Empirical study in finite correlation coefficient in two phase estimation

International Journal of Social Economics Empirical study in finite correlation coefficient in two phase estimation M. Khoshnevisan, F. Kaymarm, H.P. Singh, R. Singh, F. Smarandache,

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Article information: To cite this document: M. Khoshnevisan, F. Kaymarm, H.P. Singh, R. Singh, F. Smarandache, (2004) "Empirical study in finite correlation coefficient in two phase estimation", International Journal of Social Economics, Vol. 31 Issue: 10, pp.890-902, https://doi.org/10.1108/03068290410555381 Permanent link to this document: https://doi.org/10.1108/03068290410555381 Downloaded on: 27 April 2018, At: 08:14 (PT) References: this document contains references to 18 other documents.

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IJSE 31,10

Empirical study in finite correlation coefficient in two phase estimation

890

M. Khoshnevisan School of Accounting and Finance, Griffith University, Australia

F. Kaymarm Downloaded by Professor Florentin Smarandache At 08:14 27 April 2018 (PT)

Department of Mechanical Engineering, Massachusetts Institute of Technology, USA; currently at Sharif University, Tehran, Iran

H.P. Singh and R. Singh Department of Mathematics and Statistics, Vikram University, India

F. Smarandache Department of Mathematics, University of New Mexico, Gallup, USA Keywords Correlation analysis, Variance, Estimation Abstract This paper proposes a class of estimators for population correlation coefficient when information about the population mean and population variance of one of the variables is not available but information about these parameters of another variable (auxiliary) is available, in two phase sampling and analyzes its properties. Optimum estimator in the class is identified with its variance formula. The estimators of the class involve unknown constants whose optimum values depend on unknown population parameters. In earlier research it has been shown that when these population parameters are replaced by their consistent estimates the resulting class of estimators has the same asymptotic variance as that of optimum estimator. An empirical study is carried out to demonstrate the performance of the constructed estimators.

1. Introduction Consider a finite population U ¼ {1; 2; . . .; i; . . .; N }: Let y and x be the study and auxiliary variables taking values yi and xi, respectively, for the ith unit. The correlation coefficient between y and x is defined by

ryx ¼ S yx =ðS y S x Þ where S yx ¼ ðN 2 1Þ21

N X

i 2 XÞ;

ð yi 2 YÞðx

i¼1

S 2x ¼ ðN 2 1Þ21 International Journal of Social Economics Vol. 31 No. 10, 2004 pp. 890-902 q Emerald Group Publishing Limited 0306-8293 DOI 10.1108/03068290410555381

N X

2; ðxi 2 XÞ

i¼1

S 2y ¼ ðN 2 1Þ21

N X i¼1

2; ð yi 2 YÞ

ð1:1Þ

X ¼ N 21

N X

Finite correlation coefficient

xi ;

i¼1

Y ¼ N 21

N X

yi :

i¼1

Based on a simple random sample of size n drawn without replacement, (xi, yi), i ¼ 1; 2; . . .; n; the usual estimator of ryx is the corresponding sample correlation coefficient: Downloaded by Professor Florentin Smarandache At 08:14 27 April 2018 (PT)

r ¼ syx =ðsx sy Þ

ð1:2Þ

where syx ¼ ðn 2 1Þ21

n X

ð yi 2 y Þðxi 2 x Þ;

i¼1

s2x ¼ ðn 2 1Þ21

n X

ðxi 2 x Þ2

i¼1

s2y ¼ ðn 2 1Þ21

n X

ð yi 2 y Þ2 ;

i¼1

y ¼ n 21

n X

yi ;

i¼1

x ¼ n 21

n X

xi :

i¼1

The problem of estimating ryx has been earlier taken up by various authors including Gupta and Singh (1989), Gupta et al. (1978, 1979), Koop (1970), Rana (1989), Singh et al.

(1996) and Wakimoto (1971), in different situations. Srivastava and Jhajj (1986) have further considered the problem of estimating ryx in the situations where the information on auxiliary variable x for all units in the population is available. In such situations, they have suggested a class of estimators for ryx which utilizes the known values of the population mean X and the population variance S 2x of the auxiliary variable x.

In this paper, using two-phase sampling mechanism, a class of estimators for ryx in the presence of the available knowledge (Z and S 2z ) on second auxiliary variable z is considered, when the population mean X and population variance S 2x of the main auxiliary variable x are not known.

2. The suggested class of estimators In many situations of practical importance, it may happen that no information is available on the population mean X and population variance S 2x ; we seek to estimate

891

IJSE 31,10

892

the population correlation coefficient ryx from a sample “s” obtained through a two-phase selection. Allowing simple random sampling without replacement scheme in each phase, the two-phase sampling scheme will be as follows: (1) The first phase sample s* ðs* , U Þ of fixed size n1, is drawn to observe only x in order to furnish a good estimates of X and S 2x : (2) Given s*, the second-phase sample s ðs , s* Þ of fixed size n is drawn to observe y only.

Let x ¼ ð1=nÞ

X

xi ;

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i[s

y ¼ ð1=nÞ

X

yi ;

i[s

X xi ; i[s*

x * ¼ ð1=n1 Þ

s2x ¼ ðn 2 1Þ21

X

ðxi 2 x Þ2 ;

i[s

X ðxi 2 x * Þ2 : i[s*

s*x2 ¼ ðn1 2 1Þ21

We write u ¼ x = x* ; v ¼ s2x =s*x2 : Whatever be the sample chosen let (u, v) assume values in a bounded closed convex subset, R, of the two-dimensional real space containing the point (1, 1). Let h (u, v) be a function of u and v such that: hð1; 1Þ ¼ 1

ð2:1Þ

and such that it satisfies the following conditions: (1) The function h(u, v) is continuous and bounded in R. (2) The first and second partial derivatives of h(u, v) exist and are continuous and bounded in R. Now one may consider the class of estimators of ryx, defined by:

r^hd ¼ rhðu; vÞ

ð2:2Þ

which is double sampling version of the class of estimators r~t ¼ rf ðu* ; v* Þ

v* ¼ s2 =S 2 and X; S2 Suggested by Srivastava and Jhajj (1986), where u* ¼ x =X; x x x are known.

Sometimes even if the population mean X and population variance S 2x of x are not known, information on a cheaply ascertainable variable z, closely related to x but

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compared to x remotely related to y, is available on all units of the population. This type of situation has been briefly discussed by, among others, Chand (1975) and Kiregyera (1980, 1984).

Following Chand (1975) one may define a chain ratio-type estimator for ryx as *2 2 ! sx Sz Z x * r^1d ¼ r ð2:3Þ 2 sx z * x s*z 2 where the population mean Z and population variance S 2z of second auxiliary variable z are known, and X z * ¼ ð1=n1 Þ zi ; i[s* s*z 2 ¼ ðn1 2 1Þ21

X

ðzi 2 z * Þ2

i[s*

are the sample mean and sample variance of z based on preliminary large sample s* of size n1 ð. nÞ: The estimator r^1d in equation (2.3) may be generalized as

x r^2d ¼ r x *

a1

s2x s*x2

a2 a3 *2 !a4 sz z * Z S 2z

ð2:4Þ

where ai 0 s ði ¼ 1; 2; 3; 4Þ are suitably chosen constants.

Many other generalization of r^1d is possible. We have, therefore, considered a more general class of ryx, from which a number of estimators can be generated.

The proposed generalized estimators for population correlation coefficient ryx, is defined by

r^td ¼ rtðu; v; w; aÞ

ð2:5Þ

a ¼ s*2 =S 2 and t(u, v, w, a) is a function of (u, v, w, a) such that where w ¼ z * =Z; z z tð1; 1; 1; 1Þ ¼ 1

ð2:6Þ

Satisfying the following conditions: (1) Whatever be the samples (s* and s) chosen, let (u, v, w, a) assume values in a closed convex subset S, of the four-dimensional real space containing the point P¼ (1, 1, 1, 1).

(2) In S, the function t(u, v, w, a) is continuous and bounded.

(3) The first and second order partial derivatives of t(u, v, w, a) exist and are continuous and bounded in S. To find the bias and variance of r^td we write:

Finite correlation coefficient

893

IJSE 31,10

s2y ¼ S 2y ð1 þ e1 Þ; þ e1 Þ; x ¼ Xð1 þ e* Þ; x * ¼ Xð1 1

894

s2x ¼ S 2x ð1 þ e2 Þ

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s*x2 ¼ S 2x ð1 þ e*2 Þ; þ e* Þ; z * ¼ Zð1 3 s*z 2 ¼ S 2z ð1 þ e*4 Þ; syx ¼ S yx ð1 þ es Þ such that Eðe0 Þ ¼ Eðe1 Þ ¼ Eðe2 Þ ¼ Eðe5 Þ ¼ 0 and

E e*i ¼ 0 ;

i ¼ 1; 2; 3; 4;

and ignoring the finite population correction terms, we write to the first degree of approximation:

E e20 ¼ ðd400 2 1Þ=n; E e21 ¼ C 2x =n; E e*12 ¼ C 2x =n1 ; E e22 ¼ ðd040 2 1Þ=n;

E e*22 ¼ ðd040 2 1Þ=n1 ;

E e*32 ¼ C 2z =n1 ;

o. n E e25 ¼ d220 =r2yx 2 1 n;

Eðe0 e1 Þ ¼ d210 C x =n;

E e0 e*2 ¼ ðd220 2 1Þ=n1 ;

Eðe0 e2 Þ ¼ ðd220 2 1Þ=n;

E e0 e*4 ¼ ðd202 2 1Þ=n1 ;

E e1 e*1 ¼ C 2x =n1 ;

E e*1 e2 ¼ d030 C x =n1 ;

E e0 e*1 ¼ d210 C x =n1 ;

E e0 e*3 ¼ d201 C z =n1 ;

Eðe0 e5 Þ ¼ {ðd310 =ryx Þ 2 1}=n;

Eðe1 e2 Þ ¼ d030 C x =n;

E e1 e*3 ¼ rxz C x C z =n1 ;

E e*42 ¼ ðd004 2 1Þ=n1 ;

E e1 e*2 ¼ d030 C x =n1 ;

E e1 e*4 ¼ d012 C x =n1 ;

E e*1 e*2 ¼ d030 C x =n1 ;

Eðe1 e5 Þ ¼ ðd120 C x =ryx Þ=n;

E e*1 e*3 ¼ rxz C x C z =n1 ;

E e*1 e*4 ¼ d012 C x =n1 ;

E e*1 e5 ¼ ðd120 C x =ryx Þ=n1 ;

E e2 e*2 ¼ ðd040 2 1Þ=n1 ;

E e2 e*3 ¼ d021 C z =n1 ;

Eðe2 e5 Þ ¼ {ðd130 =ryx Þ 2 1}=n;

E e*2 e*4 ¼ ðd022 2 1Þ=n1 ;

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E e*3 e*4 ¼ d003 C z =n1 ;

Finite correlation coefficient

E e2 e*4 ¼ ðd022 2 1Þ=n1 ;

E e*2 e*3 ¼ d021 C z =n1 ;

895

E e*2 e5 ¼ {ðd130 =ryx Þ 2 1}=n1 ;

E e*3 e5 ¼ ðd111 C z =ryx Þ=n1 ;

E e*4 e5 ¼ {ðd112 =ryx Þ 2 1}=n1 : where

p=2 q=2 m=2 dpqm ¼ mpqm = m200 m020 m002 ;

mpqm ¼ ð1=N Þ

N X

p ðxi 2 XÞ q ðzi 2 ZÞ m; ð yi 2 YÞ

i¼1

( p, q, m) being non-negative integers.

To find the expectation and variance of r^td ; we expand t(u, v, w, a) about the point P ¼ ð1; 1; 1; 1Þ in a second-order Taylor’s series, express this value and the value of r in terms of e’s. Expanding in powers of e’s and retaining terms up to second power, we have: Eðr^td Þ ¼ ryx þ oðn 21 Þ

ð2:7Þ

which shows that the bias of r^td is of the order n 2 1 and so up to order n 2 1, mean square error and the variance of r^td are same.

Expanding ðr^td 2 ryx Þ2 ; retaining terms up to second power in e’s, taking expectation and using the above expected values, we obtain the variance of r^td to the first degree of approximation, as:

h Varðr^td Þ ¼ VarðrÞ þ r2yx =n C 2x t 21 ðPÞ þ ðd040 2 1Þt22 ðPÞ 2 At 1 ðPÞ i h 2 Bt2 ðPÞ þ 2d030 C x t 1 ðPÞt 2 ðPÞ 2 r2yx =n1 C 2x t21 ðPÞ þ ðd040 2 1Þt 22 ðPÞ ð2:8Þ 2 C 2z t23 ðPÞ 2 ðd004 2 1Þt 24 ðPÞ 2 At 1 ðPÞ 2 Bt2 ðPÞ þ Dt 3 ðPÞ þ Ft 4 ðPÞ þ 2d030 C x t 1 ðPÞt 2 ðPÞ 2 2d003 C z t3 ðPÞt4 ðPÞ

i

where t1(P), t2(P), t3(P) and t4(P), respectively, denote the first partial derivatives of t(u, v, w, a) respect to u, v, w and a, respectively, at the point P ¼ ð1; 1; 1; 1Þ; with

h i

VarðrÞ ¼ r2yx =n d220 =r2yx þ ð1=4Þðd040 þ d400 þ 2d220 Þ 2 {ðd130 þ d310 Þ=ryx } ð2:9Þ

IJSE 31,10

A ¼ {d210 þ d030 2 2ðd120 =ryx Þ}C x ; B ¼ {d220 þ d040 2 2ðd130 =ryx Þ};

896

D ¼ {d201 þ d021 2 2ðd111 =ryx Þ}C z ;

Downloaded by Professor Florentin Smarandache At 08:14 27 April 2018 (PT)

F ¼ {d202 þ d022 2 2ðd112 =ryx Þ} Any parametric function t(u, v, w, a) satisfying equation (2.6) and the conditions 1 and 2 can generate an estimator of the class equation (2.5).

The variance of r^td at equation (2.6) is minimized for: t1 ðPÞ ¼

½Aðd040 2 1Þ 2 Bd030 C x

¼ aðsayÞ; 2C 2x d040 2 d2030 2 1

BC 2x 2 Ad030 C x t2 ðPÞ ¼ 2 ¼ bðsayÞ; 2C x d040 2 d2030 2 1 ð2:10Þ ½Dðd004 2 1Þ 2 F d003 C z t3 ðPÞ ¼

¼ gðsayÞ; 2C 2z d004 2 d2030 2 1

C 2z F 2 Dd003 C z t4 ðPÞ ¼ 2 ¼ dðsayÞ; 2C z d004 2 d2003 2 1 Thus the resulting (minimum) variance of r^td is given by: #

" 1 1 2 A 2 {ðA=C x Þd030 2 B}2

2 r þ min:Varðr^td Þ ¼ VarðrÞ 2 n n1 yx 4C 2x 4 d040 2 d2030 2 1

2 r2yx =n1

" D2

{ðD=C z Þd003 2 F}2

þ 4C 2z 4 d004 2 d2003 2 1

#

ð2:11Þ

It is observed from equation (2.11) that if optimum values of the parameters given by (2.10) are used, the variance of the estimator r^td is always less than that of r as the last two terms on the right hand sides of (2.11) are non-negative.

Two simple functions t(u, v, w, a) satisfying the required conditions are: tðu; v; w; aÞ ¼ 1 þ a1 ðu 2 1Þ þ a2 ðv 2 1Þ þ a3 ðw 2 1Þ þ a4 ða 2 1Þ tðu; v; w; aÞ ¼ u a1 v a2 w a3 a a4 and for both these functions t 1 ðPÞ ¼ a1 ; t2 ðPÞ ¼ a2 ; t 3 ðPÞ ¼ a3 and t4 ðPÞ ¼ a4 : Thus, one should use optimum values of a1, a2, a3 and a4 in r^td to get the minimum variance.

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It is to be noted that the estimated r^td attained the minimum variance only when the optimum values of the constants ai (i¼ 1, 2, 3, 4), which are functions of unknown population parameters, are known. To use such estimators in practice, one has to use some guessed values of population parameters obtained either through past experience or through a pilot sample survey. It may be further noted that even if the values of the constants used in the estimator are not exactly equal to their optimum values as given by equation (2.8) but are close enough, the resulting estimator will be better than the conventional estimator, as illustrated by Das and Tripathi (1978, Section 3).

If no information on second auxiliary variable z is used, then the estimator r^td reduces to r^hd defined in equation (2.2). Taking z ; 1 in equation (2.8), we get the variance of r^hd to the first degree of approximation, as:

1 1 2h 2 2 ^ 2 r C h ð1; 1Þ þ ðd040 2 1Þh22 ð1; 1Þ Varðrhd Þ ¼ VarðrÞ þ n n1 yx x 1 ð2:12Þ i 2 Ah1 ð1; 1Þ 2 Bh2 ð1; 1Þ þ 2d030 C x h1 ð1; 1Þh2 ð1; 1Þ which is minimized for, h1 ð1; 1Þ ¼

½Aðd040 2 1Þ 2 Bd030 C x 2C 2x ðd040 2 d2030 2 1Þ

h2 ð1; 1Þ ¼

BC 2x 2 Ad030 C x

;

ð2:13Þ

2C 2x ðd040 2 d2030 2 1Þ

Thus the minimum variance of r^hd is given by: #

" 1 1 2 A 2 {ðA=C x Þd030 2 B}2

2 r þ min:Varðr^hd Þ ¼ VarðrÞ 2 n n1 yx 4C 2x 4 d040 2 d2030 2 1 It follows from equations (2.11) and (2.14) that " #

D 2 {ðD=C Þd 2 F}2 z 003 2

þ min:Varðr^td Þ 2 min:Varðr^hd Þ ¼ ryx =n1 4C 2z 4 d004 2 d2003 2 1

ð2:14Þ

ð2:15Þ

which is always positive. Thus, the proposed estimator r^td is always better than r^hd : 3. A wider class of estimators In this section, we consider a class of estimators of ryx wider than equation (2.5) given by:

r^gd ¼ gðr; u; v; w; aÞ where g(r, u, v, w, a) is a function of r, u, v, w, a and such that

ð3:1Þ

Finite correlation coefficient

897

IJSE 31,10

898

gðr; 1; 1; 1; 1Þ ¼ r^td

and

›gð·Þ ›r

¼1 ðr;1;1;1Þ

Proceeding as in Section 2, it can easily be shown, to the first order of approximation, that the minimum variance of r^gd is same as that of r^td given in equation (2.11).

It is to be noted that the difference-type estimator r d ¼ r þ a1 ðu 2 1Þ þ a2 ðv 2 1Þ þ a3 ðw 2 1Þ þ a4 ða 2 1Þ;

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is a particular case of r^gd ; but it is not the member of r^td in equation (2.5).

4. Optimum values and their estimates The optimum values t 1 ðPÞ ¼ a; t 2 ðPÞ ¼ b; t3 ðPÞ ¼ g and t 4 ðPÞ ¼ d given at equation (2.10) involve unknown population parameters. When these optimum values are substituted in equation (2.5), it no longer remains an estimator since it involves unknown (a, b, g, d), which are functions of unknown population parameters, say, dpqm ðp; q; m ¼ 0; 1; 2; 3; 4Þ; Cx, Cz and ryx itself. Hence, it is advisable to replace them by their consistent estimates from sample values. Let ða^; b^; g^; d^Þ be consistent estimators of t1(P), t2(P ), t3(P) and t4(P ), respectively, where ^ ^ 2 1Þ 2 B^ d^030 C^ x ^t1 ðPÞ ¼ a^ ¼ ½Aðd040

; 2 2 2C^ x d^040 2 d^030 2 1 h 2 i ^ d^030 C^ x B^ C^ x 2 A ^t2 ðPÞ ¼ b^ ¼

; 2 2 2C^ x d^040 2 d^030 2 1 ð4:1Þ ^ ^ 2 1Þ 2 F^ d^003 C^ z ^t3 ðPÞ ¼ g^ ¼ ½Dðd004

; 2 2 2C^ d^004 2 d^ 2 1 z

003

h 2 i C^ z F^ 2 D^ d^003 C^ z ^t4 ðPÞ ¼ d^ ¼

; 2 2 2C^ z d^004 2 d^003 2 1 with ^ ¼ ½d^210 þ d^030 2 2ðd^120 =rÞ C^ x ; A B^ ¼ ½d^220 þ d^040 2 2ðd^130 =rÞ ; D^ ¼ ½d^201 þ d^021 2 2ðd^111 =rÞ C^ z ; F^ ¼ ½d^202 þ d^022 2 2ðd^112 =rÞ ;

C^ x ¼ sx = x;

Finite correlation coefficient

C^ z ¼ sz = z;

p=2 q=2 m=2 d^pqm ¼ m^pqm = m^200 m^020 m^002

m^pqm

899

n X ¼ ð1=nÞ ð yi 2 y Þp ðxi 2 x Þq ðzi 2 z Þm

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i¼1 n X zi ; z ¼ ð1=nÞ i¼1

s2x ¼ ðn 2 1Þ21

n X

ðxi 2 x Þ2 ;

i¼1

x ¼ ð1=nÞ

n X

xi ;

i¼1

r ¼ syx =ðsy sx Þ; s2y ¼ ðn 2 1Þ21

n X

ð yi 2 y Þ2 ;

i¼1

s2z ¼ ðn 2 1Þ21

n X

ðxi 2 z Þ2 :

i¼1

We then replace (a, b, g, d) by ða^; b^; g^; d^Þ in the optimum r^td resulting in the estimator r^*td say, which is defined by:

r^*td ¼ rt* ðu; v; w; a; a^; b^; g^; d^Þ;

ð4:2Þ

where the function t*(U ), U ¼ ðu; v; w; a; a^; b^; g^; d^Þ is derived from the function t(u, v, w, a) given at equation (2.5) by replacing the unknown constants involved in it by the consistent estimates of optimum values. The condition (2.6) will then imply that t* ðP* Þ ¼ 1 where P* ¼ ð1; 1; 1; 1; a; b; g; dÞ: We further assume that

ð4:3Þ

IJSE 31,10

›t* ðU Þ t1 ðP* Þ ¼ ›u

900

›t* ðU Þ t3 ðP* Þ ¼ ›w

¼ a;

›t* ðU Þ ¼b t 2 ðP* Þ ¼ ›v U ¼P *

¼ g;

›t* ðU Þ ¼d t 4 ðP* Þ ¼ ›a U ¼P *

U ¼P *

U ¼P *

ð4:4Þ

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t5 ðP* Þ ¼

›t* ðU Þ ›a^

›t* ðU Þ t7 ðP* Þ ¼ ›g^

¼ 0; U ¼P *

¼ 0; U ¼P *

t6 ðP* Þ ¼

›t* ðU Þ ›b^

›t* ðU Þ t8 ðP* Þ ¼ ›d^

¼0 U ¼P *

¼0 U ¼P *

Expanding t*(U) about P* ¼ ð1; 1; 1; 1; a; b; g; dÞ; in Taylor’s series, we have:

r^*td ¼ r t* ðP* Þ þ ðu 2 1Þt *1 ðP* Þ þ ðv 2 1Þt*2 ðP* Þ þ ðw 2 1Þt*3 ðP* Þ þða 2 1Þt *4 ðP* Þ þ ða^ 2 aÞt*5 ðP* Þ þ ðb^ 2 bÞt*6 ðP* Þ þ ðg^ 2 gÞt *7 ðP* Þ þðd^ 2 dÞt*8 ðP* Þ þ second order terms

ð4:5Þ

Using equation (4.4) in (4.5) we have:

r^*td ¼ r½1 þ ðu 2 1Þa þ ðv 2 1Þb þ ðw 2 1Þg þ ða 2 1Þd þ second order terms ð4:6Þ Expressing equation (4.6) in term of e’s squaring and retaining terms of e’s up to second degree, we have:

2 * 2

1 r^td 2 ryx ¼ r2yx ð2e5 2 e0 2 e2 Þ þ a e1 2 e*1 þ b e2 2 e*2 þ ge*3 þ de*4 ð4:7Þ 2 Taking expectation of both sides in equation (4.7), we get the variance of r^*td to the first degree of approximation, as: Var

r^*td

# " 1 1 2 A 2 {ðA=C x Þd030 2 B}2

2 ¼ VarðrÞ 2 r þ n n1 yx 4C 2x 4 d040 2 d2030 2 1

þ r2yx =n1

" D2 4C 2z

2

{ðD=C z Þd003 2 F}

þ 4 d004 2 d2003 2 1

#

which is same as equation (2.11), we thus have established the following result.

ð4:8Þ

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Result 4.1. If optimum values of constants in equation (2.10) are replaced by their consistent estimators and conditions (4.3) and (4.4) hold good, the resulting estimator r^*td has the same variance to the first degree of approximation, as that of optimum r^td : Remark 4.1. It may be easily examined that some special cases: ^ ^ (1) r^*td1 ¼ ru a^v b w g^a d ;

(2) r^*td2 ¼ r{1 þ a^ðu 2 1Þ þ g^ðw 2 1Þ} {1 2 b^ðv 2 1Þ 2 d^ða 2 1Þ} (3) r^*td3 ¼ r½1 þ a^ðu 2 1Þ þ b^ðu 2 1Þ þ g^ðw 2 1Þ þ d^ða 2 1Þ (4) r^*td4 ¼ r½1 2 a^ðu 2 1Þ 2 b^ðu 2 1Þ 2 g^ðw 2 1Þ 2 d^ða 2 1Þ 21

Finite correlation coefficient

901

of r^*td satisfy the conditions (4.3) and (4.4) and attain the variance (4.8).

Remark 4.2. The efficiencies of the estimators discussed in this paper can be compared for fixed cost, following the procedure given in Sukhatme et al. (1984).

5. Empirical study To illustrate the performance of various estimators of population correlation coefficient, we consider the data given in Murthy (1967, p. 226). The variates are: y ¼ output; x ¼ number of workers; z ¼ fixed capital; N ¼ 80; n ¼ 10; n1 ¼ 25; X ¼ 283:875;

Y ¼ 5182:638;

C z ¼ 0:7460;

d003 ¼ 1:030;

d030 ¼ 1:295;

d040 ¼ 3:65;

Z ¼ 1126;

C x ¼ 0:9430;

d004 ¼ 2:8664; d102 ¼ 0:7491;

C y ¼ 0:3520;

d021 ¼ 1:1859; d120 ¼ 0:9145;

d022 ¼ 3:1522; d111 ¼ 0:8234;

d130 ¼ 2:8525;

d112 ¼ 2:5454;

d210 ¼ 0:5475;

d220 ¼ 2:3377;

d202 ¼ 2:2208;

d300 ¼ 0:1301;

d400 ¼ 2:2667;

ryx ¼ 0:9136;

d201 ¼ 0:4546; rxz ¼ 0:9859;

ryz ¼ 0:9413:

The percent relative efficiencies (PREs) of r^1d ; r^hd ; r^td with respect to conventional estimator r have been computed and compiled in Table I. Table I clearly shows that the proposed estimator r^td (or r^*td ) is more efficient than r and r^hd :

Estimator

r

r^hd

r^td (or r^*td )

PRE(., r)

100

129.147

305.441

Table I. The PRE’s of different estimators of ryx

IJSE 31,10

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902

References Chand, L. (1975), “Some ratio-type estimators based on two or more auxiliary variables”, PhD dissertation, Iowa State University, Ames, Iowa.

Das, A.K. and Tripathi, T.P. (1978), “Use of auxiliary information in estimating the finite population variance”, Sankhya, Ser. C, 40, pp. 139-48.

Gupta, J.P. and Singh, R. (1989), “Usual correlation coefficient in PPSWR sampling”, Journal of Indian Statistical Association, Vol. 27, pp. 13-16.

Gupta, J.P., Singh, R. and Lal, B. (1978), “On the estimation of the finite population correlation coefficient-I”, Sankhya C, Vol. 40, pp. 38-59.

Gupta, J.P., Singh, R. and Lal, B. (1979), “On the estimation of the finite population correlation coefficient-II”, Sankhya C, Vol. 41, pp. 1-39.

Kiregyera, B. (1980), “A chain-ratio type estimators in finite population, double sampling using two auxiliary variables”, Metrika, Vol. 27, pp. 217-23.

Kiregyera, B. (1984), “Regression type estimators using two auxiliary variables and the model of double sampling from finite populations”, Metrika, Vol. 31, pp. 215-26.

Koop, J.C. (1970), “Estimation of correlation for a finite Universe”, Metrika, Vol. 15, pp. 105-9.

Murthy, M.N. (1967), Sampling Theory and Methods, Statistical Publishing Society, Calcutta, India.

Rana, R.S. (1989), “Concise estimator of bias and variance of the finite population correlation coefficient”, Jour. Ind. Soc. Agr. Stat., Vol. 41 No. 1, pp. 69-76.

Singh, S., Mangat, N.S. and Gupta, J.P. (1996), “Improved estimator of finite population correlation coefficient”, Jour. Ind. Soc. Agr. Stat., Vol. 48 No. 2, pp. 141-9.

Srivastava, S.K. and Jhajj, H.S. (1986), “On the estimation of finite population correlation coefficient”, Jour. Ind. Soc. Agr. Stat., Vol. 38 No. 1, pp. 82-91.

Sukhatme, P.V., Sukhatme, B.V., Sukhatme, S. and Asok, C. (1984), Sampling Theory of Surveys with Applications, Indian Society of Agricultural Statistics, New Delhi.

Wakimoto, K. (1971), “Stratified random sampling (III): estimation of the correlation coefficient”, Ann. Inst. Statist. Math., Vol. 23, pp. 339-55.

Further reading Singh, R.K. (1982), “On estimating ratio and product of population parameters”, Cal. Stat. Assoc.

Bull., Vol. 32, pp. 47-56.

Srivastava, S.K. (1967), “An estimator using auxiliary information in sample surveys”, Cal. Stat.

Assoc. Bull., Vol. 16, pp. 121-32.

Srivastava, S.K. and Jhajj, H.S. (1983), “A class of estimators of the population mean using multi-auxiliary information”, Cal. Stat. Assoc. Bull., Vol. 32, pp. 47-56.

Srivenkataremann, T. and Tracy, D.S. (1989), “Two-phase sampling for selection with probability proportional to size in sample surveys”, Biometrika, Vol. 76, pp. 818-21.