Columbia International Publishing Journal of Advanced Computing (2013) 1: 70-78 doi:10.7726/jac.2013.1006

Research Article

______________________________________________________________________________________________________________________________ *Corresponding e-mail: ramkssolanki@gmail.com 1 School of Studies in Statistics, Vikram University, Ujjain-456010, M.P., India

70

Improved Estimation of Finite Population Mean in Sample Surveys

Ramkrishna S. Solanki1*, Housila P. Singh1, and Surya Kant Pal1

Received 1 May 2013; Published online 29 June 2013 © The author(s) 2013. Published with open access at www.uscip.org

Abstract The present paper discusses the problem of estimating the finite population mean in simple random sampling. A class of estimators has been proposed using information on an auxiliary variable and its properties are studied. It has been shown that the proposed class of estimators is more efficient than the usual unbiased, ratio and product estimators. An empirical study has been carried out to examine the merits of the proposed class of estimators over others. Keywords: Ratio estimator; Product estimator; Bias; Mean square error; Simple random sampling.

1. Introduction It is well established in sample surveys that auxiliary information is often used to improve the precision of estimators of population parameters. It is well known that for estimating population

mean Y of the study variable y using information on auxiliary variable x (say population mean X

of x is known in advance) at the estimation stage, the ratio and product methods are extensively used [see Cochran (1977), Upadhyaya and Singh (1999), Singh and Ruiz-Espejo (2003), Singh (2003), Singh and Solanki (2011a, b, 2012a, b, 2013a, b), Solanki et al. (2013)]. The ratio method of estimation which has been developed by Cochran (1940) is used to estimate

the population mean Y of the study variable y which is positively (high) correlated with an

auxiliary variable x . For example

(i) y may be the production of wheat and x is the area cultivated.

(ii) y may denote the number of bullocks on a holding and x its area in acres.

(iii) y may be the total production and x is the number of workers.

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However, when auxiliary variable x is negatively correlated (high) with the study variable y the

product method of estimation envisaged by Robson (1957) and rediscovered by Murthy (1964) can be employed quite effectively. For example

(i) Price and demand of a commodity. (ii) The volume and pressure of a gas.

(iii) Sale of woolen cloth and temperature. Survey statisticians seem more inclined to use the ratio and product estimators due to their stronger intuitive appeal. Perhaps, this is why prevalent work has been done in the direction of improving the performance of these estimators. However, a large number of attempts have produced ratio and product-type estimators which are more efficient than the sample mean estimator and the ancestor ratio and product estimators. In this article we have proposed the class

of estimators of population meanY , which is more efficient than the usual unbiased, ratio and product estimators, by using known information on an auxiliary variable x . Numerical illustrations are also given to through light on the merits of the proposed study. Consider a finite population NUUUU ,...,, 21 of N identifiable units. Let y and x denote the

study variable and auxiliary variable taking values iy and ix respectively on the thi unit iU of the

population U and a sample of size )( Nn is drawn by simple random sampling without

replacement (SRSWOR) from the population U . Let ][1

1

n

i iyny and ][1

1

n

i ixnx denote the

sample mean of the study variable y and auxiliary variable x respectively.

The remaining part of the paper is organized as follows: Sec. 2 give a brief review of usual estimators of the population mean. In Sec. 3, the new class of estimator is described and the expressions of its asymptotic bias and the mean square error are obtained. Sec. 4 addresses the problem of efficiency comparisons, while in Sec. 5 we have optimized the proposed class. Sec. 6 has focused on empirical study of proposed study. Sec. 7 finished off the paper with final remarks.

2. The Traditional Estimators

It is very well known that sample mean y is an unbiased estimator of population mean Y and

under simple random sampling without replacement (SRSWOR) its variance(or mean square error) is given by

222yy CYSyMSEyVar , (2.1)

where fn 11 , Nnf / (sample fraction),

N

i iy YyNS1

212 1 (population mean

square of y ) and 222 /YSC yy (coefficient of variation of y ).

The usual ratio [Cochran (1940)] and product [Robson (1957) and Murthy (1964)] estimators of

population mean Y have been defined respectively as

x

XyyR , (2.2)

Ramkrishna S. Solanki, Housila P. Singh and Surya Kant Pal / Journal of Advanced Computing (2013) 1: 70-78

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X

xyyP , (2.3)

where ,X the population mean of auxiliary variable x is assumed to be known.

To the first degree of approximation the biases and mean square errors (MSEs) of the ratio estimator Ry and the product estimator Py are respectively given by

KCYyBias xR 1)( 2 , (2.4)

KCYyBias xP2)( , (2.5)

KCCYyMSE xyR 21)( 222 , (2.6)

KCCYyMSE xyP 21)( 222 , (2.7)

where XSC xx / (coefficient of variation of x ),

N

i ix XxNS1

212 1 (population mean

square of x ), yxxy SSS / (correlation coefficient between y and x ) ,

N

i iixy YyXxNS1

11 (covariance between x and y ) and xy CCK / .

3. The Suggested Class of Estimators

We have suggested the following class of estimators of population mean Y in simple random sampling as

Xaxa

xaXaxXyyd

1

1̂ , (3.1)

where 2/ˆxxy ss ,

n

i

iixy yyxxns1

11 ,

n

i

ix xxns1

212 1 and a is the suitably

chosen scalar. To obtain the bias and MSE of suggested class of estimators dy we define the following quantities

01 eYy , 11 eXx , ˆ1ˆ , 21 eSs xyxy and 322 1 eSs xx

such that

03210 eEeEeEeE , 1ˆ nOE , 220

1yC

n

feE

, 22

1

1xC

n

feE

,

2

10

1xKC

n

feeE

,

xySXnNN

NNeeE

1

21

1 2121

,

2

3031

1

21

1

xSXnNN

NNeeE

,

where

YyXxEii

11 , YyXxE

ii

2

21 , 330 XxE i , 220 XxE i .

Expressing (3.1) in terms of e’s, we have

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1

1131210 1111

aeeaeeeeKeYyd . (3.2)

We assume that 11 ae , so that 1

11

ae is expandable. Now expanding the right hand side of

(3.2) and neglecting the terms of e’s having power greater than two, we have

3121

2

110102121211 eeeeKeaaeeaeaey

d . (3.3)

or

3121

2

11010212121 eeeeKeaaeeaeaeYYy

d . (3.4)

Taking expectation on both sides of (3.4), we get the bias of proposed class of estimators dy , up to

first degree of approximation as

20

30

11

212

221

X

K

N

NCaaKYyBias xd . (3.5)

Squaring both sides of (3.4) and neglecting terms of e’s having power greater than two, we have

10

2

1

22

0

2221221 eeaeaeYYy

d . (3.6)

Taking expectation on both sides of (3.5), we get the MSE of proposed class of estimators dy , up to

first degree of approximation as

22222 211 KaCCYyMSE xyd . (3.7)

4. Efficiency comparisons In this section we have derived the conditions under which the proposed class of estimators is more efficient than the usual unbiased estimator y , usual ratio estimator Ry and usual product

estimator Py .

From (2.1), (2.6), (2.7) and (3.7), we have

2

)21(,

2

1.max

2

21,

2

1.min..

21 22

Ka

Kifei

KKKif

yMSEyMSE d

, (4.1)

KaKifei

KKaif

yMSEyMSE Rd

,1.max,1.min..

12122

, (4.2)

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KaKifei

KKaif

yMSEyMSE Pd

1,0.max1,0.min..

12122 . (4.3)

5. Optimum choice of the scalar a Differentiating (3.6) with respect to a and equating it to zero, we get the optimum value of a (i.e. the value which is minimize the MSE) as

oaK

a

2

1 . (5.1)

Substituting (5.1) in (3.1), we get the optimum estimator of Y as

XKxK

xKXKxXyy

oa11

11ˆ)( . (5.2)

It is observed from (5.2) that the optimum estimator (OE) )( oay depends on the unknown

parameter K which lacks the practical utility of the OE )( oay . To overcome this difficulty we replace

K by its consistent estimate RK ˆ/ˆˆ based on sample data at hand, where xyR /ˆ . Thus

replacing K by K̂ in (5.2) we get an estimator of population mean Y based on estimated value of K as

XRxR

xRXRxXyy

oa

ˆˆˆˆ

ˆˆˆˆˆ

)ˆ( . (5.3)

It can be easily shown up to first degree of approximation that the MSE of the estimator )ˆ( oay is

given by

222ˆ 1 ya CYyMSEo

. (5.4)

The expression in (5.4) is equal to the MSE of linear regression estimator xXyylr ̂ . Thus

the proposed estimator )ˆ( oay in (5.3) can be used as an alternative to the usual regression estimator

lry in practice.

6. Empirical study

In this subsection, we study the preceding theoretical results empirically on two population data sets I and II. The description of population data sets are summarized in Table 1. Using the

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conditions which we have obtained in section 4, we calculated the ranges of scalar a (see Table 2)

in which proposed class of estimators dy is more efficient than the usual unbiased estimator y ,

usual ratio estimator Ry and usual product estimator Py in the population data sets I and II

described in Table 1. To evaluate the performance of the proposed class of estimators dy over the

other existing estimators, we have computed the percent relative efficiencies (PREs) of dy with

respect to y , Ry and Py in certain range (0.00, 1.75) of a for the population data sets I and II using

the respective formulae

100

211

100,2222

2

KaCC

C

yMSE

yMSEyyPRE

xy

y

d

d

, (6.1)

100

211

21100,

2222

22

KaCC

KCC

yMSE

yMSEyyPRE

xy

xy

d

RRd

, (6.2)

100

211

21100,

2222

22

KaCC

KCC

yMSE

yMSEyyPRE

xy

xy

d

PPd

. (6.3)

Table 1 Population data sets

Population Variables

Parameters

yC

xC

K

I [Murthy (1967) y : Number of workers.

x : Output.

0.3542 0.9484 0.9150 0.4441

II [Das (1988)] y : Number of agricultural

laborers for 1961.

x : Number of agricultural

laborers for 1971.

1.4451 1.6198 0.7213 0.6435

Table 2 The ranges of a in which dy is more efficient than others

Data set The range of a in which dy more efficient than

y Ry Py

I (0.50, 0.94) (0.45, 1.00) (0.00, 1.44)

II (0.50, 1.14) (0.65, 1.00) (0.00, 1.64)

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Table 2 exhibits that when a (0.50, 0.94) [0.50, 1.14], a (0.45, 1.00) [0.65, 1.00], a (0.00,

1.44) [0.00, 1.64] the proposed class of estimators dy is more efficient than the estimators y , Ry

and Py in the data set I [II] respectively. The common range of the scalar a is (0.50, 0.94) [0.65,

1.00] in which dy is to be superior than the estimators y , Ry and Py for data set I [II]. Thus, if the

scalar a (0.65, 0.94), the proposed class of estimators dy is more efficient than the estimators y ,

Ry and Py in both the population data sets I and II.

Table 3 PREs of dy with respect to y , Ry and Py

a yyPRE d , Rd yyPRE , Pd yyPRE ,

I II I II I II

0.00 10.55 25.82 15.84 16.51 100.00 100.00

0.25 24.28 47.11 36.47 30.12 230.26 182.48

0.45 69.26 85.16 104.02 54.45 656.68 329.86

0.50 100.00 100.00 150.19 63.94 948.20 387.34

0.65 482.47 159.24 724.63 101.82 4574.75 616.81

0.72 876.96 188.05 1317.13 120.24 8315.28 728.41

0.75 781.40 197.78 1173.61 126.46 7409.21 766.10

0.82 349.30 208.45 524.62 133.28 3312.04 807.39

0.94 103.38 181.82 155.26 116.26 980.20 704.26

1.00 66.58 156.40 100.00 100.00 631.32 605.79

1.14 30.75 101.14 46.18 64.67 291.54 391.76

1.25 19.52 71.36 29.32 45.63 185.13 276.40

1.44 10.67 41.66 16.02 26.63 101.14 161.35

1.50 9.10 35.82 13.67 22.90 86.30 138.75

1.64 6.56 26.01 9.85 16.63 62.16 100.75

1.75 5.24 20.79 7.86 13.29 49.64 80.53

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It is observed from Table 3 that the performance of proposed class of estimators dy is better than

the estimators y , Ry and Py , if a lies between the corresponding range of the inferior estimators

which we have calculated in Table 2 for both the population data set I and II. Further it is observed from Table 3 that there is larger gain in efficiency by using the proposed class of estimators dy

over usual unbiased estimator y , usual ratio estimator Ry and usual product estimator Py in

certain range of a . The maximum gain in efficiency is seen at the optimum value 72.0oa (for

population data set I) and 82.0oa (for population data set II) of scalar a .

7. Conclusion In this article we have considered the problem of estimating the population mean of the study variable when the population mean of an auxiliary variable is known in simple random sampling without replacement (SRSWOR). The class of estimators has been proposed and the bias and mean square error expressions of the proposed class of estimators have been obtained up to first degree of approximation. The realistic conditions under which the proposed class of estimators is better than the usual unbiased, ordinary ratio and product estimators have been obtained. The estimator based on optimum values of the scalar has been obtained. It has been found theoretically as well as empirically that the proposed class of estimators is more efficient than the usual unbiased, ratio and product estimators. However this conclusion should not be extrapolated due to limited empirical study.

Acknowledgements The authors are thankful to the Editor-in-Chief of Journal of Advanced Computing for earlier draft of the article.

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