1. Introduction - Science Direct

Journal

of Statistical

Planning

and Inference

23 (1989) 217-225

217

North-Holland

METHOD

OF ESTIMATION

RANDOM

SAMPLE

Randhir

SINGH

FROM

SAMPLES

WITH

SIZES

and P. NARAIN

Indian Agricultural Statistics Research Institute, Library Avenue, New Delhi 110012, India Received

5 March

Recommended

1985; revised manuscript

Abstract: In sample surveys, pens mainly realized

received

19 July 1988

by D. Raghavarao

quite often,

in two situations,

sample

In the present

namely,

size may be random, investigation

the sample

size becomes

(i) when the planned and (ii) when planned

we propose

a method

a random

sample sample

of estimation

variable.

This hap-

size may be fixed but the size itself may be random.

which is quite efficient

in both

these situations.

AIMS Subject Classification: 62D05-62D08. Key words and phrases: Random

sample

size; conditional

inclusion

probability;

probability

of

response.

1. Introduction In large scale sample surveys quite often situations arise when the sample size becomes a random variable. This happens mainly in two types of situations: (i) When the planned sample size is fixed but the realized sample size is random. This may occur either due to non-response from some of the selected units or may arise due to non-existence of some selected units in the target population because of use of some out-dated frame for selecting the sample or due to non-converage of some units. (ii) When the planned sample size itself is random, e.g. binomial sampling (Goodman, 1949), Poisson sampling (Hajek, 1964) or 3-P sampling (Grosenbaugh, 1967). A random sample size may also arise in multi-stage sampling when all the last stage units, or all units possessing some particular characteristics are included in the sample. The problem of non-response has been considered at great length by various workers. Iachan (1983) has presented a good review of non-response errors in surveys. In the present investigation we propose a method of estimation which can be useful in both the above situations resulting in random sample size.

0378.3758/89/$3.50

0

1989, Elsevier

Science Publishers

B.V. (North-Holland)

218

R. Singh, P. Narain / Estimation with random sample sizes

2. Method of estimation

when planned sample size is fixed but realized sample size

is random

Consider a finite population of size N. Let yi be the value of the character under study, y, for the i-th unit. Now each unit of the population has a definite probability of providing the necessary information with respect to the character under study under the particular given field method. Let pi denote this probability of obtaining required information from the i-th unit and let qi= 1 -pi. Select a sample of size n by simple random sampling without replacement. Out of these n units let only m units provide the required information. In repeated samples, 112is a random variable which takes different integral values 0,1,2, . . . , n. For the sampling scheme defined above, an unbiased estimator of the population total for the character under study y is provided by

c 0

if m = 0, if m>O,

(1)

where ui = 1 or 0 according as the unit i is or is not in the sample and where Ei= 1 or 0 according as the unit i does or does not provide the required information. Further we assume (i) e i, . . . , EN are independent and E(Ei) = Pi. (ii) The vectors (E,, . . . ,&N) and (u,, . . . , UN) are independent, and (iii) E(Ui) = n/N and E(UiUj) = n(n - l)/N(N- 1). The variance of T, can be shown to be l’(T,)

= N2

(2)

It is of interest to note that in (2) above the second term represents the contribution due to non-response. An unbiased estimator of the variance of T,may be easily shown to be given by

P(T,)= (3)

3. The new estimator Godambe (1955) pointed out that the use of the estimator proposed by Horvitz and Thompson (1952) is not desirable when the sample size is random. Therefore, an alternative estimator in place of Tl is proposed here. But before proposing the new estimator we define the following:


of m responses

Let .X0 be the sum over all (k) combinations

219

from the sample

of

size n; Zr be the sum over all (:I\) combinations of m - 1 responses out of (n - 1) units selected in the sample excluding unit, i; & be the sum over all (:I’,) combinations of (m - 2) responses from (n - 2) units excluding units i and j ; and p(m) be the probability of obtaining the realized sample of size m from the sample n given by P(W) = &I to, Pt “ii” 41, t,+t

and po=p(0)=q~q2-..q,,, the probability sample of Further obtaining sample of

of obtaining zero response from the size n. [For brevity we will use p. for p(O).] let ni(m)be the conditional inclusion probability, i.e. the probability of response from the i-th unit, given that m units respond from a fixed size n: m-1 Mm)

=Pi

&

n-m

II Pt II 4t, p(m), tfi t’ft,i I/

[

and xv(m)be the conditional probability of response of the pair of units i, j, given that m units respond from a fixed sample of size n: m-2 x&4

=

PiPj

&

[

n - In

fl Pt n 4t, t+i,j t’+t,i,j

Ii

p(m).

Here pt pertains to the t-th unit among respondents in the sample to the t’-th unit among the non-respondents in the sample. Now consider the estimator

c 0

T, =

if m = 0, N

n(l -PO) The expected

FL i=l

= ErEAT2)

(4)

ifm>O. ni(m)

value and variance

E(C)

and qt, pertains

and

of T2 are given by v(7’A = Er V2(T2)+ V,E2(W

where E,, E, and V2, Vr denote expected possible samples of size n respectively.

value and variance

(5)

for fixed m and for all

Now for given m, ni(m)is the probability of inclusion for the i-th unit and hence Cz, _Y;/Xi(m) estimates unbiasedly the sample total Cn=r yi. Thus we have

&Vi) = On taking

0

NJ,

further

expectation

E(T,)

= E,E2(T2)

if m = 0, if m>O. over all possible = NY = Y,

samples

we obtain

220


the population

total for y. To obtain the variance of T2 we see that

V,E,(T,) = I’, (NJ,) = N2

S2,

(6)

N2 v,(T2) =

2 2 ii (1 -po) n I