y}}_{p s}$ , combined ratio estimator ${\hat{\bar{Y}}}_{R (p s) C}$ , difference estimator ${\hat{T}}_{d}$ and the estimators ${\hat{T}}_{1}$ , ${\hat{T}}_{2} (for α = - 1, 1)$ and ${\hat{T}}_{3} (for δ = - 1, 1)$ . The niceness of the suggested class of estimators is that it unifies results of several estimators at one place. This will help researchers to get results (i.e., bias and MSE) of an estimator belong to the proposed class of estimators easily. An empirical study is also carried out in the favor of proposed study. Thus we recommend the use of proposed class of estimators in practice.

Footnotes

Acknowledgements

The authors are grateful to the learned referee for his valuable suggestions regarding improvement of the paper.

ORCID iD

Pragati Nigam

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

References

Agarwal

M. C.

Panda

K. B.

(1993). An efficient estimator in post stratification. Metron, 51(3-4), 179–188.

Bacanli

Aksu

E. E.

(2012). Ratio estimators in post stratified sampling. Turkish Journal of Population Studies, 34(1), 3–15.

Bahl

Tuteja

R. K.

(1991). Ratio and product-type exponential estimators. Inform. Optimize. Sci, 12(1), 159–164.

Bandyopadhyaya

(1980). Improved ratio and product estimators. Sankhya, 12(C, 142), 45–49.

Chami

P. S.

Singh

Thomas

(2012). A two parameter ratio-product-ratio estimator using auxiliary information. ISRN Probability and Statistics, 2012(1), 1–15.

Chouhan

(2012). Improved estimation of parameters using auxiliary information in sample surveys [PhD Thesis]. Vikram University, Ujjain, M.P., India.

Espejo

M. R.

Pineda

(1997). On variance estimation for post-stratification: A review. Metron, 55, 209–220.

Holt

Smith

T. M. F.

(1979). Post stratification. Journal of Royal Statistical Society A, 142(1), 33–46.

Ige

A. F.

Tripathi

T. P.

(1989). Estimation of population mean using post stratification and auxiliary information. Abacus, 18(2), 265–276.

10.

Jagers

Oden

Trulsson

(1985). Post stratification and ratio estimation, usages of auxiliary information in survey sampling and opinion polls. International Statistical Review / Revue Internationale de Statistique, 221–238.

11.

Khoshnevisan

Singh

Chauhan

Sawan

Smarandache

(2007). A general family of estimators for estimating population mean using known value of some population parameter(s). Far East Journal of Theoretical Statistics, 22, 189–191.

12.

Koyuncu

Kadilar

(2009). Ratio and product estimators in stratified random sampling. Journal of Statistical Planning and Inference, 139(8), 2552–2558.

13.

Lone

H. A.

Tailor

(2014). Dual to separate ratio type exponential estimator in post stratification. Journal of Statistics Applications and Probability, 3(3), 425–432.

14.

Lone

H. A.

Tailor

(2015). Improved separate ratio and product exponential type estimators in the case of post stratification. Statistics in Transition, 16(1), 53–64.

15.

Lone

H. A.

Tailor

(2016). A class of estimators of population mean in case of post stratification. Pakistan Journal of Statistics and Operation Research, 12(1), 111–124.

16.

Mehta

Tailor

(2024). Ratio and ratio estimator for population mean in demonstration of post-stratification. Journal of Statistics Applications & Probability Letters, 11(1), 21–26.

17.

Onyeka

A. C.

(2012). Estimation of population mean in post stratified sampling using known value of some population parameter(s). Statistics in Transition-new Series, 13(1), 65–78.

18.

Onyeka

A. C.

(2013). Dual to ratio estimators of population mean in post stratified sampling using known value of some population parameters. Global Journal Inc. (USA), 13, 2. version 1.0.

19.

Onyeka

A. C.

Izunobi

C. H.

Iwueze

I. S.

(2015a). Estimation of population ratio in post stratified sampling using Variable transformation. Open Journal of Statistics, 5(1), 1–9.

20.

Onyeka

A. C.

Izunobi

C. H.

Iwueze

I. S.

(2015b). Separate type estimators for estimating population ratio in post stratified sampling using Variable transformation. Open Journal of Statistics, 5(1), 27–34.

21.

Pandey

B. N.

Dubey

(1988). Modified product estimator using coefficient of variation of auxiliary variate. Assam Statistical Review, 2(2), 64–66.

22.

Rather Ul Islam

Jeelani Iqbal

Shah

Y. M.

Rizvi

S. E. H.

Sharma

(2022). A new ratio type estimator for computation of population mean under post-stratification. Journal of Applied Mathematics, Statistics and Informatics, 18(1), 29–42.

23.

Singh

M. P.

(1967). Ratio-Cum-Product method of estimation. Metrika, 12(1), 34–42.

24.

Singh

G. N.

(2003). On the improvement of product method of estimation in sample surveys. Journal of Indian Society of Agriculture Statistics, 56(3), 267–275.

25.

Singh

H. P.

Kakran

M. S.

(1993). A modified ratio estimator using known coefficient of kurtosis of an auxiliary character (unpublished).

26.

Singh

H. P.

Nigam

(2022). A two parameter ratio-product-ratio estimator in post stratification. Pakistan Journal of Statistics and Operation Research, 18(1), 273–296.

27.

Singh

H. P.

Tailor

(2003). Use of known correlation coefficient in estimating the finite population mean. Statistics in Transition, 6(4), 555–560.

28.

Singh

H. P.

Tailor

Kakran

M. S.

(2004). An improved estimator of population mean using power transformation. Journal of Indian Society of Agriculture Statistics, 58(2), 223–230.

29.

Singh

H. P.

Vishwakarma

(2010). Estimation of mean using auxiliary information and post stratification. Communications in Statistics-Theory and Methods, 39(19), 3456–3471.

30.

Sisodia

B. V. S.

Dwivedi

V. K.

(1981). A modified ratio estimator using coefficient of variation of auxiliary Variable. Journal of Indian Society Agricultural Statistics, 33(1), 13–18.

31.

Srivastava

S. K.

(1971). A generalized estimator for the mean of a finite population using multi auxiliary information. Journal of American Statistical Association, 66(334), 404–407.

32.

Srivenkataramana

(1980). A dual to ratio estimator in sample surveys. Biometrical Journal, 67(1), 199–204.

33.

Stephan

(1945). The expected value and variance of the reciprocal and other negative powers of a positive Bernoulli an Variable. The Annals of Mathematical Statistics, 16(1), 50–61.

34.

Tailor

Lone

H. A.

Pandey

Kumar

(2016). Ratio-cum-product type estimator of finite population mean in case of post stratification. Mathematical Sciences Letters, 5(1), 103–106.

35.

Tailor

Mehta

(2018). Estimation of ratio of two population means in case of post-stratification. International Journal of Computer Sciences and Engineering, 6(11), 233–236.

36.

Tailor

Mehta

(2019). A ratio and ratio exponential estimator for finite population mean in case of post-stratification. Journal of Statistics Applications & Probability, 8(3), 241–246.

37.

Tailor

Chouhan

(2017). Improved ratio and product-type exponential estimators for population mean in case of post-stratification. Communications in Statistics-Theory and Methods, 46(21), 10387–10393.

38.

Tuteja

R. K.

Bahl

Tripathi

T. P.

(1995). Sampling strategies for population mean based on post stratification and multivariable auxiliary information [Paper presentation]. Proc.-III, Int. Symp. Optimiz. Statist, Aligarh Muslim University, India, 56–61.

39.

Upadhyaya

L. N.

Singh

H. P.

(1999). Use of transformed auxiliary Variable in estimating the finite population mean. Biometrical Journal, 41(5), 627–636.

40.

Upadhyaya

L. N.

Singh

H. P.

Chatterjee

Yadav

(2011). Improved ratio and product exponential type estimators. Journal of Statistical Theory and Practice, 5, 285–302.

41.

Vishwakarma

G. K.

(2017). An alternative to ratio estimator in post stratification. Communications in Statistics-Theory and Methods, 47(8), 1989–2000.

42.

Vishwakarma

G. K.

Singh

H. P

. (2017). Estimation in post stratification [Paper presentation]. Proceedings The National Academy of Sciences, India.

A Generalized Class of Exponential-type Estimators for Estimating the Population Mean in Post Stratified Sampling

Abstract

Keywords

1. Introduction

1.1 Notations and Definitions

2. Review of Some Existing Estimators of Population Mean Using Auxiliary Information in Post Stratified Random Sampling

2.1 Combined Ratio and Product Estimators in Post Stratified Random Sampling

2.2 Separate Ratio and Product Estimators in Post Stratified Random Sampling

2.3 Combined Ratio-type and Product-Type Exponential Estimators in Post Stratified Sampling

2.4 Separate Ratio-Type and Product-Type Exponential Estimators in Post Stratified Sampling

2.5 Combined Dual to Ratio and Product Estimators in Post Stratified Sampling

2.6 Dual to Separate Ratio and Product Estimators in Post Stratified Sampling

2.7 Dual to Combined Ratio-Type and Product-Type Exponential Estimators of the Population Mean Y ¯ in Post Stratified Sampling

2.8 Dual to Separate Ratio-Type and Product-Type Exponential Estimators for Population Mean Y ¯ in Post Stratified Sampling

2.9 A Family of Combined Estimators in Post Stratified Sampling

2.10 Separate Ratio-Type Estimators in Post Stratified Sampling When the Coefficient of Variation ( C x h ) and Kurtosis ( β 2 h ( x ) ) of the h t h ( h = 1 , 2 , … , L ) Stratum in the Population are known

2.11 Dual to Ratio Estimators of Population Mean in Post Stratified Sampling Using Known Value of Some Population Parameters

2.12 Improved Separate Ratio and Product-Type Exponential Estimators in Case of Post Stratification

2.13 Classes of Combined and Separate Estimators of Population Mean Y ¯ in Post Stratified Sampling

2.14 Classes of Separate Estimators of Population Mean Y ¯ in Post Stratified Sampling

3. Ratio-Cum-Product-Type Estimators for Population Mean in Post Stratified Sampling

3.1 Ratio-Cum-Product-Type Estimators of Finite Population Mean in Post Stratified Sampling

3.2 A Class of Combined Ratio-Cum-Product Type Estimators

3.3 Ratio and Ratio-Type Exponential Estimator

3.4 A Two Parameter Ratio-Product-Ratio Estimator in Post Stratified Sampling

3.5 A Ratio-Type Exponential Estimator for Population Mean Under Post Stratification

3.6 Ratio and Ratio Estimators for Population Mean in Demonstration of Post-Stratification

4. Estimation of Population Ratio Using Auxiliary Information

4.1 Estimation of Population Ratio in Post Stratified Sampling Using variable Transformation

4.2 Separate-Type Estimators for Estimating Population Ratio in Post Stratified Sampling Using Variable Transformation

4.3 Estimation of Ratio of Two Population Means Using Auxiliary Information in Case of Post Stratification

5. Proposed Generalized Class of Estimators for Population Mean in Post Stratified Sampling

Footnotes

Acknowledgements

ORCID iD

Funding

Declaration of conflicting interests

References

2.7 Dual to Combined Ratio-Type and Product-Type Exponential Estimators of the Population Mean $\bar{Y}$ in Post Stratified Sampling

2.8 Dual to Separate Ratio-Type and Product-Type Exponential Estimators for Population Mean $\bar{Y}$ in Post Stratified Sampling

2.10 Separate Ratio-Type Estimators in Post Stratified Sampling When the Coefficient of Variation $(C_{x h})$ and Kurtosis $(β_{2 h} (x))$ of the $h^{t h} (h = 1, 2, \dots, L)$ Stratum in the Population are known

2.13 Classes of Combined and Separate Estimators of Population Mean $\bar{Y}$ in Post Stratified Sampling

2.14 Classes of Separate Estimators of Population Mean $\bar{Y}$ in Post Stratified Sampling