# IMPROVED EXPONENTIAL ESTIMATOR IN STRATIFIED RANDOM SAMPLING

**ABSTRACT** In this article we have considered the problem of estimating the population mean in the stratified random sampling using the information of an auxiliary variable x which is correlated with y and suggested improved exponential ratio estimators in the stratified random sampling. The mean square error (MSE) equations for the proposed estimators have been derived and it is shown that the proposed estimators under optimum condition performs better than estimators suggested by Singh et al. (2008). Theoretical and empirical findings are encouraging and support the soundness of the proposed estimators for mean estimation.

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**ABSTRACT:**In this article, we suggest a new ratio estimator in stratified random sampling based on the Prasad (1989) estimator. Theoretically, we obtain the mean square error (MSE) for this estimator and compare it with the MSE of traditional combined ratio estimate. By this comparison, we demonstrate that proposed estimator is more efficient than combined ratio estimate in all conditions. In addition, this theoretical result is supported by a numerical example.Communication in Statistics- Theory and Methods 01/2005; 34:597-602. · 0.30 Impact Factor - SourceAvailable from: Rajesh Singh[Show abstract] [Hide abstract]

**ABSTRACT:**Kadilar and Cingi (2003) have introduced a family of estimators using auxiliary information in stratified random sampling. In this paper, we propose the ratio estimator for the estimation of population mean in the stratified random sampling by using the estimators in Bahl and Tuteja (1991) and Kadilar and Cingi (2003). Obtaining the mean square error (MSE) equations of the proposed estimators, we find theoretical conditions that the proposed estimators are more efficient than the other estimators. These theoretical findings are supported by a numerical example.01/2013; - SourceAvailable from: Cem Kadilar[Show abstract] [Hide abstract]

**ABSTRACT:**This paper considers some ratio-type estimators and their properties are studied in stratified random sampling. The results are supported by an application with original data.Biometrical Journal 02/2003; 45(2):218 - 225. · 1.15 Impact Factor

Page 1

Pak.j.stat.oper.res. Vol.V No.2 2009 pp67-82

Improved Exponential Estimator in

Stratified Random Sampling

Rajesh Singh

Department of Statistics

B.H.U., Varanasi (U.P.)

India

rsinghstat@yahoo.com

Mukesh Kumar

Department of Statistics

B.H.U., Varanasi (U.P.)

India

Manoj K. Chaudhary

Department of Statistics

B.H.U., Varanasi (U.P.)

India

Cem Kadilar

Department of Statistics

Hacettepe University

Beytepe 06800, Ankara

Turkey

kadilar@hacettepe.edu.tr

Abstract

In this article we have considered the problem of estimating the population mean ? ?

stratified random sampling using the information of an auxiliary variable x which is correlated with

y and suggested improved exponential ratio estimators in the stratified random sampling. The

mean square error (MSE) equations for the proposed estimators have been derived and it is

shown that the proposed estimators under optimum condition performs better than estimators

suggested by Singh et al. (2008). Theoretical and empirical findings are encouraging and support

the soundness of the proposed estimators for mean estimation.

Y

in the

Keywords: Auxiliary variable, exponential ratio-type estimates, mean square

error, stratified random sampling.

2000 AMS Classification: 62D05

1. Introduction

In survey sampling, it is well established that the use of auxiliary information

results in substantial gain in efficiency over the estimators which do not use such

information. However, in planning surveys, the stratified sampling has often

proved needful in improving the precision of estimates over simple random

sampling. In some cases, in addition to mean of auxiliary variable, various

parameters related to auxiliary variable such as coefficient of variation, kurtosis,

correlation coefficient, etc. may also be known. For these cases, many authors,

Page 2

Rajesh Singh, Mukesh Kumar, Manoj K. Chaudhary, Cem Kadilar

Pak.j.stat.oper.res. Vol.V No.2 2009 pp67-82

68

such as Upadhayaya and Singh (1999), Sisodia and Dwivedi (1981), Singh and

Tailor (2003), Singh et al. (2007), developed various estimators to improve the

ratio estimators in the simple random sampling. Kadilar and Cingi (2003) adapted

the estimators in Upadhyaya and Singh (1999) to the stratified random sampling.

In this study, under stratified random sampling without replacement scheme, we

suggest two improved exponential estimators which are more efficient than

estimators proposed by Singh et al. (2008). There are some recent studies

proposing the family of estimators without using the exponential function in

literature, such as Koyuncu and Kadilar (2009(a,b)), Khoshnevisan et al. (2007),

Singh and Vishwakarma (2006, 2008) etc. However, in this article, the proposed

family of estimators depends on the exponential function.

We assume that the population comprises N units, which can be uniquely

partitioned into L strata of size N1, N2,…, NL such that ?

?

?

L

1h

h

.NN

The strata

weights Wh= Nh/ N (h=1, 2,…, L) are assumed known. Let (yhi, xhi) (i=1, 2,…, Nh)

denote the values of variates (y, x) respectively for the i-th unit in the h-th stratum

and

h

Y and

h

X

denote stratum means.

When the population mean of the auxiliary variable, X, is known, Hansen et al.

(1946) suggested a combined ratio estimator for estimating the population mean

of the study variable ? ?

Y :

X

x

y

t

st

st

1?

,(1.1)

where

h

L

?

?

1

1h

hst

ywy

?

,

h

L

?

?

1h

hst

xwx

?

, and

.Xw

1

?

X

h

L

?

h

h

?

Here

?

?

i

?

h

n

1

hi

h

h

y

n

y

, and

?

?

i

?

h

n

1

hi

h

h

x

n

1

x

.

The MSE of t1to a first degree of approximation is given by

? ?

1

t

?

S

?

?

?

h

????

L

1

yxh

2

xh

22

yh

h

2

h

RS2SRwMSE

,(1.2)

where

??

?

?

??

?

?

???

hh

h

N

1

n

1

,

st

st

X

Y

X

Y

R

??

is the population ratio,

2

yh

S

is the

population variance of a study variable,

auxiliary variable and

auxiliary variables in the stratum h.

2

xh

S

is the population variance of the

yxh

S

is the population covariance between study and

Page 3

Improved Exponential Estimator in Stratified Random Sampling

Pak.j.stat.oper.res. Vol.V No.2 2009 pp67-82

69

2. Kadilar and Cingi estimator

Kadilar and Cingi (2003) introduced an estimator for population mean using

known value of some population parameters in stratified random sampling given

by

y

2t =

b , a , st

b , a , st

st

X

x

(2.1)

where

??

hh

L

1h

hh b , a ,st

bxawx

?? ?

?

,

??

hh

L

1h

hhb , a ,st

bXawX

?? ?

?

and ah, bh are the

functions of the known parameters of the auxiliary variable such as coefficient of

variation Cxh, coefficient of kurtosis

2

?

stratum .

) x (

h

etc., or some constants in the hth

The MSE of the estimator t2is given by

MSE(

2t ) = ?

?

?

L

1h

h

2

h

W

??

yxhh b ,a

2

xh

2

h

2

ab ,

2

yh

SaR2SaRS

??

, (2.2)

where

.

X

Y

R

b , a ,st

st

b , a

?

Bahl and Tuteja (1991) suggested an exponential ratio type estimator for

population mean in simple random sampling as

??

?

??

?

?

?

?

xX

xX

expyt3

.(2.3)

Motivated by Bahl and Tuteja (1991), Singh et al. (2008) adapted this estimator

to the stratified random sampling as

??

?

??

?

?

?

?

i 4i 4

i 4i 4

sti 4

t

xX

xX

expy

,i = 0, 1, 2, 3, 4.(2.4)

where

h

L

?

?

1h

h40

xwx

?

,

,Xw

1

?

X

h

L

?

h

h 40

?

??

xhh

L

1h

h 41

Cxwx

???

?

,

??

xhh

L

1h

h 41

CXwX

???

?

,

?? ) x (

h

xwx

2h

L

1h

h 42

????

?

,

?? ) x (

h

XwX

2h

L

1h

h 42

????

?

,

??

xhh2h

L

1h

h43

C ) x (xwx

????

?

,

??

xhh2h

L

1h

h 43

C) x (XwX

????

?

,

?? ) x (

h

Cxwx

2 xhh

L

1h

h 44

????

?

,

?? ) x (

h

CXwX

2 xhh

L

1h

h 44

????

?

.

Page 4

Rajesh Singh, Mukesh Kumar, Manoj K. Chaudhary, Cem Kadilar

Pak.j.stat.oper.res. Vol.V No.2 2009 pp67-82

70

The bias and MSE of the estimators t4i( i=0,1,2,3,4) can be obtained respectively

by following expressions:

?

??

1h

i 4

8X

?

???

1h

where,

?

?

?

?

?

?

?

?

L

yxh hi

2

xh

2

hi

i 4h

2

h

i 4

Sa

2

1

SaR

3

w

1

)t (Bias

, (i=01,2,3,4)(2.5)

?

?

?

?

?

?

?

?

?

?

L

2

xh

2

hi

2

i 4

4

yxh hii 4

2

yh

h

2

h

i 4

Sa

R

SaRSw)t (MSE

, (i=01,2,3,4)(2.6)

,

Xw

1

?

Y

R

h

L

?

h

h

st

40

?

,

)CX(w

1

?

Y

R

L

?

h

xhhh

st

41

?

?

,

))x (

h

X(w

1

?

Y

R

L

?

h

2hh

st

42

??

?

,

)C) x (

h

X(w

1

?

Y

R

L

?

h

L

?

?

xh2hh

st

43

??

?

,

))x (

h

CX(w

1

?

Y

R

L

?

h

2 xhhh

st

44

??

?

.YYw

1

Y

h

hh st

??

We would like to remark that for various values of parameters in (2.4), we get five

estimators (for I = 0, 1, 2, 3, 4) as shown in Table 1. The MSE expressions for

these estimators are also given in the Table 1.

Table 1: Some members of family of estimators t4i

The values

of ah,bh

Estimator MSE of the Estimator

ah=1, bh=0

??

?

??

?

?

?

?

st

st

st 40

xX

xX

expyt

?

?

?

?

?

?

?

?

?

?

?

???

?

?

h

2

xh

2

40

4

yxh

S

40

2

yh

h

L

1

2

h

S

R

RSw

ah=1,

bh=Cxh

??

?

??

?

?

?

?

4141

4141

st 41

xX

xX

expyt

?

?

?

?

?

?

?

???

?

?

h

2

xh

2

41

4

yxh

S

41

2

yh

h

L

1

2

h

S

R

RSw

ah=1,

bh=?2h(x)

??

?

??

?

?

?

?

42 42

4242

st 42

xX

xX

expyt

?

?

?

?

?

?

?

?

???

?

?

h

2

xh

2

42

4

yxh

S

42

2

yh

h

L

1

2

h

S

R

RSw

ah=?2h(x),

bh=Cxh

?

?

?

?

?

?

?

?

?

4343

43 43

st43

xX

xX

expyt

? ?

x

?

?

h

?

?

?

?

?

?

?

?

?????

L

1

2

xh

2

2h

2

43

4

yxhh243

2

yh

h

2

h

S ) x (

R

SRSw

ah=Cxh,,

bh=?2h(x)

??

?

??

?

?

?

?

4444

44 44

st44

xX

xX

expyt

?

?

h

?

?

?

?

?

?

?

?

???

L

1

2

xh

2

xh

2

44

4

yxhxh 44

2

yh

h

2

h

SC

R

SCRSw

Page 5

Improved Exponential Estimator in Stratified Random Sampling

Pak.j.stat.oper.res. Vol.V No.2 2009 pp67-82

71

Remark 1 -

depends on the availability and values of the various parameter(s) used (for

choice of the parameters ah and bh refer to Singh et al. (2008) and Singh and

Kumar(2009)).

Singh et al. (2008) showed that all of these modified estimators (t4i) have a

smaller MSE than the MSE of the stratified version of exponential ratio type

estimator t3stunder certain conditions:

?

??

?

st

xX

Here we would like to mention that the choice of the estimator

??

?

?

?

st

st st3

xX

expyt

(2.7)

Note that t3stis the same estimator with t40.

Singh et al. (2008) reported the minimum value of mean square error of the

L

h

h

min i 4

?

where,

c

? is combined correlation coefficient in stratified sampling across all

estimator t4i as

) 1 (Sw) t (MSE

2

c

2

yh

1i

2

???? ?

(2.8)

strata. It is calculated as

.

SwSw

1

?

SSw

L

?

1

L

?

?

1i

2

xh

h

2

h

2

yh

h

2

h

2

L

?

?

1i

xh yhhh

2

h

2

c

??

??

?

?

??

?

?

??

??

3. Suggested modified exponential ratio estimator

In stratified random sampling, Kadilar and Cingi (2005) introduced a ratio

estimator as

tkt ?

,

where the constant k is obtained by minimizing the MSE of t5.

15

(3.1)

Motivated by Kadilar and Cingi (2005), we propose the following modified

estimator

tkt

?

?

??

?

i 4 i 4

xX

i 4 i 6

??

?

?

?

i 4i 4

st

xX

expyk

,i = 0, 1, 2, 3, 4.(3.2)

To obtain the MSE of t6i to the first degree of approximation, let us define

*

i

*

i

*

i

*i 1

e

st

0

A

Aa

and

Y

Yy

e

?

?

?

?

. Using these notations we have

??

0 st

e1Yy

??

and

?

1

?

*i 1

e

*

i

*

i

Aa

??

,

Page 6

Rajesh Singh, Mukesh Kumar, Manoj K. Chaudhary, Cem Kadilar

Pak.j.stat.oper.res. Vol.V No.2 2009 pp67-82

72

where

?

?

h

?

L

1

h hih

*

i

XawA

,

,xaw

1

?

a

L

?

h

h hih

*

i

?

??

? ? 0e

1i

EeE

*

0

??

,

? ?

0

e

2

yh

L

1h

h

2

h

2

2

Sw

Y

1

E

?

?

??

,

? ?

i 1

2

xh

L

?

?

1h

hih

2

h

2*

i

2

Saw

A

1

eE

??

,

??

yxh

L

?

?

1h

hih

2

h

*

i

*i 1

e

0

Saw

AY

1

eE

??

.

Expressing (3.2) in terms of e’s, we have

??

?

?

*i 1

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

???

?1

*i 1

e

* *i 1

e

*

0i i 6

t

2

1

2

expe1Yk

?

?

?

?

?

?

?

?

?

??

?

???

2

ee

8

e3θ

2

e

e1Yk

0

*

2*

1i

2*

*i 1*

0i

.(3.3)

where

i 4

*

i

*

i

X

A

??

.

)Y(tE) (t Bias

6i 6i

??

)

2

) ee(E

8

) e (E3

(Yk ) 1

?

k (Y

*i 1*

i

2*

i 1

*

i

stii st

?

?

?

??

??

?

??

?

?????

?

?

h

yxhhi

2

xh

2

hi

i 4

L

1

h

2

h

i 4

i

i st

Sa

2

1

SaR

8

3

w

X

k

) 1k (Y

)t ( Biask ) 1

?

k (Y

i 4ii st

??

(3.4)

From (3.3), we have

??

?

?

?

?

?

?

?

?

??????

?

?

h

2

xh

2

hi

2

i 4

4

yxh hii 4

2

yh

L

1

h

2

h

2

i

2

2

i i 6

Sa

R

SaRSwkY1k) t ( MSE

?

?

?

?

?

?

????

?

?

h

yxhhi i 4

2

xh

2

hi

2

i 4

L

1

h

2

h

ii

SaR

2

1

SaR

8

3

w ) 1k (k2

. (3.5)

Page 7

Improved Exponential Estimator in Stratified Random Sampling

Pak.j.stat.oper.res. Vol.V No.2 2009 pp67-82

73

MSE of the estimator t6ican be rewritten as

? ? ?

i i 6

ktMSE

??

???????

i 4

t

stii i 4

t

2

i

2

2

BiasY1kk2 MSEkY1

???

(3.6)

To obtain the optimum value of k, we partially differentiate the expression (3.6)

with respect to k and put it equal to zero as follows:

?

,

??

0t MSE

k

i 6

?

?

??

?? 0

?

) t (BiasY ) 1

?

k (

i

k2Y ) 1

?

k (

?

t MSEk

k

i 4sti

22

ii 4

2

?

?

?

,

??

ii 4

2

i

2

*

i

A2tMSEY

AY

k

??

?

?

,i = 0, 1, 2, 3, 4.(3.7)

where

.SaR

2

1

SaR

8

3

w

1

) (tBiasYA

yxh hii 4

2

xh

2

hi

2

i 4

L

?

?

h

h

2

h

4isti

?

?

?

?

?

?

????

Note that 0<

*

i k <1.

Remark 2

Shahbaz and Hanif (2009) proposed a general class of shrinkage estimator in

survey sampling as

tˆ

tˆ

2

s

?

?

) tˆ (MSET1

?

(3.8)

where tˆ is any available estimator of parameter T. The minimum MSE of

reported by Shahbaz and Hanif (2009) as

) tˆ (MSE

)tˆ ( MSE

2

s

?

?

stˆ

) tˆ ( MSET1

?

(3.9)

Following Shahbaz and Hanif (2009), the estimator t6iis written as

tˆ

tˆ

i 4

?

,

)tˆ ( MSEY1

2

i 4

iS6

?

?

i=0,1,2,3,4.(3.10)

and the minimum MSE of

iS6 tˆ

is given by

)tˆ ( MSEY1

)tˆ (MSE

?

) t (MSE

i 4

2

i 4

iS6

?

?

, i=0,1,2,3,4. (3.11)

Page 8

Rajesh Singh, Mukesh Kumar, Manoj K. Chaudhary, Cem Kadilar

Pak.j.stat.oper.res. Vol.V No.2 2009 pp67-82

74

4. Improved estimator

We propose classes of estimators for estimating the population mean ? ?

stratified random sampling as

????

i = 0, 1, 2, 3, 4.

Y , in the

,tyt

i 4i 2st i 1i 7

??

?

??

?

?

?

????

i 4 i 4

i 4i 4

sti 2 st i 1

xX

xX

expyy

.(4.1)

where ?1iand ?2iare real constants. There are two choices in terms of how to

select ?1i and ?2i. Some authors, such as Kadilar and Cingi (2006), use the

constraint ?1i + ?2i= 1, while others use an unconstrained selection of ?1iand

?2i. This latter group includes Upadhyaya et al. (1985), Singh et al. (1988) and

Shabbir and Gupta (2207). These authors choose ?1iand ?2iwhich minimize the

MSE for the proposed estimator and do not insist on having

In this paper we have made unconstrained selection of ?1iand ?2i.

?1i + ?2i= 1.

Expressing (4.1) in terms of e’s by using the same notations as defined earlier

we have

?

?

??????

i 20 i 1 i 7

expY)e 1 (Yt

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?1

i 1

e

*

i

0

*

i

2

1

2

e

(4.2)

Expanding the right hand side of (4.2), and retaining the terms to the first order of

approximation, we have

?

?

???????

e1Y)e1 (Yt

0 i 20 i 1i 7

?

?

?

?

?

?

?

?

?

?

?

2

ee

8

e3

2

e

*i 1

0

*

i

2*

i 1

2*

i

*i 1*

i

(4.3)

The bias of the estimator t7i, to the first order of approximation, is obtained as

)Yt (E) (tBias

7i 7i

??

L

2

h

i 2st i 2 i 1

?

?

?

?

?

?

?????????

?

?

h

yxhhi i 4

2

xh

2

hi

2

i 4

1

h

SaR

2

1

SaR

8

3

wY ) 1(

) t (BiasYY ) 1

?

(

i 4st i 2st i 2 ` i 1

??????

(4.4)

From equation (4.4), we observe that to the first order of approximation the

proposed estimator is biased.

From (4.3), squaring and than retaining the terms of e’s upto power two, we have

?

i 7 i 7

YtE)t ( MSE

??

?

2

i i 2sti 2 ` i 1

?

AY) 1

?

(

?????

Page 9

Improved Exponential Estimator in Stratified Random Sampling

Pak.j.stat.oper.res. Vol.V No.2 2009 pp67-82

75

2

st

*i 1

e

0

2

*

i

2*

i 1

2*

i

*i 1

e

*

i

0 i 20 i 1

Y

e

8

e3

2

e1Y)e 1 (YE

??

?

?

?

??

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

???????

2

*i 1

e

0

2

*

i

2*

i 1

2*

i

*i 1

e

*

i

0 i 20i 1 i 2 i 1

e

8

e3

2

eYYe) 1

?

(YE

??

?

?

?

??

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?????????

??

) t (MSED1Y

i 4

2

i 2

2

i 1

2

i 2i 1

2

?????????

??

i i 2i 2 1 ii i 2i 1

A12C2

?????????

(4.5)

where

?

?

?

?

?

?

??? ?

h

?

yxh hi

i 4

2

2

yh

L

1

h

2

h

i

Sa

R

SwC

and

2

yh

h

L

1h

2

h

SwD

?? ?

?

.

In order to find optimum value of estimator t7i, we differentiate MSE (t7i) with

respect to

i 1

?

and

.i 2

?

The optimum values of

?

?

ii

YDYACY

????

i 1

?

and

i 2

?

are

?

???

?

?????

i i 4

22

2

2

i i 4

)

22

ii

2

i

2

*

i 1

A2 t (MSE

A2)

?

t (MSEYYACYAY

?

??????

??

(4.6)

and

??

i i 4

222

ii

2

i

?

222

ii

)

2

*

i 2

A2) t ( MSEY)DY(ACY(

)AY )(DY(Y)C

?

AY(

????

?????

??

.(4.7)

Using these optimum values,

respectively, we get the minimum MSE of the estimator t7i.

*

i 1

?

and

*

i 2

? , in the place of

i 1

? and

i 2

?

in (4.5),

Remark 3 - Here we would like to mention that the optimum values of

?

require information about population parameters. However, in applications,

i 1

? and

i 2

to have this information is sometimes impossible. However in repeated surveys

or studies based on multiphase sampling, where information is gathered on

several occasions (or based on past experience) it may be possible to guess the

value of these parameters quite accurately. Even though this approach may

reduce the precision, it may be satisfactory provided the relative decrease in

precision is marginal. (For further details refer to Sukhatme et. al. ((1984),

pp. 260), Singh and Vishwakarma (2006) and Koyuncu and Kadilar (2009(b)).

Page 10

Rajesh Singh, Mukesh Kumar, Manoj K. Chaudhary, Cem Kadilar

Pak.j.stat.oper.res. Vol.V No.2 2009 pp67-82

76

5. Efficiency comparisons

For the estimators, efficiency comparisons are given below.

In this section, we first compare modified estimator t6i given in (3.2), with the

estimator t4igiven in (2.4).

MSE (t6i) < MSE (t4i), i = 0, 1, 2, 3, 4,

??

?

?

?

?

?

?

?

?

?????

?

?

h

2

yxh

2

h

2

i 4

4

yxhh i 4

2

yh

L

1

h

2

h

2

i

2

2

i

Sa

R

SaRSwkY1k

?

?

?

?

?

?

?

?

????

?

?

h

xyh hii 4

2

xh

2

hi

2

i 4

8

l

1

h

2

h

ii

SaR

2

1

Sa

R3

w) 1k (k2

?

?

?

?

?

?

t (

?

?

???? ?

h

?

2

xh

2

hi

2

i 4

4

yxhhi i 4

2

yh

h

L

1

2

h

Sa

R

SaRSw

,

ii 4

2

i 4

?

2

i

A2) t (MSEY

) MSEY

?

k

?

?

. (5.1)

As the condition (5.1) is always satisfied, we can say that the suggested

estimator is more efficient than exponential ratio estimator in stratified random

sampling in all conditions. Note that when i = 0, MSE (t40) = MSE (t3st).

Second, we compare the improved estimator t7i, given in (4.1), with estimator t4i,

given in (2.4) as follows:

MSE (t7i) < MSE (t4i),

?

i 1

Y

???

i = 0, 1, 2, 3, 4,

???

i i

i 2 i 2i 1i

?

i 2i 1 i 4

2

i 2

2

i 1

2

i 2

2

A12C2) t ( MSED1

?????????????

?

?

?

?

?

?

?

?

???? ?

h

?

?

2

xh

2

hi

2

i 4

4

yxh hi i 4

2

yh

h

L

1

?

2

h

Sa

R

SaRSw

.

Let

ii 2i 1

B) 1(

???

. Then the expressions can be obtained on solving as

*i 3

*i 2*i 1

2

Y

?

???

?

, (5.2)

where

????

2

i 2

i 4

t

*i 1

1 MSE

????

,

??

i i 2 i 1

2

i 1

i i 2i

*i 2

C2DBA2

?????????

,

2

i

i

*

3

B

??

.

Page 11

Improved Exponential Estimator in Stratified Random Sampling

Pak.j.stat.oper.res. Vol.V No.2 2009 pp67-82

77

Next, we compare the improved estimator t7igiven in (4.1) with the estimator t6i

given in (3.2).

MSE (t7i) < MSE (t6i),

?

i 1

Y

???

i = 0, 1, 2, 3, 4,

???

i i 2i 2 i 1i i 2

2

i 4

8

i 1

?

i 4

2

i 2

2

i 1

2

i 2

2

A

?

12C2) t (MSED1

??????????????

????

?

?

?

?

?

?

???????

?

?

h

xyh hii 4

2

xh

2

hi

l

1

h

2

h

iii 4

t

2

i

2

i

2

SaR

2

1

Sa

R3

w) 1k (k2MSEk1kY

,

i 3

i 2 i 1

2

Y

?

???

?

,(5.3)

where

????

2

i 2

2

i

i 4

t

i 1

kMSE

????

,

??

??

i i 2i 1

2

i 1

i 2iii i 2

C2DB1kkA2

??????????

,

??2

i

2

i i 3

1kB

????

.

Finally, we compare the improved estimator t7i, given in (4.1), with

estimator

min

t (MSE

given in (2.8).

i 4)

min i 4 i 7

) t (MSE) t ( MSE

?

,i = 0, 1, 2, 3, 4,

????

). 1 (Sw

A12C2) t ( MSED1Y

2

c

L

?

?

1h

2

yh

h

2

h

i i 2i 2 i 1i i 2 i 1i 4

2

i 2

2

i 1

2

i 2i 1

2

????

?????????????????

or

*i 3

*i 2 *i 1

?

*

2

Y

?????

?

(5.4)

where,

min

??

?

i 4

?

i 4

t

*

)t ( MSE

??

?

2

i 2

?

*i 1

MSE

???

,

?

i i

i 2i 1

2

i 1

i i 2i

*i 2

C2DBA2

?????????

,

2

i

i

*

3

B

??

.

Page 12

Rajesh Singh, Mukesh Kumar, Manoj K. Chaudhary, Cem Kadilar

Pak.j.stat.oper.res. Vol.V No.2 2009 pp67-82

78

In all above expressions we used optimum values of

*i 1

i 1

???

,

i 2

???

, and

i 1

? ,

i 2

?

, and

i k , i.e.,

*i 2

i k =

*

i k , respectively.

Table 2: Data Statistics

N=854

Cx=3.85

Sy=17106

N1=106

N4=171

n1=9

n4=67

X1?

X4?

iY1?

5588Y4?

x??

x ??

Cx1=2.02

Cx4=3.84

Cy1=4.18

Cy4=5.13

Sx1=49189

Sx4=285603

Sy1=6425

Sy4=28643

. 0

?

. 0

?

. 0

1??

. 0

4??

. 0w2

. 0w2

n=140

Cy=5.84

. 0

??

N2=106

N5=204

n2=17

n5=7

X2?

X5?

Y2?

Y5?

x ??

x ??

Cx2=2.10

Cx5=1.72

Cy2=5.22

Cy5=2.47

Sx2=57461

Sx5=45403

Sy2=11552

Sy5=2390

. 0

?

. 0

?

. 0

2??

. 0

5??

. 0w2

. 0w2

07. 312

x??

Sx=144794

92

N3=94

N6=173

n3=38

n6=2

X3?

X6?

Y3?

Y6?

x ??

x ??

Cx3=2.22

Cx6=1.91

Cy3=3.19

Cy6=2.34

Sx3=160757

Sx6=18794

Sy3=29907

Sy6=946

. 0

?

. 0

?

. 0

3??

. 0

6??

. 0w2

. 0w2

X

=37642

X

=102815

R42=0.07780

24375

74365

536

27421

26441

2212

967

. 34

27

72409

9844

9384

404

26

28

71.

60.

25

97

1

57

47.

2

14

10.

.

3

45

6

82

99

102

009

015

04

1?

4?

86

71

049

138

015

057

2?

5?

90

89

016

006

012

041

3?

6?

1?

4?

2?

5?

3?

6?

37600

=37602

R40=0.07790

R43=0.00140

A40= -68606.40

A43= -23961.80

C40= 476871.32

C43= 439626.58

X ?

X

2930

=2086644

R41=0.07789

R44=0.02850

A41= -68611.10

A44= -63674.00

C41= 476884.00

C44= 457109.01

Y ?

X

41

42

43

44

A42= -68692.40

C42= 477111.29

D = 691035.70

Page 13

Improved Exponential Estimator in Stratified Random Sampling

Pak.j.stat.oper.res. Vol.V No.2 2009 pp67-82

79

6. Numerical study

For empirical study, we use the data set earlier presented by Kadilar and Cingi

(2003). In this data set, Y is the apple production amount and X is the number of

apple trees in 854 villages of Turkey in 1999. The population information about

this data set is given in Table 2.

By using this data, we have calculated MSE values of suggested estimators and

compared them with MSE values of Singh et al. (2008) estimators. For the

different values of ahand bh, we can find the MSE equations of all members of

the improved estimator t7iby simply changing

i 4

R , respectively.

Table 3: MSE Values of Estimators

Values of R

(Bias)

) MSE(t4i

(Bias)

) MSE(t6i

(Bias)

) MSE(t7i

Values of

*

i k ,

*

1i

ω

and

*

2i

ω

R40=0.07790359745.60

(-23.5173)

350039.68

(-119.9887)

217839.68

(-74.6724)

*

0

k = 0.9667

? = -1.1917

? = 2.1837

*

10

*

20

R41=0.07789359760.09

(-23.5189)

350053.50

(-119.9934)

217839.99

(-74.6725)

*

1 k = 0.9667

*

11

? = -1.1918

*

21

? = 2.1839

R42=0.07780360008.22

(-23.4940)

350290.60

(-120.0747)

217845.36

(-71.3671)

*

2

k = 0.9665

*

12

? = -1.1942

*

22

? = 2.1863

R43=0.00140 339848.98

(-8.2138)

328512.90

(-112.6096)

267874.93

(-91.8238)

*

3

k = 0.9641

? = -0.6360

? = 1.6091

*

13

*

23

R44=0.0285 336684.09

(-21.8266)

328136.37

(-112.4806)

208197.14

(-74.6743)

*

4

*

14

*

24

k = 0.9687

? = -1.0460

? = 2.0368

In Table 3 we have reported the bias and MSE values of the different estimators.

Here we would like to mention that the MSE of the different estimators depend on

choice of ahand bh. The minimum MSE attained by Singh et al. (2008) estimator

was 218374.8898. From Table 3 we observe that the estimator t7i (i=0,1,2,3,4)

Page 14

Rajesh Singh, Mukesh Kumar, Manoj K. Chaudhary, Cem Kadilar

Pak.j.stat.oper.res. Vol.V No.2 2009 pp67-82

80

under optimum conditions performs better than all other estimators proposed by

Singh et al. (2008). We also observe from the table that the estimator t73 is

having larger MSE than the minimum MSE attained by Singh et al. (2008). So,

for this data set choice of ah=?2h(x) and bh=Cxhis not a good choice. Also, for the

choice ah=Cxh and bh=

) x (

h2

?

the MSEs of the estimators t44, t64 and t74 is

minimum.

7. Conclusion

As shown in theory, we also observe from Table 3 that suggested estimators t6i

(including optimal value) performs better than corresponding estimators without

optimal values, t4i. Besides, we see that improved estimator t7iand its members

are always more efficient than corresponding estimators (without optimal value of

kiand with

*

ik ) for this data set.

Acknowledgements

The authors are thankful to the referee for his valuable comments and

suggestions regarding the improvement of the paper. The second author

(Mukesh Kumar) is grateful to UGC, New Delhi, India, for providing financial

assistance.

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