Generalized Estimator of Population Mean Using Auxiliary Information in Presence of Measurement Errors

Peeyush Misra

Department of Statistics, D.A.V.(P.G.) College, Dehradun – 248001, Uttarakhand, India
E-mail: dr.pmisra.dav@gmail.com

Received 07 March 2023; Accepted 02 October 2023; Publication 23 January 2024

Abstract

It is assumed in survey research that the respondent’s reported response is precise. More often, due to prestige bias, the data provided by respondents frequently include estimates that are significantly different from the genuine values. As a consequence, measurement error is present in the sample estimates that may affect the results. Therefore, this study illustrates an improved generalized estimator that utilizes auxiliary data under measurement error. A numerical study to establish its effectiveness is also conducted.

Keywords: Auxiliary variable, bias, mean squared error, efficiency and measurement errors.

1 Introduction

Many survey scientists have focused on the issue of parameter estimation in the face of measurement errors. The characteristics of estimators based on data in survey sampling normally presumes that the observations of the attributes being researched is accurate. This premise is not always met in practise, as measurement problems such non-response errors, recording errors, and calculation errors pollute data leading to invalid results. The statistical conclusions drawn from the observed data remain valid if measurement errors are negligibly small and can be disregarded. On the other hand, if they are not comparatively small and inconsequential, the deductions might not only be inaccurate and invalid but frequently have unintended, regrettable, and unfortunate results. For more details one may refer Cochran, W.G. (1968), Lessler Judith. T. and Kalsbeek, William, D. (1992), Paul P. Biemer, Robert M. Groves, Lars E. Lyberg, Nancy A. Mathiowetz and Seymour Sudman (1991), Sukhatme, P. V. and G.R. Seth (1952) etc. Numerous statisticians have addressed the issue of determining population mean when measurement errors are present, including Shalabh (1997), Singh and Karpe (2009), Misra and Yadav (2015), Manisha and Singh R. K (2002).

Let $U=U_1,U_2,\mathrm{\dots },U_N$ be a finite population of N distinct and identifiable units with $Y$ being the study variable and $X$ being the auxiliary variable taking the value Y${}_{\mathrm{i}}$ and X${}_{\mathrm{i}}$ for the unit $i$th of the population $U$ respectively. Additionally, suppose that n paired observations of characteristics X and Y were gathered using a basic random sampling process without replacement. Further, for size $n$ simple random sample, let $(x_i,y_i)$ be the observed values rather than true values $(X_i,Y_i)$ for the ith $(i=1,2,\mathrm{\dots }n)$ sampling unit in the sample. Also, $u_i=y_i-Y_i$ and $v_i=x_i-X_i$ where $u_i$ and $v_i$ are related measurement errors that are stochastic in character with mean zero and variance ${\sigma }_u^2$ and ${\sigma }_v^2$ respectively. Additionally, assume that $u_i^{\prime }s$ and $v_i^{\prime }s$ are uncorrelated while $X_i^{\prime }s$ and $Y_i^{\prime }s$ are correlated. Let ${\mu }_X,{\mu }_Y,{\sigma }_X^2,{\sigma }_Y^2$ and $\rho$ represents the population mean, variances and correlation coefficient between $X$ and $Y$.

Let $\widehat{x}=\frac{1}{n}{\sum }_{i=1}^n{x}_i$&$\widehat{y}=\frac{1}{n}{\sum }_{i=1}^n{y}_i$ denotes the unbiased estimators of populations means ${\mu }_X$ and ${\mu }_Y$ i.e. $E(\widehat{x})={\mu }_X$ and $E(\widehat{y})={\mu }_Y$.

But in presence of measurement errors, $s_X^2=\frac{1}{n-1}{\sum }_{i=1}^n{(x_i-\widehat{x})}^2$ and $s_Y^2=\frac{1}{n-1}{\sum }_{i=1}^n{(y_i-\widehat{y})}^2$ are biased estimators of the population variances ${\sigma }_X^2$ and ${\sigma }_Y^2$. $E(s_Y^2)={\sigma }_Y^2+{\sigma }_u^2$ provides the expected value of $s_y^2$, in the presence of measurement errors.

In the event where error variances ${\sigma }_u^2$ and ${\sigma }_v^2$ are known in advance, then ${\widehat{\sigma }}_Y^2=s_y^2-{\sigma }_u^2>0$ and ${\widehat{\sigma }}_X^2=s_x^2-{\sigma }_v^2>0$ are unbiased estimators of population variances under measurement errors.

Further let,

$$C_Y=\frac{{\sigma }_Y}{{\mu }_Y}\mathrm{and}{C}_X=\frac{{\sigma }_X}{{\mu }_X},{\gamma }_{2Y}={\beta }_{2Y}-3,$$

$${\gamma }_{2X}={\beta }_{2X}-3,{\gamma }_{2u}={\beta }_{2u}-3,{\gamma }_{2v}={\beta }_{2V}-3.$$

$${\gamma }_{1(X)}=\sqrt{{\beta }_1(X)}$$

$${\beta }_2(X)=\frac{{\mu }_4(X)}{{\mu }_2^2(X)},$$

$${\beta }_2(u)=\frac{{\mu }_4(u)}{{\mu }_2^2(u)},$$

$${\beta }_2(v)=\frac{{\mu }_4(v)}{{\mu }_2^2(v)},$$

$${\mu }_{qrst}=E[{(X-{\mu }_X)}^q{(Y-{\mu }_Y)}^r{v}^s{u}^t]$$

$${\mu }_{2000}={\sigma }_X^2,{\mu }_{0200}={\sigma }_Y^2,{\mu }_{0020}={\sigma }_v^2\mathrm{and}{\mu }_{0002}={\sigma }_u^2.$$

Now suppose that sample values are collected with measurement errors i.e. the observed values (x${}_{\mathrm{i}}$,y${}_{\mathrm{i}}$) as different from the values (X${}_{\mathrm{i}}$,Y${}_{\mathrm{i}}$). It is suggested to use the following generalised estimator to estimate population mean when measurement errors are present.

$${\widehat{\overline{y}}}_{g_{\mathit{ME}}}=g(\overline{y},b,\overline{x},{\widehat{\sigma }}_x^2)$$

(1)

where ‘b’ is defined as an estimate of the change in y caused by a unity increase in x and $g(\overline{y},b,\overline{x},{\widehat{\sigma }}_x^2)$ is bounded in such a way that at the point $({\mu }_Y,\beta ,{\mu }_X,{\sigma }_X^2)$, we have

$$g({\mu }_Y,\beta ,{\mu }_X,{\sigma }_X^2)={\mu }_Y$$	(2)
$$g_0=\frac{\partial }{\partial \overline{y}}g(\overline{y},b,\overline{x},{\widehat{\sigma }}_x^2)]{}_{({\mu }_Y,\beta ,{\mu }_X,{\sigma }_X^2)}=1$$	(3)
$$g_1=\frac{\partial }{\partial b}g(\overline{y},b,\overline{x},{\widehat{\sigma }}_x^2)]{}_{({\mu }_Y,\beta ,{\mu }_X,{\sigma }_X^2)}=0$$	(4)
$$g_2=\frac{\partial }{\partial \overline{x}}g(\overline{y},b,\overline{x},{\widehat{\sigma }}_x^2)]{}_{({\mu }_Y,\beta ,{\mu }_X,{\sigma }_X^2)}=-\beta$$	(5)
$$g_{00}=\frac{\partial }{\partial {\overline{y}}^2}g(\overline{y},b,\overline{x},{\widehat{\sigma }}_x^2)]{}_{({\mu }_Y,\beta ,{\mu }_X,{\sigma }_X^2)}=0$$	(6)
$$g_{22}=\frac{\partial }{\partial {\overline{x}}^2}g(\overline{y},b,\overline{x},{\widehat{\sigma }}_x^2)]{}_{({\mu }_Y,\beta ,{\mu }_X,{\sigma }_X^2)}=0$$	(7)

2 Bias and Mean Squared Error

Here we consider the approximations as

$$\widehat{y}={\mu }_Y(1+e_0)$$
$$\widehat{x}={\mu }_X(1+e_1)$$
$${\widehat{\sigma }}_y^2={\sigma }_Y^2(1+e_2)$$
$${\widehat{\sigma }}_x^2={\sigma }_X^2(1+e_3)$$
$${\widehat{\sigma }}_{xy}={\sigma }_{XY}(1+e_4)$$
$$\text{so that}E(e_i)=0,i=1,2,3,4$$	(8)

Using the results from Singh and Karpe (2009), we have

$$E(e_0^2)=\frac{C_Y^2}{n{\theta }_Y}\mathit{\hspace{1em}}\mathrm{and}\mathit{\hspace{1em}}E(e_1^2)=\frac{C_X^2}{n{\theta }_X}$$

(9)

where, ${\theta }_Y=\frac{{\sigma }_Y^2}{{\sigma }_Y^2+{\sigma }_u^2}$ and ${\theta }_X=\frac{{\sigma }_X^2}{{\sigma }_X^2+{\sigma }_v^2}$

$$E(e_3^2)=\frac{A_X}{n},\mathrm{where}{A}_X={\gamma }_{2X}+{\gamma }_{2\nu }\frac{{\sigma }_{\nu }^4}{{\sigma }_X^4}+2{\left(1+\frac{{\sigma }_{\nu }^2}{{\sigma }_X^2}\right)}^2$$	(10)
$$\mathrm{E}({\mathrm{e}}_2^2)=\frac{{\mathrm{A}}_Y}{\mathrm{n}},\mathrm{where}{\mathrm{A}}_{\mathrm{Y}}={\gamma }_{2\mathrm{Y}}+{\gamma }_{2\mathrm{u}}\frac{{\sigma }_u^4}{{\sigma }_{\mathrm{Y}}^4}+2{\left(1+\frac{{\sigma }_{\mathrm{u}}^2}{{\sigma }_{\mathrm{Y}}^2}\right)}^2$$	(11)
$$\mathrm{E}({\mathrm{e}}_1{\mathrm{e}}_3)=\frac{{\mu }_{3000}}{\mathrm{n}{\sigma }_{\mathrm{X}}^2{\mu }_{\mathrm{X}}}$$	(12)
$$\mathrm{E}({\mathrm{e}}_0{\mathrm{e}}_2)=\frac{{\mu }_{0300}}{\mathrm{n}{\sigma }_{\mathrm{Y}}^2{\mu }_{\mathrm{Y}}}$$	(13)
$$E(e_0{e}_3)=\frac{{\mu }_{2100}}{n{\sigma }_X^2{\mu }_Y}$$	(14)
$$\mathrm{E}({\mathrm{e}}_1{\mathrm{e}}_2)=\frac{{\mu }_{1200}}{\mathrm{n}{\sigma }_{\mathrm{Y}}^2{\mu }_{\mathrm{X}}}$$	(15)
$$\mathrm{E}({\mathrm{e}}_0{\mathrm{e}}_1)=\frac{{\sigma }_{\mathrm{XY}}}{\mathrm{n}{\mu }_{\mathrm{X}}{\mu }_{\mathrm{Y}}}=\frac{\rho {\mathrm{C}}_{\mathrm{X}}{\mathrm{C}}_{\mathrm{Y}}}{\mathrm{n}}$$	(16)
$$\mathrm{E}({\mathrm{e}}_1{\mathrm{e}}_4)=\frac{{\mu }_{2100}}{\mathrm{n}{\sigma }_{\mathrm{XY}}{\mu }_{\mathrm{X}}}$$	(17)
$$E(e_3{e}_4)=\frac{{\mu }_{3100}}{n{\sigma }_X^2{\sigma }_{XY}}$$	(18)
$$E(e_2{e}_3)=\frac{\delta -1}{n},\mathrm{where}\delta =\frac{{\mu }_{2200}}{{\sigma }_X^2{\sigma }_Y^2}$$	(19)
$$E(e_2{e}_4)=\frac{{\mu }_{1200}}{n{\sigma }_Y^2{\sigma }_{XY}}$$	(20)

Using Taylor’s Series expansion, we now expand $g(\overline{y},b,\overline{x},{\widehat{\sigma }}_x^2)$ as

${\widehat{\overline{y}}}_{g_{\mathit{ME}}}$	$=g({\mu }_Y,\beta ,{\mu }_X,{\sigma }_X^2)+(\overline{y}-{\mu }_Y)g_0+(b-\beta )g_1+(\overline{x}-{\mu }_X)g_2$
	$\mathrm{\hspace{1em}}+({\widehat{\sigma }}_x^2-{\sigma }_X^2)g_3+{\displaystyle \frac{1}{2}}\{{(\overline{y}-{\mu }_Y)}^2{g}_{00}+{(b-\beta )}^2{g}_{11}+{(\overline{x}-{\mu }_X)}^2{g}_{22}$
	$\mathrm{\hspace{1em}}+{({\widehat{\sigma }}_x^2-{\sigma }_X^2)}^2{g}_{33}+2(\overline{y}-{\mu }_Y)(b-\beta )g_{01}$
	$\mathrm{\hspace{1em}}+2(\overline{y}-{\mu }_Y)(\overline{x}-{\mu }_X)g_{02}+2(\overline{y}-{\mu }_Y)({\widehat{\sigma }}_x^2-{\sigma }_X^2)g_{03}$
	$\mathrm{\hspace{1em}}+2(b-\beta )(\overline{x}-{\mu }_X)g_{12}+2(b-\beta )({\widehat{\sigma }}_x^2-{\sigma }_X^2)g_{13}$
	$\mathrm{\hspace{1em}}+2(\overline{X}-{\mu }_X)({\widehat{\sigma }}_x^2-{\sigma }_X^2)g_{23}\}$
	$\mathrm{\hspace{1em}}+{\displaystyle \frac{1}{3!}}\{(\overline{y}-{\mu }_Y){\displaystyle \frac{\partial }{\partial \overline{y}}}+(b-\beta ){\displaystyle \frac{\partial }{\partial b}}+(\overline{x}-{\mu }_X){\displaystyle \frac{\partial }{\partial \overline{x}}}$
	$\mathrm{\hspace{1em}}+({\widehat{\sigma }}_x^2-{\sigma }_X^2){\displaystyle \frac{\partial }{\partial {\widehat{\sigma }}_x^2}}\}{}^{3}g({\overline{y}}^{*},b^{*},{\overline{x}}^{*},{\widehat{\sigma }}_x^{2*})$	(21)

where $g_0,g_1,g_2,g_{00}\&g_{22}$ are already defined and

$$g_3=\frac{\partial }{\partial {\widehat{\sigma }}_x^2}g(\overline{y},b,\overline{x},{\widehat{\sigma }}_x^2)]{}_{({\mu }_Y,\beta ,{\mu }_X,{\sigma }_X^2)}$$

$$g_{02}=\frac{\partial }{\partial \overline{y}\partial \overline{x}}g(\overline{y},b,\overline{x},{\widehat{\sigma }}_x^2)]{}_{({\mu }_Y,\beta ,{\mu }_X,{\sigma }_X^2)}$$

$$g_{12}=\frac{\partial }{\partial b\partial \overline{x}}g(\overline{y},b,\overline{x},{\widehat{\sigma }}_x^2)]{}_{({\mu }_Y,\beta ,{\mu }_X,{\sigma }_X^2)}$$

$$g_{23}=\frac{\partial }{\partial \overline{x}\partial {\widehat{\sigma }}_x^2}g(\overline{y},b,\overline{x},{\widehat{\sigma }}_x^2)]{}_{({\mu }_Y,\beta ,{\mu }_X,{\sigma }_X^2)}$$

$$g_{01}=\frac{\partial }{\partial \overline{y}\partial b}g(\overline{y},b,\overline{x},{\widehat{\sigma }}_x^2)]{}_{({\mu }_Y,\beta ,{\mu }_X,{\sigma }_X^2)}$$

$$g_{03}=\frac{\partial }{\partial \overline{y}\partial {\sigma }_x^2}g(\overline{y},b,\overline{x},{\widehat{\sigma }}_x^2)]{}_{({\mu }_Y,\beta ,{\mu }_X,{\sigma }_X^2)}$$

$$g_{13}=\frac{\partial }{\partial b\partial {\widehat{\sigma }}_x^2}g(\overline{y},b,\overline{x},{\widehat{\sigma }}_x^2)]{}_{({\mu }_Y,\beta ,{\mu }_X,{\sigma }_X^2)}$$

$$\overline{y}*={\mu }_Y+h(\overline{y}-{\mu }_Y)$$

$$b*=\beta +h(b-\beta )$$

$$\overline{x}*={\mu }_X+h(\overline{x}-{\mu }_X)$$

$${\widehat{\sigma }}_x^2*={\sigma }_X^2+h({\widehat{\sigma }}_x^2-{\sigma }_X^2)\mathit{\hspace{1em}}\mathrm{for}\mathrm{\hspace{0.25em}0}<\mathrm{h}<1.$$

Under the above mentioned conditions given from (2) to (7), the expression (2) in terms of $e_i$’s reduces to,

${\widehat{\overline{y}}}_{g_{\mathit{ME}}}$	$={\mu }_Y+{\mu }_Y{e}_0+{\mu }_X{e}_1(-\beta )+{\sigma }_X^2{e}_3{g}_3$
	$\mathrm{\hspace{1em}}+{\displaystyle \frac{1}{2}}\{{\sigma }_X^4{e}_3^2{g}_{33}+2{\mu }_Y{e}_0{\sigma }_X^2{e}_3{g}_{03}$
	$\mathrm{\hspace{1em}}+2\beta (e_4-e_3+e_3^2-e_3{e}_4){\mu }_X{e}_1{g}_{12}$
	$\mathrm{\hspace{1em}}+2\beta (e_4-e_3+e_3^2-e_3{e}_4){\sigma }_X^2{e}_3{g}_{13}+2{\mu }_X{e}_1{\sigma }_X^2{e}_3{g}_{23}\}$

or

${\widehat{\overline{y}}}_{g_{\mathit{ME}}}-{\mu }_Y$	$={\mu }_Y{e}_0-\beta {\mu }_X{e}_1+{\sigma }_X^2{e}_3{g}_3$
	$\mathrm{\hspace{1em}}+{\displaystyle \frac{1}{2}}\{{\sigma }_X^4{e}_3^2{g}_{33}+2{\mu }_Y{\sigma }_X^2{e}_0{e}_3{g}_{03}+2\beta {\mu }_X(e_1{e}_4-e_1{e}_3)g_{12}$
	$\mathrm{\hspace{1em}}+2\beta {\sigma }_X^2(e_3{e}_4-e_3^2)g_{13}+2{\mu }_X{\sigma }_X^2{e}_1{e}_3{g}_{23}\}$	(22)

The expression of bias is now derived as

$\mathit{Bias}({\widehat{\overline{y}}}_{g_{\mathit{ME}}})$	$=E({\widehat{\overline{y}}}_{g_{\mathit{ME}}}-{\mu }_Y)$
	$={\displaystyle \frac{1}{2n}}\{{\sigma }_X^4{A}_X{g}_{33}+2{\mu }_Y{\sigma }_X^2{\displaystyle \frac{{\mu }_{2100}}{{\sigma }_X^2{\mu }_Y}}{g}_{03}$
	$\mathrm{\hspace{1em}}+2\beta {\mu }_X\left({\displaystyle \frac{{\mu }_{2100}}{{\sigma }_{XY}{\mu }_X}}-{\displaystyle \frac{{\mu }_{3000}}{{\sigma }_X^2{\mu }_X}}\right)g_{12}$
	$\mathrm{\hspace{1em}}+2\beta {\sigma }_X^2({\displaystyle \frac{{\mu }_{31}}{{\sigma }_X^2{\sigma }_{XY}}}-A_X)g_{13}+2{\mu }_X{\sigma }_X^2{\displaystyle \frac{{\mu }_{3000}}{{\sigma }_X^2{\mu }_X}}{g}_{23}\}$

Now on squaring (22) on both the sides and approximating to the first degree, we have

$MSE({\widehat{\overline{y}}}_{g_{\mathit{ME}}})$	$=E{({\widehat{\overline{y}}}_{g_{\mathit{ME}}}-{\mu }_Y)}^2$
	$=E{({\mu }_Y{e}_0-\beta {\mu }_X{e}_1+{\sigma }_X^2{e}_3{g}_3)}^2$
	$=E({\mu }_Y^2{e}_0^2+{\displaystyle \frac{{\sigma }_{XY}^2}{{\sigma }_X^4}}{\mu }_X^2{e}_1^2+{\sigma }_X^4{e}_3^2{g}_3^2$
	$\mathrm{\hspace{1em}}-2{\mu }_Y{\displaystyle \frac{{\sigma }_{XY}}{{\sigma }_X^2}}{\mu }_X{e}_0{e}_1+2{\mu }_Y{\sigma }_X^2{e}_0{e}_3{g}_3$
	$\mathrm{\hspace{1em}}-2{\displaystyle \frac{{\sigma }_{XY}}{{\sigma }_X^2}}{\mu }_X{\sigma }_X^2{e}_1{e}_3{g}_3)$

on using values of the expectations given from (8) to (20), the above expression becomes

$\mathit{MSE}({\widehat{\overline{y}}}_{g_{\mathit{ME}}})$	$=(1-{\rho }^2){\displaystyle \frac{{\sigma }_Y^2}{n}}+{\displaystyle \frac{1}{n}}\left({\sigma }_u^2+{\rho }^2{\sigma }_Y^2{\displaystyle \frac{{\sigma }_v^2}{{\sigma }_X^2}}\right)$
	$\mathrm{\hspace{1em}}+\left({\sigma }_X^4{\displaystyle \frac{A_X}{n}}{g}_3^2+2{\displaystyle \frac{{\mu }_{2100}}{n}}{g}_3-2{\displaystyle \frac{{\sigma }_{XY}}{{\sigma }_X^{2}n}}{\mu }_{3000}{g}_3\right)$	(24)

Now for optimizing (2) w.r.t g${}_3$, the optimum value of g${}_3$ is given by

$$g_3=\frac{\left(\frac{{\sigma }_{XY}}{{\sigma }_X^2}{\mu }_{3000}-{\mu }_{2100}\right)}{{\sigma }_X^4{A}_X}.$$

(25)

Therefore, using optimum value of g${}_3$ from (25), we get

$\mathit{MSE}{({\widehat{\overline{y}}}_{g_{\mathit{ME}}})}_{\min }$	$=(1-{\rho }^2){\displaystyle \frac{{\sigma }_Y^2}{n}}+{\displaystyle \frac{1}{n}}\left({\sigma }_u^2+{\rho }^2{\sigma }_Y^2{\displaystyle \frac{{\sigma }_v^2}{{\sigma }_X^2}}\right)$
	$\mathrm{\hspace{1em}}-{{\displaystyle \frac{\left(\frac{{\sigma }_{XY}}{{\sigma }_X^2}{\mu }_{3000}-{\mu }_{2100}\right)}{n{\sigma }_X^4{A}_X}}}^2$	(26)

3 Theoretical Comparison

To establish the effectiveness and superiority of the recommended estimator we now contrast it to the usual estimator of mean when measurement errors are present.

$${\overline{y}}_m=\frac{1}{n}\sum _{E_i}^n{y}_i$$

(27)

Expressing above in terms of e${}_{\mathrm{i}}$’s, we have

$${\overline{y}}_m={\mu }_Y(1+e_0)$$

$${\overline{y}}_m-{\mu }_Y={\mu }_Y{e}_0$$

Therefore

$$\mathit{Bias}({\overline{y}}_m)=0$$

(28)

& From Salabh (1997), we have

$$\mathit{MSE}(\overline{y}{}_m)=\frac{{\sigma }_Y^2}{n}(1+\frac{{\sigma }_u^2}{{\sigma }_Y^2})$$

(29)

If there exists measurement error, the proposed estimator ${\widehat{\overline{y}}}_{g_{\mathit{ME}}}$ will now be more effective than the conventional estimate of mean if

$$\mathit{MSE}(\overline{y}{}_m)-\mathit{MSE}{({\widehat{\overline{y}}}_{g_{\mathit{ME}}})}_{\min }>0$$

i.e.

${\displaystyle \frac{{\sigma }_Y^2}{n}}\left(1+{\displaystyle \frac{{\sigma }_u^2}{{\sigma }_Y^2}}\right)-(1-{\rho }^2){\displaystyle \frac{{\sigma }_Y^2}{n}}-{\displaystyle \frac{1}{n}}\left({\sigma }_u^2+{\rho }^2{\sigma }_Y^2{\displaystyle \frac{{\sigma }_v^2}{{\sigma }_X^2}}\right)$

$\mathrm{\hspace{1em}\hspace{1em}}+{{\displaystyle \frac{\left(\frac{{\sigma }_{XY}}{{\sigma }_X^2}{\mu }_{3000}-{\mu }_{2100}\right)}{n{\sigma }_X^4{A}_X}}}^2>0$

$\mathrm{or}h>{\rho }^2{\displaystyle \frac{{\sigma }_Y^2}{n}}\left({\displaystyle \frac{{\sigma }_v^2}{{\sigma }_X^2}}-1\right)\mathit{\hspace{1em}}\mathrm{where}h={{\displaystyle \frac{\left(\frac{{\sigma }_{XY}}{{\sigma }_X^2}{\mu }_{3000}-{\mu }_{2100}\right)}{n{\sigma }_X^4{A}_X}}}^2.$

Therefore, if condition (3) is met, the suggested estimator ${\widehat{\overline{y}}}_{g_{\mathit{ME}}}$ will be more effective than the already accepted typical estimator of mean in the presence of measurement errors.

4 Empirical Study

Using a known population data set, we compare the effectiveness of the proposed estimator to that of the conventional mean estimator in this section. The population set is described as follows. We have taken the data set for empirical study from Gujrati, Porter and Sangeetha (2012) as

Y${}_{\mathrm{i}}=$ “True Consumption Expenditure”

X${}_{\mathrm{i}}=$ “True Income”

y${}_{\mathrm{i}}=$ “Measured consumption expenditure”

x${}_{\mathrm{i}}=$ “Measure Income”

The population characteristics obtained using above data is as follows

$$n=10,{\mu }_X=170,{\mu }_Y=127,{\sigma }_X^2=3300,{\sigma }_Y^2=1278,{\sigma }_u^2=32.4001,$$

$${\sigma }_v^2=32.3998,C_Y=0.2815,C_X=0.3379,{\rho }_{XY}=0.9641,{\beta }_{2Y}=1.9026,$$

$${\beta }_{2X}=1.7758,{\beta }_{2u}=1.7186,{\beta }_{2v}=1.8409$$

Substituting the above parameters in Equations (2) and (29), the MSE’s of the usual and recommended estimator with measurement errors is given by

$\mathit{MSE}({\widehat{y}}_m)$	$=131.033$
$\mathit{MSE}{({\widehat{\widehat{y}}}_{\mathit{ME}})}_{min}$	$=13.372$

The above results justify the efficiency of the recommended estimator over the usual counterparts.

5 Conclusion

Mean squared error criterion has been used to assess the effectiveness of the estimators in both theoretical and empirical studies. The proposed estimator is contrasted with the standard mean estimator and it is discovered that the proposed estimator is more effective in terms of MSE. Based on the aforementioned MSEs, the suggested estimator’s percent relative efficiency (PRE) over the standard estimate of mean under measurement error is 979, demonstrating its improved efficiency.

Acknowledgement

I would like to extend my sincere thanks to the editor-in- chief and the anonymous referees for their invaluable input, expertise and insights in improving the paper.

References

[1] Cochran, W.G.(1968): Errors of Measurement in Statistics, Technometrics, 10, 637–666.

[2] Gujrati, D.N., Porter, D.C. and Gunasekar Sangeetha (2012): Basic Econometrics, Fifth Edition, McGraw-Hill Education (India) Private Limited, New Delhi.

[3] Lessler Judith. T. and Kalsbeek, William, D. (1992): Non-Sampling Error in Surveys, John Wiley and Sons.

[4] Maneesha and Singh R.K. (2002): Role of regression estimator involving measurement errors, Brazilian Journal of Probability and Statistics, 16, 39–46.

[5] Misra, S. and Yadav, D. K. (2015): Estimating population mean using known coefficient of variation under measurement errors, in the edited book “Statistics and Informatics in Agricultural Research”, edited by Indian Society of Agricultural Statistics (ISAS), Library Avenue, Pusa, New Delhi and published by Excel India Publishers, New Delhi, ISBN 978-93-84869-98-4, pp. 175–182.

[6] Paul P. Biemer, Robert M. Groves, Lars E. Lyberg, Nancy A. Mathiowetz and Seymour Sudman (1991): Measurement Errors in Surveys, New York: Wiley.

[7] Sukhatme, P. V. and G.R. Seth (1952): Non Sampling Errors in Surveys, Journal of the Indian Society of Agricultural Statistics, 4, 5–41.

[8] Shalabh (1997): Ratio Method of Estimation in the Presence of Measurement Errors, Journal of Indian Society of Agricultural Statistics, Vol. I, No. 2, 150–155.

[9] Singh H.P. and Karpe, N. (2009): A General Procedure for Estimating the General Parameter using Auxiliary Information in Presence of Measurement Errors, Communication of the Korean Statistical Society, 16(5), 821–840.

Biography

Peeyush Misra received the bachelor’s degree in Science, the master’s degree in Statistics and Ph.D degree in Statistics from University of Lucknow, Lucknow, Uttar Pradesh, India in 2001, 2004 and 2008, respectively. He is currently working as an Associate Professor at the Department of Statistics, DAV (PG) College, Dehradun, Uttarakhand, India. His field of specialization is sampling Theory.

Journal of Reliability and Statistical Studies, Vol. 16, Issue 2 (2023), 269–280.
doi: 10.13052/jrss0974-8024.1624
© 2024 River Publishers

Abstract

1 Introduction

2 Bias and Mean Squared Error

3 Theoretical Comparison

4 Empirical Study

5 Conclusion

Acknowledgement

References

Biography