Austral. J. Statist., 19 (2). 1977, 105-107.

A NOTE ON THE USE OF INCOMPLETE

I N SAMPLE SURVEYS' MULTI-AUXILIARY INFORMATION

RANDHIR SINGH Institwte of Agricultural Research Statistics, New Delhi, India

1. Introduction

In sampling theory the precision of estimates may be improved substantially by the use of auxiliary information available for some character, x, which is correlated with the character under study, y. It may be used for the purpose of selection of the sample, stratification of the population or estimation of the parameters. If 2 is known for the population but x is not known for every unit, then procedures of ratio and regression methods of estimation are commonly used. When such information is available for more than one character, Olkin (1958), Des Raj (1965) and Shukla (1965) have presented the use of multivariate ratio and regression methods of estimation.

Frequently there may arise situations where we may possess information about several auxiliary variables but each variable may be known for some part of population only. It is important to have techniques at hand which enable the investigator to make maximum use of such information. In the present investigation a method of estimation for such a situation is provided.

2. Statement of the Problem and Method of Estimation Consider a finite population of size N. Let xl, x 2 , . . . , xp be the

p-auxiliary vanates which are known for N,, N 2 , . . . , N, units of the population respectively. In this situation, the population may be consi- dered as consisting of different strata according to the number of auxiliary variates known. Let there be a stratum of size No for which no auxiliary variate is known, p strata each of size Nii(i = 1,2 , . . . , p) for which only the i-th variate is known. Similarly there will be pc, strata each of s u e Nii(i < j ; i, J = 1,2, . . . , p), for which the two var- iates and 3 are known, Pc, strata each of size N&(i<j< k ; i, j , k = 1,2,. . . , p ) for which the 3 vanates xi, xi and X k are known, and so on. Ultimately we will have a stratum of sue Nl.2,. . . , , €or which all the p vanates are known. Thus we will have the population consisting of a total number of 2p strata since pQ+ pc, + pc, . . . + pc, = 2p. * Manuscript received June 24, 1976; revised December 22, 1976.

106 R. SINGH

Now select a stratifed sample of size n from the population With simple random sampling without replacement within each stratum. Let n, denote the sample size from the s-th stratum, s = 1,2, . . . ,2', such that 1:' n, = n. Let y be the character under study which is observed from the sample.

Let F, and ji, denote the population mean and sample mean respectively for y from the s-th stratum. Let xsi and .fs, denote the population mean and sample mean respectively for the i-th auxiliary variable in the s-th stratum, where i may take the values 0,1,2,. . . , p within different strata.

Now consider the j-th stratum in which k auxiliary variables are known. We may obtain the multivariate unbiased linear regression estimate based on k auxiliary variables for as

k

ylr . jk = yj + 1 p y . j I ( x . l - ZjI) 1-1

(2.1)

where py.p is the regression coefficient of y on x1 in the j-th stratum. The variance of jilr.jk may be obtained as (Shukla, 1965)

Where S: is the variance of y for the j-th stratum, and Ri:, is the square of multiple correlation coefficient of y with the k variables known for the j-th stratum.

written as Now an unbiased estimate of the population mean, may

where k may take values 0,1,2, . . . , p and Wi is the proportion units in the j-th stratum. The variance of jilr may be written as

(2.4)

be

of

These may be easily obtained by using the values of j i lr . j~ and v ( g 1 r . j k )

for dif€erent values of k from (2.1) and (2.2) respectively.

s-ary In this paper an estimation procedure has been suggested making

use of multivariate auxiliary information when information on some of the auxiliary variables is not availahle for the whole population, instead it is known for some parts of the population only.

MULTI-AUXILIARY INFORMATION 107

References

[1] Olkin, I. (1958). “Multivariate ratio estimation for finite populations” Biornetrika,

[2] Raj, D. (1965). “On a method of using multi-auxiliary information in sample

[3] Shukla, G. R. (1965). “Multivariate regression estimate” J. Indian Statist. Assoc. 3.

45, 154-165.

s w e y s ” I. Amer. Statist. Assoc., 60, 270-277.