Improvement Over General And Wider Class Of Estimators Using Ranked Set Sampling

Improvement Over General And Wider Class of

Estimators Using Ranked Set Sampling

V. L. Mandowara, Nitu Mehta (Ranka)

Abstract: In this paper, Improvement over general and wider class of estimators of finite population means using ranked set sampling is investigated.

Ranked set sampling (RSS) was first suggested to increase the efficiency of estimator of the population mean. The first order approximation to the bias and mean square error (MSE) of the investigated estimators are obtained. Theoretically, it is shown that these suggested estimators are more efficient than the general and wider class of estimators in simple random sampling.

Key Words: Ranked set sampling, General and wider class of estimator, Auxiliary variable, Mean square error, Efficiency.

1. Introduction

Ranked set sampling was first suggested by McIntyre (1952) to increase the efficiency of estimator of population mean. Kadilar et al. (2009) used this technique to improve ratio estimator given by Prasad (1989). Here we shall improve general and wider class of estimators given by Srivastava (1971, 1980) based on auxiliary variable. Srivastava (1971) proposed a general class of estimators to

estimate the population mean

Y

of the study variable

which in the case of single mean

X

of the auxiliary variable is given by

where

X

x

u



and

H

_{ }



is a parametric function.

In the Simple random sampling, the bias and minimum mean square error of the general class of estimator, see singh ( Volume 1, 2003, page 164-165) are given by

 

g

Y



H

C

x

H

xy

C

Y

C

x



n

f

t

B

1



2 2

1













 



or

_{ }

_g

Y



H

C

_x

H

_xy

C

_Y

C

_x



n

t

B

1



2 2

1





(On ignoring

N

n

f



)

or

B

_{ }

t

_g



Y



H

C

_x

H

1



_xy

C

_Y

C

_x



2

2





(1.2)

where

n

1





,

n

and

N

are the sample and population

sizes respectively ;

;

C

y2

,

C

x2

N

n

f



denote the

coefficient of variation of

Y

and

X

respectively and

yx



denote the correlation coefficient between

Y

and

X

.

Here

1 1





















u

u

H

H

and

` 1 2 2

2

2

1

























u

u

H

H

denote

the first and second order partial derivatives of

H

with respect to

u

and are the known constants.

The minimum mean square error (MSE) of the general class of estimator

t

_g defined at (1.1), to the first order of approximation is

 





2 2



2



1

1

.

g

Y

C

y xy

n

f

t

MSE

Min









or

_{ }

2



2



2

1

1

.

g

Y

C

y xy

n

t

MSE

Min







(on ignoring

N

n

f



)

or

Min

.

MSE

 

t

_g





Y

2

C

_y2



1





_xy2



(1.3)

If we attach any function of

X

x

to the sample mean

y

,

the

asymptotic minimum mean square error of the resultant estimator cannot be reduced further than that given in (1.3). Thus the usual ratio estimator, product estimator and power transformation estimator are the special cases of the class of estimators defined in (1.1). While the regression estimator and difference estimator are not special cases of the general class of estimators. Then Srivastava (1980)

t

g



y

H

_{ }

u

(1.1)

_______________________________

 V.L.Mandowara is working as Professor in the

Department of Mathematics and Statistics, University College of Science, MLSU, Udaipur.

Emai- [email protected]

 Nitu Mehta (Ranka) is currently pursuing Ph.D. in

defined another class of estimators and named a wider class of estimators as



y

u



H

t

w



,

(1.4)

where

H



y

,

u



is a function of

y

and

u

.

The asymptotic bias and minimum mean square error, see singh ( Volume 1, 2003, page 166-167) are given by

 



4



2 2

2 2 3

1

H

C

Y

H

C

H

C

C

Y

n

f

t

B

w



xy Y x



x



y









 





(on ignoring

N

n

f



)

or

_{ }



2 ₄



2

2 2

3

C

H

Y

C

H

H

C

C

Y

t

B

_w

_





_{xy Y x}

_

_x

_

_y (1.5)

Where

1 , 1

 



















u Y y

u

H

H

,

1 , 2 2

2

2

1

 























u Y y

u

H

H

,

1 , 2

3

2

1

 

























u Y y

u

y

H

H

and

1 , 2 2

2

4

2

1

 























u Y y

y

H

H

.

The minimum mean squared error of the wider class of estimator,

t

_w, is given by

 



1



2 2



1

2



.

_w

Y

C

_{y xy}

n

f

t

MSE

Min









or

.

_{ }

w

1

Y

2

C

2y

_

1

xy2

_

n

t

MSE

Min







(on ignoring

N

n

f



)

or

Min

.

MSE

_{ }

t

_w





Y

2

C

_y2



1





_xy2



(1.6)

If we take any function of

y

,

x

and

X

to estimate the

population mean,

Y

, the asymptotic minimum mean square error of the resultant estimator again cannot be reduced further than that given in (1.6). Thus the usual linear regression estimator and difference estimator are special cases of the wider class of estimator defined at (1.4).

2. The Suggested Estimators

In Ranked set sampling (RSS),

m

independent random sets, each of size

m

are selected with equal probability and without replacement from the population. The members

of each random set are ranked with respect to the characteristic of the study variable or auxiliary variable. Then, the smallest unit is selected from the ordered set and the second smallest unit is selected from the second ordered set. By this way, this procedure is continued until

the unit with the largest rank is chosen from the

m

th set. This cycle may be repeated

r

times, so

mr

( n



)

units

have been measured during this process. When we rank on

the auxiliary variable, let

(

y

[i]

,

x

(i)

)

denote a

th

i

judgment

ordering in the

i

th set for the study variable and

i

th set for the auxiliary variable. Adapting the estimator in (1.1) to the general class of estimator for the population mean proposed by Srivastava (1971), we suggest the following estimator for general class of estimator using ranked set sampling is

 

u

H

y

t

g,RSS



[_n] (2.1)

where

X

x

u



(n ) ,

_





n i

i n

y

n

y

1 ] [ ]

[

1

,

_





n i

i n

x

n

x

1 ) ( )

(

1

and

H

_{ }

_

is a parametric function, such that, it satisfies the following conditions:

a)

H

_{ }

1



1

b) The first and second order partial derivatives of

H

with respect to

u

exists and are known constants at a given point

u



1

.

Expanding

H

_{ }

u

about the value 1 in a second order Taylor’s series, we have

 

u



H



1





u



1





H

 





...

2

1

1

)

1

(

1

1 2 2 2

1



















































 u

u

u

H

u

u

H

u

H

Note that

u



1



1

thus the higher order terms can be

neglected. Using the above two conditions, we obtain











1

1

1

2

...



2 1

] [

,



y



u



H



u



H



t

gRSS _n (2.2)

where

1 1





















u

u

H

H

and

` 1 2 2

2

2

1

























u

u

H

H

To obtain bias and MSE of

t

_g,_RSS, we put

)

1

(

0

]

[



Y





y

_n and

x

(n)



X

(

1





1

)

so that

0

)

(

)

(



0



E



1



2 ] [ 2

0 0

)

(

)

(

)

(

Y

y

V

E

V









n

=















_



m i

i y y

m

S

Y

mr

1

2 ] [ 2

2

1

1

1



=





C

_y2



W

_y2_[i]



similarly,

V

(



1

)



E

(



12

)

=





2 ) ( 2

i x x

W

C





and





Y

X

x

y

Cov

E

Cov

(



₀

,



₁

)



(



₀

,



₁

)



[n]

,

(n)

=



₍₎



1 ) (

1

1

1

i yx x y yx m

i i yx

yx

C

C

W

m

S

mr

Y

X



_















_







where

mr

1





, ₂

2 2

Y

S

C

_y



y ,

2 2 2

X

S

C

_x



x

x y yx yx

C

C

Y

X

Syx

C







,

_





m i

i x i

x

X

r

m

W

1 2

) ( 2 2 2

) (

1

1









m i

i y i

y

Y

r

m

W

1 2

] [ 2 2 2

] [

1

1



&

_





m i

i yx i

yx

Y

X

r

m

W

1 ) ( 2

) (

1

1



Here we would also like to remind that



_x(_i)





_x(_i)



X

,

Y

i y i

y[]





[]





and



yx(i)



(



x(i)



X

)

(



y[i]



Y

)

. Further to validate first degree of approximation, we

assume that the sample size is large enough to get

_

₀

and

_

₁ as small so that the terms involving

_

₀ and or

_

₁

in a degree greater than two will be negligible. The proposed estimator

t

_g,RSS given in (2.1) can easily be written in terms of

_

₀ and

_

₁ as



1

₀





1

_{1 1 1}2 ₂

...



,



Y





H



H



t

_gRSS







 







H



H





H

o





Y













2 0 1 1

2 1 1 1 0

1

Now Bias and MSE of the estimator

t

_g,_RSS to the first degree of approximation are respectively given by

(

B

t

_g,_RSS)=

E

(

t

g,RSS)-

Y

Here

E

(

t

g,RSS

)

_



2 0 1 1



2 1 1 1

0

H

H

H

E

























 

() 1





2 ) ( 2 1

2 2

, Y HC H CC HW W H

t

B gRSS  x  xy y x  xi  yxi

   (2.3)

Now

MSE

(

t

_g,RSS

)

_

E



t

_g,RSS

_

Y



2

again neglecting higher order terms, we have



1 0 1



2

2 1 2 1 2 0 2

,

)

2

(

t

Y

E



H



H





MSE

_gRSS













































) ( 1 2

) ( 2 1 2

] [

1 2 2 1 2 2

,

2

2

)

(

i yx i

x i

y

x y xy x

y RSS

g

W

H

W

H

W

C

C

H

C

H

C

Y

t

MSE





(2.4)

The optimum value of

H

₁ to minimize the MSE of

t

_g,_RSS can be easily found as follows

0

)

(

1 ,







H

t

MSE

_gRSS







2



) ( 2

) ( *

1

i x x

i yx y x xy

W

C

W

C

C

H















=

X

x

V

Y

X

y

x

Cov

n n n

]

[

]

,

[

) (

] [ ) (



x y xy

C

C

H









1* [because

Cov

(

x

(n)

,

y

[n]

)





V



x

(n)



] when we replace

H

₁ by

H

₁* in (2.4), we obtain minimum MSE of the proposed estimator as follows







C



A

Y

t

MSE

Min

_gRSS



_y



_xy





2 2

2

,

)

1

(

.





(2.5)

where



_{[] 1}* ₍₎



2

2

i x i

y

H

W

W

Y

A

_

_

By this way , we can write (2.5) as



)

(

t

g,RSS

MSE

MSE

_{ }

t

g



A

t

w,RSS



H



y

[_n]

,

u



(2.6)

where

H



y

_[n]

,

u



is a function of

y

_[n] and

u

,satisfies the following regularity conditions:

 The point



y

[n]

,

u



assumes the value in a closed convex subset

R

₂ of two dimensional real space

containing the point

 

Y

,

1

.

 The function

H



y

[n]

,

u



is continuous and bounded in

R

₂.



 

Y

,

1

=

Y

and

H

0

 

Y

,

1



1

, where

H

0

 

Y

,

1

denotes the first order partial derivative of

H

with

respect to

y

_[n].

 The first and second order partial derivatives of



y

u



H

_[n]

,

exist and are continuous and bounded in

R

₂.

Expanding

H



y

_[n]

,

u



about the point in a second order Taylor series, we have



y

u



H

t

w,RSS



[n]

,

=

H



Y





y

[n]



Y



,

1





u



1





Using the above regularity conditions and

1 , ] [ ] [  





















u Y y n n

y

H

=1, we have



























































....

1

.

1

1

4 2 ] [ 3 ] [ 2 2 1 ] [ ,

H

Y

y

H

u

Y

y

H

u

H

u

y

t

n n n RSS

w (2.7)

where 1 , ] [ 1 ] [  























u Y y n n

y

H

H

,

1 , 2 2 2 ] [

2

1

 























u Y y_n

u

H

H

1 , ] [ 2 3 ] [

2

1

 

























u Y y n n

u

y

H

H

& 1 , 2 ] [ 2 2 4 ] [

2

1

 























u Y y n n

y

H

H

Now Bias and MSE of the estimator

t

_w,_RSS to the first degree of approximation are respectively given by

(

B

t

_w,RSS)=

E

(

t

_w,RSS)-

Y

The estimator

t

_w,RSS can easily be written in terms of

_

₀ and

_

₁ as

t

_w,_RSS



1



₀





_{1 1}



₁2 ₂



_{0 1 3}



2 ₀2 ₄



...



Y





H



H

Y





H

Y



H

Now





 





 

02

2 4 1 0 3 2 1 2

,

H

E



H

Y

E





H

Y

E



t

B

_wRSS

_

_

_











































) ( 3 2 ] [ 2 4 2 ) ( 2 2 2 4 2 2 , i yx i y i x y x xy y x RSS w

W

Y

H

W

Y

H

W

H

C

C

Y

C

Y

H

C

H

t

B





(2.8)

and

MSE

(

t

_w,RSS

)



E



t

_w,RSS



Y



2=

E



Y



₀





_1`

H

₁



2







































) ( 1 2 ) ( 2 1 2 ] [ 2 1 2 2 1 2 2 ,

2

2

)

(

i yx i x i y x y xy x y RSS w

W

Y

H

W

H

W

Y

C

C

Y

H

C

H

C

Y

t

MSE





(2.9)

On differentiating (2.9) with respect to

H

₁and equating to

zero, we obtain the optimum value of

H

₁denoted by

H

₁**

is

0

)

(

1 ,







H

t

MSE

_wRSS







2



) ( 2 ) ( * * 1 i x x i yx y x xy

W

C

W

C

C

Y

H















x y xy

C

C

Y

H









1**

Replacing

H

₁ by

H

₁** in (2.9), we obtain minimum MSE of

the estimator

t

_w,_RSS as follows





















































2 ) ( ] [ 2 2 2 ,

1

)

(

.

i x x y xy i y xy y RSS w

W

C

C

W

C

Y

t

MSE

Min













C



A

Y

t

MSE

Min

_wRSS

_

_y

_

_xy

_



2 2

2

,

)

1

(

.





(2.10)

Again we can write (2.10) as

MSE

(

t

_w,RSS

)

_

MSE

_{ }

t

_w



A

of estimator, suggested by Srivastava given in (1.6), because

A

is a non negative value. As a result, show that the suggested estimator

t

_w,RSS is more efficient than the estimator

t

_w.

References

1) Kadilar, C.,Unyazici, Y. and Cingi H.(2009), Ratio

estimator for the population mean using ranked set sampling, Stat. papers,50,301-309.

2) McIntyre, G.A.(1952), A method of unbiased

selective sampling using ranked sets, Australian Journal of Agricultural Research, 3, 385-390.

3) Prasad,B. (1989), Some improved ratio type estimators of population mean and ratio in finite population sample surveys. Commun Stat Theory Methods 18:379–392

4) Singh, Sarjinder (2003), Advanced Sampling

Theory with Application, Vol. 1, Kluwer Academic Publishers, Netherlands.

5) Srivastava, S.K.(1971), A generalized estimator for

the mean of a finite population using mutli auxiliary information, J. Amer. Statist. Assoc., 66, 404-407.

6) Srivastava, S.K.(1980), A class of estimator using