Enhanced estimation of population mean in the presence of auxiliary information

Muhammad Irfan; Maria Javed; Zhengyan Lin

doi:10.1016/j.jksus.2018.12.005

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31 (

); 1373-1378

doi:

10.1016/j.jksus.2018.12.005

Enhanced estimation of population mean in the presence of auxiliary information

Muhammad Irfan^{^a,⁎}, Maria Javed^{^a,^b}, Zhengyan Lin^{^b}

a Department of Statistics, Government College University, Faisalabad, Pakistan

b Department of Mathematics, Institute of Statistics, Zhejiang University, Hangzhou 310027, China

⁎Corresponding author. mirfan@zju.edu.cn (Muhammad Irfan)

Received: 2018-9-8, Accepted: 2018-12-19, Published: 2018-10-01

Disclaimer:
This article was originally published by Elsevier and was migrated to Scientific Scholar after the change of Publisher.

Peer review under responsibility of King Saud University.

Abstract

In this article, we propose a general family of exponential-type estimators for enhanced estimation of population mean in simple random sampling. These estimators are based on the available parameters of the auxiliary variable such as coefficient of skewness, coefficient of kurtosis, standard deviation and coefficient of variation etc. Expressions for bias, mean squared error and minimum mean squared error of the proposed family are derived up to first degree of approximation. Five natural populations are considered to assess the performance of the proposed estimators. Numerical findings confirm that the proposed estimators dominate over the existing estimators such as sample mean, ratio, regression, Singh et al. (2008, 2009), Upadhyaya et al. (2011), Yadav and Kadilar (2013) and Kadilar (2016) in terms of mean squared error.

Keywords

Auxiliary variable

Bias

Efficiency

Exponential type estimators

Mean squared error

Simple random sampling

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1 Introduction

Survey sampling is broadly used in agriculture, business management, demography, economics, education, engineering, industry, medical sciences, political science, social sciences and many others. The main objective of sample survey theory is to make inferences about the unknown population parameters like population total, population proportion, population mean or population variance etc. One of the hottest issues in survey sampling is to enhanced the efficiency of ratio, product and regression type estimators in the presence of known auxiliary information for estimating the unknown population parameters of the study variable under different sampling techniques. Ratio method of estimation proposed by Cochran (1940) is at its best than the usual estimators of mean/total when the correlation between study variable and auxiliary variable is positively high and regression line of study variable on auxiliary variable is linear and passes through or nearly through the origin. In many real situations, auxiliary variables are highly correlated with the study variable for instance, person’s age and duration of sleep, income and expenditure, person’s age and blood pressure etc. Product method of estimation was first suggested by Robson (1957) and rediscovered by Murthy (1964). It is preferably used when the correlation between study variable and auxiliary variable is negatively high. Regression method of estimation proposed by Watson (1937) is the appropriate choice when the regression line is linear and passes through a point away from the origin.

Many authors have suggested estimators for the estimation of population mean based on auxiliary information in simple random sampling without replacement (SRSWOR). Readers may refer to Singh and Tailor (2003), Kadilar and Cingi (2004, 2006a, 2006b, 2006c), Koyuncu and Kadilar (2009), Singh et al. (2009), Yan and Tian (2010), Upadhyaya et al. (2011), Yadav and Kadilar (2013), Khan et al. (2015), Kadilar (2016) and the references cited therein.

The enthusiasm behind this article is to propose an improved general family of exponential-type estimators for the estimation of finite population mean under SRSWOR. A brief introduction of some conventional and exponential-type estimators for population mean is provided in section 2. In section 3, we proposed a general exponential-type family of estimators for estimating finite population mean $\bar{Y}$ under SRSWOR scheme and derived their properties up to first order of approximation. An empirical study using five real data sets is performed to compare the proposed and existing estimators in terms of MSE in section 4. Finally, concluding remarks are addressed in the last section.

2 Existing estimators

Consider a sample of size $" n "$ is selected from a population of size $" N "$ subject to the constraint $n < N$ under SRSWOR. Let $n$ pair of observations ( $y_{i}, x_{i}), i = 1, 2, 3, \dots, n$ for the study $(y)$ and auxiliary variable $(x)$ respectively. Let $\bar{Y} = N^{- 1} \sum_{i = 1}^{N} y_{i}$ and $\bar{X} = N^{- 1} \sum_{i = 1}^{N} x_{i}$ be the population means of the study and auxiliary variables respectively. Similarly, $\bar{y} = n^{- 1} \sum_{i = 1}^{n} y_{i}$ and $\bar{x} = n^{- 1} \sum_{i = 1}^{n} x_{i}$ be the respective sample means of the study and auxiliary variables.

To drive the expressions for bias and mean squared error (MSE) of the existing and proposed estimators, we consider $ζ_{0} = \frac{\bar{y} - \bar{Y}}{\bar{Y}} a n d ζ_{1} = \frac{\bar{x} - \bar{X}}{\bar{X}}$ such that $E (ζ_{0}) = E (ζ_{1}) = 0$ $V (ζ_{0}) = E (ζ_{0}^{2}) = \frac{V (\bar{y})}{{\bar{Y}}^{2}} = λ C_{y}^{2}$ $V (ζ_{1}) = E (ζ_{1}^{2}) = λ C_{x}^{2}$ $C o v (ζ_{0}, ζ_{1}) = E (ζ_{0} ζ_{1}) = \frac{C o v (\bar{y}, \bar{x})}{\bar{Y} \bar{X}} = λ ρ_{yx} C_{y} C_{x}$ where $λ = (\frac{1}{n} - \frac{1}{N}), C_{y}^{2} = \frac{S_{y}^{2}}{{\bar{Y}}^{2}}, C_{x}^{2} = \frac{S_{x}^{2}}{{\bar{X}}^{2}}, ρ_{yx} = \frac{S_{yx}}{S_{y} S_{x}}, S_{y}^{2} = \frac{\sum_{i = 1}^{N} {(y_{i} - \bar{Y})}^{2}}{N - 1}, S_{x}^{2} = \frac{\sum_{i = 1}^{N} {(x_{i} - \bar{X})}^{2}}{N - 1} a n d$ $S_{yx} = \frac{\sum_{i = 1}^{N} (y_{i} - \bar{Y}) (x_{i} - \bar{X})}{N - 1} .$

The conventional unbiased estimator without utilizing auxiliary information is defined by

(1)

η_{0} = \bar{y}

Variance/MSE of $η_{0}$ is given by

(2)

M S E (η_{0}) = V a r (η_{0}) = λ {\bar{Y}}^{2} C_{y}^{2}

The conventional ratio estimator proposed by Cochran (1940) is defined as

(3)

{\hat{\bar{Y}}}_{R} = \bar{y} (\frac{\bar{X}}{\bar{x}}), \bar{x} \neq 0

Expressions for the bias and MSE of the ${\hat{\bar{Y}}}_{R}$ estimator, up to first degree of approximation, are respectively given by

(4)

B i a s ({\hat{\bar{Y}}}_{R}) ≅ λ \bar{Y} [C_{x}^{2} - ρ_{yx} C_{y} C_{x}]

and

(5)

M S E ({\hat{\bar{Y}}}_{R}) ≅ λ {\bar{Y}}^{2} [C_{y}^{2} + C_{x}^{2} (1 - 2 A)]

where

A = ρ_{yx} \frac{C_{y}}{C_{x}}

Exponential ratio type estimator initiated by Bahl and Tuteja (1991) of population mean is as below

(6)

{\hat{\bar{Y}}}_{BT} = \bar{y} exp (\frac{\bar{X} - \bar{x}}{\bar{X} + \bar{x}})

Bias and MSE of the ${\hat{\bar{Y}}}_{BT}$ estimator, up to first order of approximation, are respectively given as

(7)

B i a s ({\hat{\bar{Y}}}_{BT}) ≅ \frac{λ}{2} \bar{Y} [\frac{3 C_{x}^{2}}{4} - ρ_{yx} C_{y} C_{x}]

and

(8)

M S E ({\hat{\bar{Y}}}_{BT}) ≅ \frac{λ}{4} {\bar{Y}}^{2} [4 C_{y}^{2} + C_{x}^{2} - 4 ρ_{yx} C_{y} C_{x}]

Singh et al. (2008) suggested a ratio-product exponential type estimator for population mean given as

(9)

{\hat{\bar{Y}}}_{SI} = \bar{y} [μ e x p (\frac{\bar{X} - \bar{x}}{\bar{X} + \bar{x}}) + (1 - μ) e x p (\frac{\bar{x} - \bar{X}}{\bar{X} + \bar{x}})]

where

μ

is a real constant.

The optimal value of $μ = \frac{1}{2} + \frac{ρ_{yx} C_{y}}{C_{x}} .$

Minimum MSE of ${\hat{\bar{Y}}}_{SI}$ estimator, up to first degree of approximation, as follows

(10)

MS E_{\min} ({\hat{\bar{Y}}}_{SI}) ≅ λ {\bar{Y}}^{2} C_{y}^{2} (1 - ρ_{yx}^{2}) ≅ M S E ({\hat{\bar{Y}}}_{Reg})

Motivated by Khoshnevisan et al. (2007), Singh et al. (2009) proposed an exponential family of estimators for $\bar{Y}$ as

(11)

η = \bar{y} e x p [\frac{(α \bar{X} + β) - (α \bar{x} + β)}{(α \bar{X} + β) + (α \bar{x} + β)}]

where

α a n d β

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

(β_{1 (x)})

, coefficient of kurtosis

(β_{2 (x)})

, standard deviation

(S_{x})

, coefficient of variation

(C_{x})

and coefficient of correlation

(ρ_{yx})

of the population.

Following are the bias and MSE up to first degree of approximation

(12)

B i a s (η) ≅ λ \bar{Y} [\frac{3}{2} θ_{i}^{2} C_{x} - θ_{i} ρ_{yx} C_{y}] C_{x}

and

(13)

M S E (η) ≅ λ {\bar{Y}}^{2} [C_{y}^{2} + θ_{i}^{2} C_{x}^{2} - 2 θ_{i} ρ_{yx} C_{y} C_{x}]

where

θ_{i} = \frac{α \bar{X}}{2 (α \bar{X} + β)}, i = 1, 2, 3, . . ., 10

and

θ_{1} = 0.5; θ_{2} = \frac{\bar{X}}{2 (\bar{X} + β_{2 (x)})}; θ_{3} = \frac{\bar{X}}{2 (\bar{X} + C_{x})}; θ_{4} = \frac{\bar{X}}{2 (\bar{X} + ρ_{yx})}; θ_{5} = \frac{β_{2 (x)} \bar{X}}{2 (β_{2 (x)} \bar{X} + C_{x})}; θ_{6} = \frac{C_{x} \bar{X}}{2 (C_{x} \bar{X} + β_{2 (x)})}; θ_{7} = \frac{C_{x} \bar{X}}{2 (C_{x} \bar{X} + ρ_{yx})}; θ_{8} = \frac{ρ_{yx} \bar{X}}{2 (ρ_{yx} \bar{X} + C_{x})}; θ_{9} = \frac{β_{2 (x)} \bar{X}}{2 (β_{2 (x)} \bar{X} + ρ_{yx})}; θ_{10} = \frac{ρ_{yx} \bar{X}}{2 (ρ_{yx} \bar{X} + β_{2 (x)})} .

Some members of the η- family of estimators are given in Table 1. Furthermore, Singh et al. (2009) proposed a generalized estimator by combining the estimator η₁ and $η_{i} (i = 2, 3, 4, \dots, 10)$ as follows

(14)

η^{*} = γ η_{1} + (1 - γ) η_{i}, i = 2, 3, 4, \dots, 10

where

γ

is a suitable chosen weight.

Table 1 Members of

η

- family of estimators.

Estimators	$α$	$β$
Sample Mean
$η_{0} = \bar{y}$	$0$	$0$
Bahl and Tuteja (1991)
$η_{1} = \bar{y} e x p [\frac{\bar{X} - \bar{x}}{\bar{X} + \bar{x}}]$	1	$0$
Singh et al. (2009)
$η_{2} = \bar{y} e x p [\frac{[\bar{X} - \bar{x}]}{[\bar{X} + \bar{x}] + 2 β_{2 (x)}}]$	1	$β_{2 (x)}$
$η_{3} = \bar{y} e x p [\frac{[\bar{X} - \bar{x}]}{[\bar{X} + \bar{x}] + 2 C_{x}}]$	1	$C_{x}$
$η_{4} = \bar{y} e x p [\frac{[\bar{X} - \bar{x}]}{[\bar{X} + \bar{x}] + 2 ρ_{yx}}]$	1	$ρ_{yx}$
$η_{5} = \bar{y} e x p [\frac{β_{2 (x)} [\bar{X} - \bar{x}]}{β_{2 (x)} [\bar{X} + \bar{x}] + 2 C_{x}}]$	$β_{2 (x)}$	$C_{x}$
$η_{6} = \bar{y} e x p [\frac{C_{x} [\bar{X} - \bar{x}]}{C_{x} [\bar{X} + \bar{x}] + 2 β_{2 (x)}}]$	$C_{x}$	$β_{2 (x)}$
$η_{7} = \bar{y} e x p [\frac{C_{x} [\bar{X} - \bar{x}]}{C_{x} [\bar{X} + \bar{x}] + 2 ρ_{yx}}]$	$C_{x}$	$ρ_{yx}$
$η_{8} = \bar{y} e x p [\frac{ρ_{yx} [\bar{X} - \bar{x}]}{ρ_{yx} [\bar{X} + \bar{x}] + 2 C_{x}}]$	$ρ_{yx}$	$C_{x}$
$η_{9} = \bar{y} e x p [\frac{β_{2 (x)} [\bar{X} - \bar{x}]}{β_{2 (x)} [\bar{X} + \bar{x}] + 2 ρ_{yx}}]$	$β_{2 (x)}$	$ρ_{yx}$
$η_{10} = \bar{y} e x p [\frac{ρ_{yx} [\bar{X} - \bar{x}]}{ρ_{yx} [\bar{X} + \bar{x}] + 2 β_{2 (x)}}]$	$ρ_{yx}$	$β_{2 (x)}$

The optimal value of $γ = \frac{2 (A - θ_{i})}{(1 - 2 θ_{i})} .$

Minimum MSE of $η^{*}$ estimator is equal to the MSE of the regression estimator ${\hat{\bar{Y}}}_{Reg} .$

(15)

{MSE}_{\min} (η^{*}) ≅ λ {\bar{Y}}^{2} C_{y}^{2} (1 - ρ_{yx}^{2}) ≅ M S E ({\hat{\bar{Y}}}_{Reg})

Upadhyaya et al. (2011) proposed ratio exponential type estimator as

(16)

{\hat{\bar{Y}}}_{UP} = \bar{y} e x p (1 - \frac{2 \bar{x}}{\bar{X} + \bar{x}})

MSE of ${\hat{\bar{Y}}}_{UP}$ estimator, up to first degree of approximation, as follows

(17)

{MSE}_{\min} ({\hat{\bar{Y}}}_{UP}) ≅ λ {\bar{Y}}^{2} C_{y}^{2} (1 - ρ_{yx}^{2}) ≅ M S E ({\hat{\bar{Y}}}_{Reg})

Yadav and Kadilar (2013) suggested an improved exponential family of estimators for population mean $\bar{Y}$ by utilizing Singh et al. (2009) as

(18)

ω = k \bar{y} e x p [\frac{(α \bar{X} + β) - (α \bar{x} + β)}{(α \bar{X} + β) + (α \bar{x} + β)}]

where

k

is a suitable constant and

α a n d β

are defined earlier. Some members of the

ω

- family of estimators are given in Table 2.

Table 2 Members of

ω

- family of estimators.

Estimators	$α$	$β$	k
Bahl and Tuteja (1991)
$ω_{0} = \bar{y} e x p [\frac{\bar{X} - \bar{x}}{\bar{X} + \bar{x}}]$	$1$	$0$	$1$
Yadav and Kadilar (2013)
$ω_{1} = k \bar{y} e x p [\frac{\bar{X} - \bar{x}}{\bar{X} + \bar{x}}]$	$1$	$0$	$k$
$ω_{2} = k \bar{y} e x p [\frac{[\bar{X} - \bar{x}]}{[\bar{X} + \bar{x}] + 2 β_{2 (x)}}]$	1	$β_{2 (x)}$	$k$
$ω_{3} = k \bar{y} e x p [\frac{[\bar{X} - \bar{x}]}{[\bar{X} + \bar{x}] + 2 C_{x}}]$	$1$	$C_{x}$	$k$
$ω_{4} = k \bar{y} e x p [\frac{[\bar{X} - \bar{x}]}{[\bar{X} + \bar{x}] + 2 ρ_{yx}}]$	1	$ρ_{yx}$	$k$
$ω_{5} = k \bar{y} e x p [\frac{β_{2 (x)} [\bar{X} - \bar{x}]}{β_{2 (x)} [\bar{X} + \bar{x}] + 2 C_{x}}]$	$β_{2 (x)}$	$C_{x}$	$k$
$ω_{6} = k \bar{y} e x p [\frac{C_{x} [\bar{X} - \bar{x}]}{C_{x} [\bar{X} + \bar{x}] + 2 β_{2 (x)}}]$	$C_{x}$	$β_{2 (x)}$	$k$
$ω_{7} = k \bar{y} e x p [\frac{C_{x} [\bar{X} - \bar{x}]}{C_{x} [\bar{X} + \bar{x}] + 2 ρ_{yx}}]$	$C_{x}$	$ρ_{yx}$	$k$
$ω_{8} = k \bar{y} e x p [\frac{ρ_{yx} [\bar{X} - \bar{x}]}{ρ_{yx} [\bar{X} + \bar{x}] + 2 C_{x}}]$	$ρ_{yx}$	$C_{x}$	$k$
$ω_{9} = k \bar{y} e x p [\frac{β_{2 (x)} [\bar{X} - \bar{x}]}{β_{2 (x)} [\bar{X} + \bar{x}] + 2 ρ_{yx}}]$	$β_{2 (x)}$	$ρ_{yx}$	$k$
$ω_{10} = k \bar{y} e x p [\frac{ρ_{yx} [\bar{X} - \bar{x}]}{ρ_{yx} [\bar{X} + \bar{x}] + 2 β_{2 (x)}}]$	$ρ_{yx}$	$β_{2 (x)}$	k

Following are the expressions of bias and MSE of the $ω$ - family of estimators

(19)

B i a s (ω) ≅ λ k \bar{Y} [\frac{3}{2} θ_{i}^{2} C_{x} - θ_{i} ρ_{yx} C_{y}] C_{x} + \bar{Y} (k - 1)

and

(20)

M S E (ω) ≅ λ {\bar{Y}}^{2} [{k^{2} C}_{y}^{2} + k θ_{i}^{2} C_{x}^{2} (4 k - 3) - 2 k θ_{i} ρ_{yx} C_{y} C_{x} (2 k - 1)] + {\bar{Y}}^{2} {(k - 1)}^{2}

where

θ_{i}

is defined earlier.

Partially differentiating Eq. (20) w.r.t. $" k "$ and setting $\frac{\partial M S E (t)}{\partial k} = 0$ , the optimal value of $" k "$ is given by

(21)

k = \frac{1 + λ [\frac{3}{2} θ_{i}^{2} C_{x}^{2} - θ_{i} ρ_{yx} C_{y} C_{x}]}{1 + λ [C_{y}^{2} + 4 θ_{i}^{2} C_{x}^{2} - 4 θ_{i} ρ_{yx} C_{y} C_{x}]} = \frac{C_{1 i}}{C_{2 i}}

where

C_{1 i} = 1 + λ [\frac{3}{2} θ_{i}^{2} C_{x}^{2} - θ_{i} ρ_{yx} C_{y} C_{x}]

and

C_{2 i} = 1 + λ [C_{y}^{2} + 4 θ_{i}^{2} C_{x}^{2} - 4 θ_{i} ρ_{yx} C_{y} C_{x}]

Substituting Eq. (21) in Eq. (20), we get minimum MSE of the $ω$ -family of estimators as

(22)

{MSE}_{\min} (ω) ≅ {\bar{Y}}^{2} [1 - \frac{C_{1 i}^{2}}{C_{2 i}}]

Kadilar (2016) suggested a modified exponential type estimator for population mean as

(23)

{\hat{\bar{Y}}}_{K} = \bar{y} {(\frac{\bar{x}}{\bar{X}})}^{δ} exp (\frac{\bar{X} - \bar{x}}{\bar{X} + \bar{x}})

where

δ

is a real constant.

The optimal value of $δ = \frac{(C_{x} - 2 ρ_{yx} C_{y})}{2 C_{x}} .$

Minimum MSE of ${\hat{\bar{Y}}}_{K}$ estimator, up to first degree of approximation, as follows

(24)

{MSE}_{\min} ({\hat{\bar{Y}}}_{K}) ≅ λ {\bar{Y}}^{2} C_{y}^{2} (1 - ρ_{yx}^{2}) ≅ M S E ({\hat{\bar{Y}}}_{Reg})

Remark 2.1

Minimum MSE of ${\hat{\bar{Y}}}_{SI}, η^{*}, {\hat{\bar{Y}}}_{UP} a n d {\hat{\bar{Y}}}_{K}$ estimators, up to first order of approximation is exactly equal to the variance of the usual regression estimator ${\hat{\bar{Y}}}_{Reg} .$ Regression estimator suggested by Watson (1937) is given by

(25)

{\hat{\bar{Y}}}_{Reg} = \bar{y} + b_{y . x} (\bar{X} - \bar{x})

where

b_{y . x} = \frac{S_{yx}}{S_{x}^{2}}

is the sample regression coefficient.

3 The suggested family of estimators

In this section, general exponential-type estimators for estimating finite population mean $\bar{Y}$ under SRSWOR is proposed. Some members of proposed family are given in Table 3. Expressions for the bias, MSE and minimum MSE are obtained to the first degree of approximation.

(26)

ξ = t_{1} \bar{y} (\frac{{\bar{X}}^{*}}{{\bar{x}}^{*}}) + t_{2} (\bar{X} - \bar{x}) e x p (\frac{{\bar{X}}^{*} - {\bar{x}}^{*}}{{\bar{X}}^{*} + {\bar{x}}^{*}})

where

{\bar{X}}^{*} = α \bar{X} + β a n d {\bar{x}}^{*} = α \bar{x} + β

Table 3 Some members of

ξ

- family of estimators.

Estimators	$α$	$β$
$ξ_{1} = t_{1} \bar{y} (\frac{\bar{X}}{\bar{x}}) + t_{2} (\bar{X} - \bar{x}) e x p (\frac{(\bar{X} - \bar{x})}{(\bar{X} + \bar{x})})$	$1$	$0$
$ξ_{2} = t_{1} \bar{y} (\frac{β_{2 (x)} \bar{X} + β_{1 (x)}}{β_{2 (x)} \bar{x} + β_{1 (x)}}) + t_{2} (\bar{X} - \bar{x}) e x p (\frac{β_{2 (x)} (\bar{X} - \bar{x})}{β_{2 (x)} (\bar{X} + \bar{x}) + 2 β_{1 (x)}})$	$β_{2 (x)}$	$β_{1 (x)}$
$ξ_{3} = t_{1} \bar{y} (\frac{S_{x} \bar{X} + β_{2 (x)}}{S_{x} \bar{x} + β_{2 (x)}}) + t_{2} (\bar{X} - \bar{x}) e x p (\frac{S_{x} (\bar{X} - \bar{x})}{S_{x} (\bar{X} + \bar{x}) + 2 β_{2 (x)}})$	$S_{x}$	$β_{2 (x)}$
$ξ_{4} = t_{1} \bar{y} (\frac{C_{x} \bar{X} + β_{1 (x)}}{C_{x} \bar{x} + β_{1 (x)}}) + t_{2} (\bar{X} - \bar{x}) e x p (\frac{C_{x} (\bar{X} - \bar{x})}{C_{x} (\bar{X} + \bar{x}) + 2 β_{1 (x)}})$	$C_{x}$	$β_{1 (x)}$
$ξ_{5} = t_{1} \bar{y} (\frac{\bar{X} + β_{1 (x)}}{\bar{x} + β_{1 (x)}}) + t_{2} (\bar{X} - \bar{x}) e x p (\frac{(\bar{X} - \bar{x})}{(\bar{X} + \bar{x}) + 2 β_{1 (x)}})$	$1$	$β_{1 (x)}$
$ξ_{6} = t_{1} \bar{y} (\frac{ρ_{yx} \bar{X} + β_{2 (x)}}{ρ_{yx} \bar{x} + β_{2 (x)}}) + t_{2} (\bar{X} - \bar{x}) e x p (\frac{ρ_{yx} (\bar{X} - \bar{x})}{ρ_{yx} (\bar{X} + \bar{x}) + 2 β_{2 (x)}})$	$ρ_{yx}$	$β_{2 (x)}$
$ξ_{7} = t_{1} \bar{y} (\frac{β_{1 (x)} \bar{X} + β_{2 (x)} C_{x}}{β_{1 (x)} \bar{x} + β_{2 (x)} C_{x}}) + t_{2} (\bar{X} - \bar{x}) e x p (\frac{β_{1 (x)} (\bar{X} - \bar{x})}{β_{1 (x)} (\bar{X} + \bar{x}) + 2 β_{2 (x)} C_{x}})$	$β_{1 (x)}$	$β_{2 (x)} C_{x}$
$ξ_{8} = t_{1} \bar{y} (\frac{S_{x} \bar{X} + β_{1 (x)}}{S_{x} \bar{x} + β_{1 (x)}}) + t_{2} (\bar{X} - \bar{x}) e x p (\frac{S_{x} (\bar{X} - \bar{x})}{S_{x} (\bar{X} + \bar{x}) + 2 β_{1 (x)}})$	$S_{x}$	$β_{1 (x)}$
$ξ_{9} = t_{1} \bar{y} (\frac{C_{x} \bar{X} + ρ_{yx}}{C_{x} \bar{x} + ρ_{yx}}) + t_{2} (\bar{X} - \bar{x}) e x p (\frac{C_{x} (\bar{X} - \bar{x})}{C_{x} (\bar{X} + \bar{x}) + 2 ρ_{yx}})$	$C_{x}$	$ρ_{yx}$
$ξ_{10} = t_{1} \bar{y} (\frac{β_{2 (x)} \bar{X} + ρ_{yx}}{β_{2 (x)} \bar{x} + ρ_{yx}}) + t_{2} (\bar{X} - \bar{x}) e x p (\frac{β_{2 (x)} (\bar{X} - \bar{x})}{β_{2 (x)} (\bar{X} + \bar{x}) + 2 ρ_{yx}})$	$β_{2 (x)}$	$ρ_{yx}$
$ξ_{11} = t_{1} \bar{y} (\frac{\bar{X} + ρ_{yx}}{\bar{x} + ρ_{yx}}) + t_{2} (\bar{X} - \bar{x}) e x p (\frac{(\bar{X} - \bar{x})}{(\bar{X} + \bar{x}) + 2 ρ_{yx}})$	$1$	$ρ_{yx}$

where $t_{1} a n d t_{2}$ are the appropriate constants to be determined such the MSE of $ξ$ is minimum, $α (\neq 0)$ and $β$ are either real numbers or functions of the known parameters such as coefficient of variation $(C_{x})$ , standard deviation $(S_{x})$ , coefficient of skewness $(β_{1 (x)}),$ coefficient of kurtosis $(β_{2 (x)})$ and coefficient of correlation $(ρ_{yx})$ of the population.

3.1

3.1 Bias, MSE and minimum MSE of $ξ$

Eq. (26) can be transformed in terms of $ζ_{i}' s$ as

(27)

ξ = t_{1} \bar{Y} (1 + ζ_{0}) {(1 + ϕ_{i} ζ_{1})}^{- 1} - t_{2} \bar{X} ζ_{1} e x p ({\frac{- ϕ_{i} ζ_{1}}{2} (1 + \frac{ϕ_{i} ζ_{1}}{2})}^{- 1}), i = 1, 2, 3, \dots, 11

where

ϕ_{i} = \frac{α \bar{X}}{α \bar{X} + β}, i = 1, 2, 3, . . ., 11

and

ϕ_{1} = 1; ϕ_{2} = \frac{\bar{X} β_{2 (x)}}{\bar{X} β_{2 (x)} + β_{1 (x)}}; ϕ_{3} = \frac{\bar{X} S_{x}}{\bar{X} S_{x} + β_{2 (x)}}; ϕ_{4} = \frac{\bar{X} C_{x}}{\bar{X} C_{x} + β_{1 (x)}}; ϕ_{5} = \frac{\bar{X}}{\bar{X} + β_{1 (x)}};

ϕ_{6} = \frac{\bar{X} ρ_{yx}}{\bar{X} ρ_{yx} + β_{2 (x)}}; ϕ_{7} = \frac{\bar{X} β_{1 (x)}}{\bar{X} β_{1 (x)} + β_{2 (x)} C_{x}}; ϕ_{8} = \frac{\bar{X} S_{x}}{\bar{X} S_{x} + β_{1 (x)}}; ϕ_{9} = \frac{\bar{X} C_{x}}{\bar{X} C_{x} + ρ_{yx}};

ϕ_{10} = \frac{\bar{X} β_{2 (x)}}{\bar{X} β_{2 (x)} + ρ_{yx}} a n d ϕ_{11} = \frac{\bar{X}}{\bar{X} + ρ_{yx}} .

Subtracting $\bar{Y}$ on both sides of the Eq. (27) and expanding up to first degree of approximation, we have

(28)

ξ - \bar{Y} = t_{1} (\bar{Y} - 1) + t_{1} \bar{Y} ζ_{0} - t_{1} \bar{Y} ϕ_{i} ζ_{1} - t_{2} \bar{X} ζ_{1} + t_{1} \bar{Y} ϕ_{i}^{2} ζ_{1}^{2} + \frac{t_{2} \bar{X} ϕ_{i} ζ_{1}^{2}}{2} - t_{1} \bar{Y} ϕ_{i} ζ_{0} ζ_{1}

Taking expectation on both sides of Eq. (28), we obtained bias of the $ξ$ -family of estimators to the first degree of approximation, given as

(29)

B i a s (ξ) = E (ξ - \bar{Y}) ≅ t_{1} (\bar{Y} - 1) + t_{1} \bar{Y} ϕ_{i}^{2} λ C_{x}^{2} + \frac{t_{2} \bar{X} ϕ_{i} λ C_{x}^{2}}{2} - t_{1} \bar{Y} ϕ_{i} λ ρ_{yx} C_{y} C_{x}

Squaring and taking expectation on both sides of Eq. (28), we get MSE of the proposed $ξ$ -family of estimators to the first degree of approximation as:

(30)

M S E (ξ) = E {(ξ - \bar{Y})}^{2} ≅ {\bar{Y}}^{2} [1 + t_{1}^{2} A + t_{2}^{2} R^{2} λ C_{x}^{2} - 2 t_{1} t_{2} R B - 2 t_{1} C - t_{2} R ϕ_{i} λ C_{x}^{2}]

where

R = \frac{\bar{X}}{\bar{Y}}

To obtained minimum MSE of the proposed estimators $″ ξ^{″}$ , Eq. (30) is differentiated with respect to the unknown parameters $t_{1} a n d t_{2}$ and setting $\frac{\partial M S E (ξ)}{\partial t_{1}} = 0 a n d \frac{\partial M S E (ξ)}{\partial t_{2}} = 0$ , we obtained the optimal values of $t_{1} a n d t_{2}$ as given below $t_{1 (o p t)} = \frac{λ C_{x}^{2} (2 C + B ϕ_{i})}{2 (A λ C_{x}^{2} - B^{2})}$ and $t_{2 (o p t)} = \frac{2 B C + A ϕ_{i} λ C_{x}^{2}}{2 R (A λ C_{x}^{2} - B^{2})}$

Minimum MSE of $ξ$ -family up to first degree of approximation is obtained by substituting the optimal values of $t_{1} a n d t_{2}$ in Eq. (30) and simplifying as

(31)

{MSE}_{\min} (ξ) ≅ \frac{{\bar{Y}}^{2} λ C_{x}^{2}}{4 D^{2}} [\frac{4 D^{2}}{λ C_{x}^{2}} + E^{2} (D + B^{2}) + F^{2} - 2 B E F - 4 C E D - 2 ϕ_{i} F D]

where

A = 1 + λ C_{y}^{2} + 3 ϕ_{i}^{2} λ C_{x}^{2} - 4 ϕ_{i} λ ρ_{yx} C_{y} C_{x}, B = λ ρ_{yx} C_{y} C_{x} - \frac{3 ϕ_{i} λ C_{x}^{2}}{2}

C = 1 + ϕ_{i}^{2} λ C_{x}^{2} - ϕ_{i} λ ρ_{yx} C_{y} C_{x}, D = A λ C_{x}^{2} - B^{2}

E = 2 C + B ϕ_{i}, F = 2 B C + A ϕ_{i} λ C_{x}^{2}

Remark 3.1

We can also obtain many more exponential-type estimators by putting different choices of $α a n d β$ in the estimator given in Eq. (26).

4 Empirical study

In this section, we demonstrate the performance of the proposed estimators over simple mean, ratio, regression, Bahl and Tuteja (1991), Singh et al. (2008, 2009), Upadhyaya et al. (2011), Yadav and Kadilar (2013) and Kadilar (2016) estimators in terms of MSE. Five natural populations are considered to access the performance of the proposed estimators. Parameters of all the populations are detailed below in this section.

Population 1

Population 1 Source: Koyuncu and Kadilar, 2009

Let $y$ be the number of teachers and $x$ be the number of students in both primary and secondary schools for 923 districts of Turkey in 2007. The summary statistics are shown as follows. $N = 923, n = 180, \bar{Y} = 436.4345, \bar{X} = 11440.4984, ρ_{yx} = 0.9543, S_{y} = 749.9395,$ $C_{y} = 1.7183, S_{x} = 21331.1315, C_{x} = 1.8645, β_{1 (x)} = 3.9365, β_{2 (x)} = 18.7208$

Population 2

Population 2 Source: Srisodaphol et al., 2015

Let $y$ be the entire height in feet and $x$ be the diameter in centimeters of a breast height of conifer (Pinus palustric) trees. The statistic values are shown as follows. $N = 396, n = 30, \bar{Y} = 20.9629, \bar{X} = 52.6742, ρ_{yx} = 0.9073, S_{y} = 17.6164, C_{y} = 0.8404,$ $S_{x} = 57.1132, C_{x} = 1.0843, β_{1 (x)} = 1.6157, β_{2 (x)} = 1.7785$

Population 3

Population 3 Source: Cochran, 1977

Let $y$ be the population size in 1930 and $x$ be the population size in 1920. The statistic values are shown as follows. $N = 49, n = 20, \bar{Y} = 127.7959, \bar{X} = 103.1429, ρ_{yx} = 0.9817, S_{y} = 123.1212, C_{y} = 0.9634,$ $S_{x} = 104.4051, C_{x} = 1.0122, β_{1 (x)} = 2.2553, β_{2 (x)} = 5.1412$

Population 4

Population 4 Source: Kadilar and Cingi, 2004

Let $y$ be the number of teachers and $x$ be the number of students in both primary and secondary schools of Turkey in 2007. The summary statistics are shown as follows. $N = 106, n = 20, \bar{Y} = 2212.59, \bar{X} = 27421.70, ρ_{yx} = 0.86, S_{y} = 11549.72, C_{y} = 5.22,$ $S_{x} = 57585.57, C_{x} = 2.10, β_{1 (x)} = 5.1237, β_{2 (x)} = 34.5723$

Population 5

Population 5 Source: Kadilar and Cingi, 2003

Let $y$ be the apple production quantity and $x$ be the count of apple trees in 854 villages in Turkey. The summary statistics are shown as follows. $N = 94, n = 20, \bar{Y} = 9384, \bar{X} = 72410, ρ_{yx} = 0.901, S_{y} = 29906.810, C_{y} = 3.187,$ $S_{x} = 160750.20, C_{x} = 2.22, β_{1 (x)} = 4.611, β_{2 (x)} = 26.136$

We computed MSE of all the proposed and competing estimators including simple mean $\bar{y}$ , ratio ${\hat{\bar{Y}}}_{R}$ , Bahl and Tuteja (1991) ${\hat{\bar{Y}}}_{BT}$ , regression ${\hat{\bar{Y}}}_{Reg},$ Singh et al. (2008), Singh et al. (2009) Upadhyaya et al. (2011) , Yadav and Kadilar (2013) ${\hat{\bar{Y}}}_{SI},$ ${\hat{\bar{Y}}}_{UP}$ , and Kadilar (2016) ${\hat{\bar{Y}}}_{K}$ . Findings are presented in Tables 4–9 . It is clearly observed from Tables 4–9 that

our proposed estimators have minimum MSEs as compared to all other estimators for all data sets.
in each data set, MSEs of all proposed estimators are almost equal which indicates that all proposed estimators are equally efficient. So, researcher may be used any one of them depending upon the availability of the values of $α$ and $β$ .

Table 4 Var/MSE of the estimators.

\bar{y}, {\hat{\bar{Y}}}_{R}, {\hat{\bar{Y}}}_{BT}, {\hat{\bar{Y}}}_{Reg}, {\hat{\bar{Y}}}_{SI}, {\hat{\bar{Y}}}_{UP} a n d {\hat{\bar{Y}}}_{K}

Population	$V a r (\bar{y})$	$M S E ({\hat{\bar{Y}}}_{R})$	$M S E ({\hat{\bar{Y}}}_{BT})$	$M S E ({\hat{\bar{Y}}}_{Reg}) = {MSE}_{\min} ({\hat{\bar{Y}}}_{i}), i = S I, U P, K$
1	2515.074	267.644	651.042	224.625
2	9.562	3.093	2.348	1.691
3	448.563	18.362	109.665	16.230
4	5,411,348	2,542,740	3,758,095	1,409,115
5	35,205,782	8,096,858	17,380,652	6,625,693

Table 5 MSE of existing and proposed estimators for Population 1.

$η$ - family	MSE	$t$ - family	MSE	$ξ$ - family	MSE
$η_{0}$	2515.0740	$t_{0}$	651.0423	$ξ_{1}$	223.7713
$η_{1}$	651.0423	$t_{1}$	649.9361	$ξ_{2}$	223.7714
$η_{2}$	652.8800	$t_{2}$	651.7722	$ξ_{3}$	223.7713
$η_{3}$	651.2254	$t_{3}$	650.1190	$ξ_{4}$	223.7726
$η_{4}$	651.1360	$t_{4}$	650.0297	$ξ_{5}$	223.7737
$η_{5}$	651.0520	$t_{5}$	650.0342	$ξ_{6}$	223.7833
$η_{6}$	652.0282	$t_{6}$	650.9211	$ξ_{7}$	223.7767
$η_{7}$	651.0925	$t_{7}$	649.9863	$ξ_{8}$	223.7713
$η_{8}$	651.2341	$t_{8}$	650.1278	$ξ_{9}$	223.7716
$η_{9}$	651.0473	$t_{9}$	649.9411	$ξ_{10}$	223.7714
$η_{10}$	652.9680	$t_{10}$	651.8601	$ξ_{11}$	223.7719

Table 6 MSE of existing and proposed estimators for Population 2.

$η$ - family	MSE	$t$ - family	MSE	$ξ$ - family	MSE
$η_{0}$	9.5618	$t_{0}$	2.3479	$ξ_{1}$	1.6534
$η_{1}$	2.3479	$t_{1}$	2.3312	$ξ_{2}$	1.6565
$η_{2}$	2.4578	$t_{2}$	2.4422	$ξ_{3}$	1.6535
$η_{3}$	2.4148	$t_{3}$	2.3987	$ξ_{4}$	1.6583
$η_{4}$	2.4038	$t_{4}$	2.3877	$ξ_{5}$	1.6586
$η_{5}$	2.3854	$t_{5}$	2.3935	$ξ_{6}$	1.6597
$η_{6}$	2.4492	$t_{6}$	2.4335	$ξ_{7}$	1.6574
$η_{7}$	2.3995	$t_{7}$	2.3833	$ξ_{8}$	1.6535
$η_{8}$	2.4216	$t_{8}$	2.4057	$ξ_{9}$	1.6562
$η_{9}$	2.3793	$t_{9}$	2.3629	$ξ_{10}$	1.6551
$η_{10}$	2.4691	$t_{10}$	2.4535	$ξ_{11}$	1.6565

Table 7 MSE of existing and proposed estimators for Population 3.

$η$ - family	MSE	$t$ - family	MSE	$ξ$ - family	MSE
$η_{0}$	448.5631	$t_{0}$	109.6657	$ξ_{1}$	16.1653
$η_{1}$	109.6657	$t_{1}$	109.4151	$ξ_{2}$	16.1696
$η_{2}$	120.1576	$t_{2}$	119.8725	$ξ_{3}$	16.1658
$η_{3}$	111.7679	$t_{3}$	111.5113	$ξ_{4}$	16.1853
$η_{4}$	111.7048	$t_{4}$	111.4484	$ξ_{5}$	16.1855
$η_{5}$	110.0760	$t_{5}$	111.4860	$ξ_{6}$	16.2048
$η_{6}$	120.0335	$t_{6}$	119.7489	$ξ_{7}$	16.1859
$η_{7}$	111.6803	$t_{7}$	111.4239	$ξ_{8}$	16.1655
$η_{8}$	111.8068	$t_{8}$	111.5501	$ξ_{9}$	16.1746
$η_{9}$	110.0636	$t_{9}$	109.8119	$ξ_{10}$	16.1672
$η_{10}$	120.3483	$t_{10}$	120.0624	$ξ_{11}$	16.1747

Table 8 MSE of existing and proposed estimators for Population 4.

$η$ - family	MSE	$t$ - family	MSE	$ξ$ - family	MSE
$η_{0}$	5,411,348	$t_{0}$	3,758,095	$ξ_{1}$	1,246,075
$η_{1}$	3,758,095	$t_{1}$	2,423,766	$ξ_{2}$	1,246,075
$η_{2}$	3,759,902	$t_{2}$	2,424,194	$ξ_{3}$	1,246,075
$η_{3}$	3,758,205	$t_{3}$	2,423,792	$ξ_{4}$	1,246,075
$η_{4}$	3,758,140	$t_{4}$	2,423,777	$ξ_{5}$	1,246,075
$η_{5}$	3,758,098	$t_{5}$	2,423,779	$ξ_{6}$	1,246,070
$η_{6}$	3,758,956	$t_{6}$	2,423,970	$ξ_{7}$	1,246,073
$η_{7}$	3,758,117	$t_{7}$	2,423,771	$ξ_{8}$	1,246,075
$η_{8}$	3,758,223	$t_{8}$	2,423,797	$ξ_{9}$	1,246,075
$η_{9}$	3,758,097	$t_{9}$	2,423,767	$ξ_{10}$	1,246,075
$η_{10}$	3,760,195	$t_{10}$	2,424,264	$ξ_{11}$	1,246,075

Table 9 MSE of existing and proposed estimators for Population 5.

$η$ - family	MSE	$t$ - family	MSE	$ξ$ - family	MSE
$η_{0}$	35,205,782	$t_{0}$	17,380,652	$ξ_{1}$	6,427,017
$η_{1}$	17,380,652	$t_{1}$	15,693,026	$ξ_{2}$	6,427,018
$η_{2}$	17,385,543	$t_{2}$	15,696,853	$ξ_{3}$	6,427,017
$η_{3}$	17,381,067	$t_{3}$	15,693,351	$ξ_{4}$	6,427,033
$η_{4}$	17,380,820	$t_{4}$	15,693,158	$ξ_{5}$	6,427,053
$η_{5}$	17,380,668	$t_{5}$	15,693,173	$ξ_{6}$	6,427,246
$η_{6}$	17,382,855	$t_{6}$	15,694,750	$ξ_{7}$	6,427,116
$η_{7}$	17,380,728	$t_{7}$	15,693,085	$ξ_{8}$	6,427,017
$η_{8}$	17,381,113	$t_{8}$	15,693,387	$ξ_{9}$	6,427,020
$η_{9}$	17,380,658	$t_{9}$	15,693,031	$ξ_{10}$	6,427,017
$η_{10}$	17,386,080	$t_{10}$	15,697,273	$ξ_{11}$	6,427,024

5 Concluding remarks

In this article, we proposed a general family of exponential-type estimators for finite population mean under simple random sampling. Known information of auxiliary variable including coefficient of skewness, coefficient of kurtosis, standard deviation and coefficient of variation etc. is utilized to enhance the estimation. We derived the expressions for bias, mean squared error (MSE) and minimum MSE of the proposed estimators up to first degree of approximation. A numerical study is conducted through five real data sets to highlight the applicability of proposed estimators. This study concludes that all the proposed estimators are i) more efficient over the competing estimators and ii) are equally efficient, so anyone of them can be used depending upon the availability of the auxiliary information ( $α$ and $β$ ). Therefore, the practitioners are suggested to use the proposed estimators to get more efficient results in future applications.

References

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Cochran W.G., . The estimation of the yields of cereal experiments by sampling for the ratio gain to total produce. J. Agric. Sci.. 1940;30:262-275.
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Cochran W.G., . Sampling Techniques (3 edn). New York: John Wiley and Sons; 1977.
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Kadilar C., Cingi H., . Ratio estimators in simple random sampling. Appl. Math. Comput.. 2004;151:893-902.
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Kadilar C., Cingi H., . New ratio estimators using correlation coefficient. Interstat 2006:1-11.
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Khan S.A., Ali H., Manzoor S., Alamgir, . A class of transformed efficient ratio estimators of finite population mean. Pakistan J. Statist.. 2015;31(4):353-362.
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Murthy M.N., . Product method of estimation. Sankhya. 1964;26:294-307.
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Robson D.S., . Application of multivariate polykays to the theory of unbiased ratio type estimation. J. Am. Stat. Assoc.. 1957;52:411-422.
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Singh R., Chauhan P., Sawan N., . On linear combination of ratio and product type exponential estimator for estimating the finite population mean. Statist. Transition. 2008;9(1):105-115.
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Singh R., Chauhan P., Sawan N., Smarandache F., . Improvement in estimating the population mean using exponential estimator in simple random sampling. Bull. Statist. Econ.. 2009;3:13-18.
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Singh H.P., Tailor R., . Use of known correlation coefficient in estimating the finite population means. Statist. Transition. 2003;6(4):555-560.
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Srisodaphol W., Kingphai K., Tanjai N., . New ratio estimators of a population mean using one auxiliary variable in simple random sampling. Chiang Mai J. Sci.. 2015;42(2):523-527.
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Upadhyaya L.N., Singh H.P., Chatterjee S., Yadav R., . Improved ratio and product exponential type estimators. J. Statist. Theory Pract.. 2011;5(2):285-302.
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Watson D.J., . The estimation of leaf area in field crops. J. Agric. Sci.. 1937;27:474-483.
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Yan Z., Tian B., . Ratio method to the mean estimation using coefficient of skewness of auxiliary variable. ICICA. Part II, CCIS. 2010;106:103-110.
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Yadav S.K., Kadilar C., . Efficient family of exponential estimators for the population mean. Hacettepe J. Math. Statist.. 2013;42(6):671-677.
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Introduction

Existing estimators

The suggested family of estimators

Bias, MSE and minimum MSE of ξ

Empirical study

Concluding remarks

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[1] Bahl S., Tuteja R.K., . Ratio and product type exponential estimator. Inf. Optim. Sci.. 1991;XII(1):159-163.
[Google Scholar]

[2] Cochran W.G., . The estimation of the yields of cereal experiments by sampling for the ratio gain to total produce. J. Agric. Sci.. 1940;30:262-275.
[Google Scholar]

[3] Cochran W.G., . Sampling Techniques (3 edn). New York: John Wiley and Sons; 1977.

[4] Kadilar G.O., . A new exponential type estimator for the population mean in simple random sampling. J. Mod. Appl. Stat. Methods. 2016;15(2):207-214.
[Google Scholar]

[5] Kadilar C., Cingi H., . Ratio estimators in stratified random sampling. Biometrical J.. 2003;45(2):218-225.
[Google Scholar]

[6] Kadilar C., Cingi H., . Ratio estimators in simple random sampling. Appl. Math. Comput.. 2004;151:893-902.
[Google Scholar]

[7] Kadilar C., Cingi H., . An improvement in estimating the population mean by using the correlation coefficient. Hacettepe J. Math. Stat.. 2006;35(1):103-109.
[Google Scholar]

[8] Kadilar C., Cingi H., . New ratio estimators using correlation coefficient. Interstat 2006:1-11.
[Google Scholar]

[9] Kadilar C., Cingi H., . Improvement in estimating the population mean in simple random sampling. Appl. Math. Lett.. 2006;19:75-79.
[Google Scholar]

[10] Khan S.A., Ali H., Manzoor S., Alamgir, . A class of transformed efficient ratio estimators of finite population mean. Pakistan J. Statist.. 2015;31(4):353-362.
[Google Scholar]

[11] Khoshnevisan M., Singh R., Chauhan P., Sawan N., Smarandache F., . A general family of estimators for estimating population mean using known value of some population parameter(s) Far East J. Theor. Stat.. 2007;22(2):181-191.
[Google Scholar]

[12] Koyuncu N., Kadilar C., . Efficient estimators for the population mean. Hacettepe J. Math. Statist.. 2009;38(2):217-225.
[Google Scholar]

[13] Murthy M.N., . Product method of estimation. Sankhya. 1964;26:294-307.
[Google Scholar]

[14] Robson D.S., . Application of multivariate polykays to the theory of unbiased ratio type estimation. J. Am. Stat. Assoc.. 1957;52:411-422.
[Google Scholar]

[15] Singh R., Chauhan P., Sawan N., . On linear combination of ratio and product type exponential estimator for estimating the finite population mean. Statist. Transition. 2008;9(1):105-115.
[Google Scholar]

[16] Singh R., Chauhan P., Sawan N., Smarandache F., . Improvement in estimating the population mean using exponential estimator in simple random sampling. Bull. Statist. Econ.. 2009;3:13-18.
[Google Scholar]

[17] Singh H.P., Tailor R., . Use of known correlation coefficient in estimating the finite population means. Statist. Transition. 2003;6(4):555-560.
[Google Scholar]

[18] Srisodaphol W., Kingphai K., Tanjai N., . New ratio estimators of a population mean using one auxiliary variable in simple random sampling. Chiang Mai J. Sci.. 2015;42(2):523-527.
[Google Scholar]

[19] Upadhyaya L.N., Singh H.P., Chatterjee S., Yadav R., . Improved ratio and product exponential type estimators. J. Statist. Theory Pract.. 2011;5(2):285-302.
[Google Scholar]

[20] Watson D.J., . The estimation of leaf area in field crops. J. Agric. Sci.. 1937;27:474-483.
[Google Scholar]

[21] Yan Z., Tian B., . Ratio method to the mean estimation using coefficient of skewness of auxiliary variable. ICICA. Part II, CCIS. 2010;106:103-110.
[Google Scholar]

[22] Yadav S.K., Kadilar C., . Efficient family of exponential estimators for the population mean. Hacettepe J. Math. Statist.. 2013;42(6):671-677.
[Google Scholar]

Enhanced estimation of population mean in the presence of auxiliary information

Abstract

Keywords

Auxiliary variable

Bias

Efficiency

Exponential type estimators

Mean squared error

Simple random sampling

1 Introduction

2 Existing estimators

3 The suggested family of estimators

3.1 Bias, MSE and minimum MSE of ξ

4 Empirical study

Population 1 Source: Koyuncu and Kadilar, 2009

Population 2 Source: Srisodaphol et al., 2015

Population 3 Source: Cochran, 1977

Population 4 Source: Kadilar and Cingi, 2004

Population 5 Source: Kadilar and Cingi, 2003

5 Concluding remarks

References

Suggested read for related articles:

3.1 Bias, MSE and minimum MSE of $ξ$