  Improved ratio-cum-product estimators of finite population mean using known parameters of two auxiliary variates in double sampling

 

International Journal of Statistics and Applied Mathematics 2017; 2(6): 172-186 

ISSN: 2456-1452 Maths 2017; 2(6): 172-186

© 2017 Stats & Maths www.mathsjournal.com Received: 12-09-2017 Accepted: 14-10-2017 Arpita Lakhre

School of mathematics, Statistics and Computational Science, Central University of Rajasthan, Ajmer, Rajasthan, India Rajesh Tailor

S.S in statistics, Vikram University, Ujjain, Madhya Pradesh, India

Correspondence Arpita Lakhre

School of mathematics, Statistics and Computational Science, Central University of Rajasthan, Ajmer, Rajasthan, India

Improved ratio-cum-product estimators of finite population mean using known parameters of two

auxiliary variates in double sampling

Arpita Lakhre and Rajesh Tailor

Abstract

Use of auxiliary information has been in practice to improve the efficiency of the estimators of parameters. Ratio, product and regression methods are good examples of use of auxiliary information.

Ratio, product and regression type estimators essentially require the knowledge of population mean of auxiliary variates. But many times, the information on population mean of the auxiliary variate is not available. In this type of situations, double sampling is used. Ajagaonkar (1975) and Sisodia and Dwivedi (1982) discussed problem of estimation using single auxiliary variate whereas Khan and Tripathi (1967), Rao (1975) and Singh and Namjoshi (1988) considered the use of multi auxiliary variates in double sampling.

Singh (1967) used information on two auxiliary variates and envisaged a ratio-cum-product estimator of finite population mean of the study variate assuming that the population mean of the auxiliary variates are known. Upadhyaya and Singh (1999) proposed some ratio type estimators using coefficient of variation and coefficient of kurtosis. Tailor et al. (2011) suggested ratio-cum-product estimators using coefficient variation and coefficient of kurtosis of two auxiliary variates in simple random sampling. In this paper, authors study the Tailor et al. (2011) ratio-cum-product estimators in double sampling.

Keywords: Ratio-cum-product estimator, double sampling, population mean, Bias, Mean squared error

Introduction

This paper considers the problem of estimation of finite population mean in double sampling.

In this paper, two ratio-cum-product estimators of finite population mean, using known coefficient of variation and coefficient of kurtosis of two auxiliary variates, have been suggested. Suggested estimators have been compared with usual unbiased estimator, classical ratio and product estimators in double sampling and double sampling versions of Singh (1967)

[3] and Upadhyaya and Singh (1999) estimators. To judge the performance of the suggested estimators over other considered estimators an empirical study also has been carried out.

Let us consider a finite population

U   U

₁

, U

₂

,..., U

_N



of size

N

. Let

y

,

x

₁and

x

₂ be the study variate and auxiliary variates taking values

y

_i

x

_1i and

x

_2i respectively on

) ,..., 2 , 1

( i N

U

_i



Let the auxiliary variates

x

₁ and

x

₂ be positively and negatively correlated with the study variate y respectively.

Let us define







^N

i

y

i

Y N

1

1

: Population mean of the study variate

y

_,







^N

i

x

i

X N

1 1 1

1

: Population mean of the auxiliary variate

x

¹_,

 

 







^N

i

x

i

X N

1 2 2

1

: Population mean of the auxiliary variate

x

₂.

Let







ⁿ

i i

n y y

1

/

,







ⁿ

i i

n x x

1 1

1

/

and







ⁿ

i

i

n x x

1 2

2

/

be the unbiased estimators of population mean

Y

,

X

₁ and

X

₂ respectively.







ⁿ

i

x

i

x n

1 1 /

1

1

: First phase sample mean of the auxiliary variate

x

₁ based on sample size

n

^/,







ⁿ

i

x

i

x n

1 2 /

2

1

: First phase sample mean of the auxiliary variate

x

₂ based on sample size

n

^/,







ⁿ

i

y

i

y n

1

1

: Second phase sample mean of the study variate

y

based on sample size

n

,







ⁿ

i

x

i

x n

1 1 1

1

: Second phase sample mean of the auxiliary variate

x

₁ based on sample size

n

,







ⁿ

i

x

i

x n

1 2 2

1

: Second phase sample mean of the auxiliary variate

x

₂ based on sample size

n

,





 



^N

i i

y

y Y

S N

1

2

2

( )

1

1

: Population mean square of the study variate

y

,





 



^N

i i

x

x X

S N

1

2 1 1

2

( )

1 1

1 : Population mean square of the auxiliary variate

x

₁,





 



^N

i i

x

x X

S N

1

2 2 2

2

( )

1 1

21 : Population mean square of the auxiliary

variate

x

²_,







 



^N

i

i i

yx

y Y x X

S N

1

1

1

)

)(

1 ( 1

1 : Population covariance between the study variate

y

and auxiliary variate

x

₁







 



^N

i

i i

yx

y Y x X

S N

1

2

2

)

)(

1 ( 1

2 : Population covariance between the study variate

y

and auxiliary variate

x

₂,







 



^N

i

i i

x

x

x X x X

S N

1

2 2 1

1

)( )

1 ( 1

2

1 : Population covariance between the auxiliary variate

x

¹ _and

x

_2,

2 2

1 1 1

x y

yx

yx

S S

 S



: Population correlation coefficient between the study variate

y

and auxiliary variate

x

₁,

2 2

2 2 2

x y yx

yx

S S

 S



: Population correlation coefficient between the study variate

y

and auxiliary variate

x

₂,

2 2

2 1

2 1 2

1

x x

x x x

x

S S

 S



: Population correlation coefficient between the auxiliary variate

x

₁ and auxiliary variate

x

₂,

Y C

_y

 S

^y

: Population coefficient of variation of the study variate

y

,

1

1

1

X

C

_x

 S

^x

: Population coefficient of variation of the auxiliary variate

x

₁,

2

2

2

X

C

_x

 S

^x

: Population coefficient of variation of the auxiliary variate

x

₂,

 

 

 



 

₂

1 1

4 1 1 1

2

( )

) ) (

( X X

X x X

i



i : Coefficient of kurtosis of the auxiliary variate

x

₁,

 



 

₂

2 2

4 2 2 2

2

( )

) ) (

( X X

X x X

i



i : Coefficient of kurtosis of the auxiliary variate

x

₂,

Cochran (1940) ^[2] envisaged classical ratio estimator for estimating the population mean

Y

when study variate

y

and auxiliary variate

x

₁ are positively correlated as

 

 



 

1

ˆ

1

x y X

Y

_R (1.1)

In case of negative correlation between the study variate

y

and the auxiliary variate

x

₂, the classical product estimator was given by Robson (1957) ^[6] as

 

 



 

2

ˆ

2

X y x

Y

_P (1.2)

Assuming that the population mean

X

₁ and coefficient of variation

x1

C

of the auxiliary variate

x

₁ are known, Sisodia and Dwivedi (1981) defined a ratio type estimator as

 





 







 

1 1

1

ˆ

1

x x

SDR

x C

C y X

Y

, (1.3)

When the correlation coefficient between the study variate

y

and auxiliary variate

x

₂ is negative product type estimator using coefficient of variation

x2

C

is expressed as

 





 







 

2 2

2

ˆ

2

x x

SDP

X C

C y x

Y

. (1.4)

Singh et al. (2004) ^[10] defined ratio and product type estimators using coefficient of kurtosis



₂

( x

₁

)

and



₂

( x

₂

)

respectively as

 

 







 

) (

) ˆ (

1 2 1

1 2 1

x x

x y X

Y

_SER





, (1.5)

and

 

 







 

) (

) ˆ (

2 2 2

2 2 2

x X

x y x

Y

_SEP





. (1.6)

Upadhyaya and Singh (1999) utilized both coefficient of kurtosis as well as coefficient of variations of auxiliary variates and suggested two ratio and two product type estimators of population mean

Y

as

 





 







 

) (

) ˆ (

1 2 1

1 2 1

1

1 1

x C

x

x C

y X Y

x x R

US





, (1.7)

 





 







 

1 1

) (

) ˆ (

1 2 1

1 2 1 2

x x R

US

x x C

C x y X

Y 



(1.8)

 





 







 

) (

) ˆ (

2 2 2

2 2 2

1

2 2

x C

X

x C

y x Y

x x R

US





.

(1.9)

 





 







 

2 2

) (

) ˆ (

2 2 2

2 2 2 2

x x P

US

X x C

C x y x

Y 



(1.10)

Singh (1967) ^[3] utilized information on known population means

X

₁ and

X

₂ of auxiliary variates

x

₁ and

x

₂ respectively and envisaged a ratio-cum-product estimator of population mean

Y

as

 

 

 

 



 



 



 

2 2

1

ˆ

1

X x x y X

Y

_SRP (1.11)

The problem of estimating the population mean

Y

of y when the population means

X

₁ and

X

₂ of

x

₁ and

x

₂ respectively are known, has been discussed by many researchers including Singh and Tailor (2005) ^[9],Tailor and Tailor (2008) ^[12], Tailor et al.

(2011a) ^[13], Tailor (2012) ^[11] and many others. When information is not available on

X

₁ and

X

₂ in advance, double sampling procedure is used. The standard double sampling procedure is described as

(i) a first phase sample

S

₁ of fixed size n' is drawn form U to observe only

x

₁ and

x

₂ to estimate

X

₁ and

X

₂ respectively then (ii) a second phase sample

S

₂ of fixed size n is drawn either from

S

₁ from first phase sample or directly from the population.

These two cases may be designated as

Case I: As a sub sample from the first phase sample and Case II: Drawn independently to the first phase sample.

In double sampling, the usual ratio and product estimators of population mean

Y

are respectively defined as

 

 



  

1 ) 1

ˆ

(

x y x Y

_R^d

, (1.12)

and

 

 



 

_'

2 ) 2

ˆ

(

x y x Y

_P^d

, (1.13)

where

y

,

x

₁ and

x

₂ are sample means based on second phase sample of size n whereas











/

1 1 1

1

ⁿ

i

x

i

x n

and



^





 

ⁿ

i

x

i

x n

1 2 2

1

are the first phase sample means of

x

₁ and

x

₂, which are unbiased estimates of population means

X

₁ and

X

₂ respectively.

In double sampling, Sisodia and Dwivedi (1981) ratio type and Pandey and Dubey (1988) ^[4] product type estimators of population mean

Y

are defined as

 





 







 



1 1

1 ) 1

ˆ

(

x d x

SDR

x C

C y x

Y

, (1.14)

and

 





 





 

 

2 2

2 ) 2

ˆ

(

x d x

SDP

x C

C y x

Y

. (1.15)

In double sampling, Singh et al. (2004) ^[10] ratio and product type estimators of population mean

Y

are defined as

 

 







 

 ( )

) ˆ (

1 2 1

1 2 ) 1

(

x x

x y x

Y

_SER^d





, (1.16)

and

 

 





 

 

) (

) ˆ (

2 2 2

2 2 ) 2

(

x x

x y x

Y

_SEP^d





. (1.17)

Double sampling versions of Upadhyaya and Singh (1999) ratio type estimators are

 





 







 

) (

) ˆ (

1 2 1

1 2 /

) 1 (

1

1 1

x C

x

x C

y x Y

x d x

R

US





, (1.18)

and

 





 







 

1 1

) (

) ˆ (

1 2 1

1 2

| / ) 1

( 2

x d x

R

US

x x C

C x y x

Y 



. (1.19)

Double sampling version of Upadhyaya and Singh (1999) product type estimators are define as

 





 







 

) (

) ˆ (

2 2 2

2 2 /

) 2 (

1

2 2

x C

x

x C

y x Y

x d x

P

US





, (1.20)

 

 

and

 





 







 

2 2

) (

) ˆ (

2 2 2

2 2 / ) 2

( 2

x d x

P

US

x x C

C x y x

Y 



. (1.21)

In double sampling, Singh (1967) ^[3] ratio-cum-product estimator

Yˆ

_RP of population mean

Y

is expressed as

 

 



 



 



 

_'

2 2

1 ' ) 1

ˆ

(

x x x y x

Y

_RP^d . (1.22)

The biases and mean squared errors

ˆ

^(d)

Y

R ,

ˆ

^(d)

Y

P ,

ˆ

⁽d⁾

Y

SDR,

ˆ

⁽d⁾

Y

SDP,

ˆ

⁽d⁾

Y

SER,

ˆ

⁽d⁾

Y

SEP,

ˆ

⁽d₁⁾ R

Y

US ,

ˆ

⁽d₂⁾ R

Y

US ,

ˆ

⁽d₁⁾ P

Y

US ,

ˆ

⁽d₂⁾ P

Y

US and

ˆ

^(d)

Y

RP in both case I and case II are given below:

), 1 ( ˆ )

(

^{( )} ₃ ² ₀₁

1

K

C f Y Y

B

_R^d _I



_x



(1.23)

), 1 ( ˆ )

(

^{( )} ₃ ² ₀₂

2

K

C f Y Y

B

_P^d _I



_x



(1.24)

), (

ˆ )

(

^{( )} _{3 1} ² _{1 01}

1

t K

C t f Y Y

B

_SDR^d _I



_x



(1.25)

), (

ˆ )

(

^{( )} _{3 2} ² _{2 02}

2

t K

C t f Y Y

B

_SDP^d _I



_x



(1.26)

), (

ˆ )

(

^{( )} _{3 3} ² _{3 01}

1

t K

C t f Y Y

B

_SER^d _I



_x



(1.27)

), (

ˆ )

(

^{( )} _{3 4} ² _{4 02}

2

t K

C t f Y Y

B

_SEP^d _I



_x



(1.28)

), (

ˆ )

(

⁽₁⁾ _{3 5} ² _{5 01}

1

t K

C t f Y Y

B

_US^d_{R I}



_x



(1.29)

), (

ˆ )

(

⁽₁⁾ _{3 6} ² _{6 02}

2

t K

C t f Y Y

B

_US^d_{P I}



_x



(1.30)

), (

ˆ )

(

⁽₂⁾ _{3 7} ² _{7 01}

1

t K

C t f Y Y

B

_US^d_{R I}



_x



(1.31)

), (

ˆ )

(

⁽₂⁾ _{3 8} ² _{8 02}

2

t K

C t f Y Y

B

_US^d_{P I}



_x



(1.32)

 ^{( 1 ) ( )}  ^,

ˆ )

(

^{( )} ₃ ² ₀₁ ² _{02 12}

2

1

K C K K

C f Y Y

B

_RP^d _I



_x

 

_x



(1.33)

), 1 ( ˆ )

(

^{( )} ₁ ² ₀₁

1

K

C f Y Y

B

_R^d _II



_x



(1.34)

), (

ˆ )

(

^{( ) 2} _{2 1 02}

2

f f K

C Y Y

B

_P^d _II



_x



(1.35)

), (

ˆ )

(

^{( )} _{1 1} ² _{1 01}

1

t K

C t f Y Y

B

_SDR^d _II



_x



(1.36)

), (

ˆ )

(

^{( )} ₂ ² _{2 2 1 02}

2

f t f K

C t Y Y

B

_SDP^d _II



_x



(1.37)

), (

ˆ )

(

^{( )} _{1 3} ² _{3 01}

1

t K

C t f Y Y

B

_SER^d _II



_x



(1.38)

), (

ˆ )

(

^{( )} ₄ ² _{24 1 02}

2

f t f K

C t Y Y

B

_SEP^d _II



_x



(1.39)

), (

ˆ )

(

⁽₁⁾ _{1 5} ² _{5 01}

1

t K

C t f Y Y

B

_US^d_{R II}



_x



(1.40)

), (

ˆ )

(

⁽₁⁾ _{1 6} ² _{2 6 1 02}

2

f t f K

C t f Y Y

B

_US^d_{P II}



_x



(1.41)

), (

ˆ )

(

⁽₂⁾ _{3 7} ² _{7 01}

1

t K

C t f Y Y

B

_US^d_{R II}



_x



(1.42)

), (

ˆ )

(

⁽₂⁾ _{1 8} ² _{28 1 02}

2

f t f K

C t f Y Y

B

_US^d_{P II}



_x



(1.43)

 ^{( 1 ) ( )}  ^,

ˆ )

(

^{( )} ₁ ² ₀₁ ² _{2 12 1 02}

2

1

K C f K f K

C f Y Y

B

_RP^d _II



_x

 

_x

 

(1.44)

 ^{( 1 2 )} 

ˆ )

(

^{( ) 2} ₁ ² ₃ ² ₀₁

1

K

C f C f Y Y

MSE

_R^d _I



_y



_x



, (1.45)

 ^{( 1 2 )} 

ˆ )

(

^{( ) 2} ₁ ² ₃ ² ₀₂

2

K

C f C f Y Y

MSE

_P^d _I



_y



_x



, (1.46)

 ^{( 2 )} 

ˆ )

(

^{( ) 2} ₁ ² _{3 1} ² _{1 01}

1

t K

C t f C f Y Y

MSE

_SDR^d _I



_y



_x



, (1.47)

 

 

 ^{( 2 )} 

ˆ )

(

^{( ) 2} ₁ ² _{3 2} ² _{2 02}

2

t K

C t f C f Y Y

MSE

_SDP^d _I



_y



_x



, (1.48)

 ^{( 2 )} 

ˆ )

(

^{( ) 2} ₁ ² _{3 3} ² _{3 01}

1

t K

C t f C f Y Y

MSE

_SER^d _I



_y



_x



, (1.49)

 ^{( 2 )} 

ˆ )

(

^{( ) 2} ₁ ² _{3 4} ² _{4 02}

2

t K

C t f C f Y Y

MSE

_SEP^d _I



_y



_x



, (1.50)

 ^{( 2 )} 

ˆ )

(

⁽₁^{) 2} ₁ ² _{3 5} ² _{5 01}

1

t K

C t f C f Y Y

MSE

_US^d_{R I}



_y



_x



, (1.51)

 ^{( 2 )} 

ˆ )

(

⁽₁^{) 2} ₁ ² _{3 5} ² _{6 01}

2

t K

C t f C f Y Y

MSE

_US^d_{P I}



_y



_x



, (1.52)

 ^{( 2 )} 

ˆ )

(

⁽₂^{) 2} ₁ ² _{3 7} ² _{7 01}

1

t K

C t f C f Y Y

MSE

_US^d_{R I}



_y



_x



, (1.53)

 ^{( 2 )} 

ˆ )

(

⁽₂^{) 2} ₁ ² _{3 8} ² _{8 01}

2

t K

C t f C f Y Y

MSE

_US^d_{P I}



_y



_x



, (1.54)

 ^{( 1 2 ) ( 1 2 2 )} 

ˆ )

(

^{( ) 2} ₁ ² ₃ ² _{01 3} ² _{02 12}

2

1

K f C K K

C f C f Y Y

MSE

_SRP^d _I



_y



_x

 

_x

 

, (1.55)

 



1 2 1 01



2 2 1 2 )

(

) ( ) 2

( ˆ

1

f f f K

C C f Y Y

MSE

_R^d _II



_y



_x

 

, (1.56)

 



1 2 1 02



2 2 1 2 )

(

) ( ) 2

( ˆ

2

f f f K

C C f Y Y

MSE

_P^d _II



_y



_x

 

, (1.57)

 



1 1 2 1 01



2 1 2 1 2 )

(

) ( ) 2

( ˆ

1

t f f f K

C t C f Y Y

MSE

_SDR^d _II



_y



_x

 

, (1.58)

 



2 1 2 1 02



2 2 2 1 2 )

(

) ( ) 2

( ˆ

2

t f f f K

C t C f Y Y

MSE

_SDP^d _II



_y



_x

 

, (1.59)

 



3 1 2 1 01



2 3 2 1 2 )

(

) ( ) 2

( ˆ

1

t f f f K

C t C f Y Y

MSE

_SER^d _II



_y



_x

 

, (1.60)

 



4 1 2 1 02



2 4 2 1 2 )

(

) ( ) 2

( ˆ

2

t f f f K

C t C f Y Y

MSE

_SEP^d _II



_y



_x

 

, (1.61)

 



5 1 2 1 01



2 5 2 1 2 )

(

1

) ( ) 2

( ˆ

1

t f f f K

C t C f Y Y

MSE

_US^d_{R II}



_y



_x

 

, (1.62)

 



6 1 2 1 02



2 6 2 1 2 )

(

1

) ( ) 2

( ˆ

2

t f f f K

C t C f Y Y

MSE

_US^d_{P II}



_y



_x

 

, (1.63)

 



7 1 2 1 01



2 7 2 1 2 )

(

2

) ( ) 2

( ˆ

1

t f f f K

C t C f Y Y

MSE

_US^d_{R II}



_y



_x

 

, (1.64)

 



8 1 2 1 02



2 8 2 1 2 )

(

2

) ( ) 2

( ˆ

2

t f f f K

C t C f Y Y

MSE

_US^d_{P II}



_y



_x

 

, (1.65)

and

^{( ˆ )}   ^{2 )}   ⁽

1 2

²

1 02

²

2 12

⁾   ^.

2 01 1 2 1 2 2 1 2 )

(

2

1

f f f K C f f f K f K

C C f Y Y

MSE

_SRP^d _II



_y



_x

  

_x

  

(1.66)

where

)

(

1 1

1 1

C

x

X t X

 

,

)

(

2 2

2 2

C

x

X t X

 

,

)) ( (

_{1 2 1}

1

3

X x

t X



 

,

)) (

(

_{2 2 2}

2

4

X x

t X



 

,

)) (

(

_{1 2 1}

1 5

1 1

x C

X C t X

x x



 

,

)) (

(

_{2 2 2}

2 6

2 2

x C

X C t X

x x



 

,

) ) ( (

) (

1 1

2 1

1 2 1 7

C

x

x X

x t X

 





,

) ) ( (

) (

2 2

2 2

2 2 2 8

C

x

x X

x t X

 





.

x y yx

C K

₀₁

  C

,

z y yz

C K

₀₂

  C

,

z x xz

C K

₁₂

  C

 

 

  

 n N

f 1 1

1 ,





 

  

 

N f n 1 1

2 and

f

₃

 f

₁

 f

₂. 2. Suggested Ratio-Cum-Product Estimators

Tailor et al. (2011 a) proposed ratio-cum-product estimators of population mean

Y

using information on coefficient of variations and coefficient of kurtosis of auxiliary variates

x

₁ and

x

₂ as