Apartado 19207, 28080 Madrid, Spain. 2 School of Studies in Statistics, Vikram University, Ujjain - 456 010,. Madhya Pradesh, India. 3 Institute of Management, ...
Statistical Papers49, 133-137(2008)
Statistical Papers © Springer-Verlag2008
On inverse sampling without replacement M. Ruiz Espejo l, H. P. Singhz, S. Saxena 3 Departamento de Matemfiticas Fundamentales, UNED and UPCO,
Apartado 19207, 28080 Madrid, Spain 2 School of Studies in Statistics, Vikram University, Ujjain - 456 010, Madhya Pradesh, India 3 Institute of Management, Nirma University of Science and Technology, Ahmedabad- 382 481, Gujarat, India
Received: July 30, 2004; revised version: May 19, 2005
The inverse sampling (due to Haldane, 1945) was proposed for random sampling with replacement draws when "the number r of successes in the sample until the process of drawing is stopped" is prefixed. In this paper an extension of that result is proposed when the draws are obtained via random sampling without replacement.
Key Words: Finite population; Inverse sampling; Sampling without replacement.
1. I n t r o d u c t i o n . Haldane (1945) has considered the estimation of a population proportion in inverse sampling. A population proportion p (of successes) is considered when sampling is with replacement. Let r be the "prefixed (before sampling) number of successes" and n = x + r is the "random size of the sample with replacement until the r t h prefixed success appears"; then x -- n - r is the "random number of failures until the r t h prefixed success appears". The random variable X (which takes the random values x) is named negative binomial distribution of parameters (r,p) when the units are drawn with replacement,
134
and the quantity function of X is given by P r ( X = x ) (=x + r -
x
1) pr (1 - p)X for x = 0, 1,2,...
The mathematical expectation of X is E ( X ) - r(1 - p ) , P and an "observable unbiased estimator of E ( X ) " is x; from the equation that provides the method of the moments, we obtain X--
r(X - p) P
and then the estimator of p (for random sampling with replacement) is given by ID~ = r/n = r/(x + r), where r is prefixed and n = x + r is a random and observable number, because x is random and observable, and r is prefixed and known (see also Cochran, 1977).
2. N e g a t i v e h y p e r g e o m e t r i c d i s t r i b u t i o n . When the random sampling is without replacement of a finite population of size N = NI+N2, where N1 is the number of success units, and N2 is the number of failure units in the finite population; if r (the number of successes in the random sample without replacement) is prefixed, with r = 1, 2, ..., N1 and p = N1/N, the random variable "number x of failures in the sample until the r t h success appears" has got the quantity function
(x rx x)(N xr r) Pr (X = x) =
for x = 0, 1,2,...,N2;
it is named by us the negative hypergeometric distribution of parameters (N, r,p). The event (X = x) has a probability which
135
is the product of the two following probabilities of the events A and B. A = (We obtain r - 1 successes and x failures) and B = (We obtain the success r in the last draw I A). Their probabilities are respectively
(N-x-r+l) Pr(A)=(x+r-1) Nl-r+l x (N)~ and
Nl-r+l Pr(B)=
N-x-r+1"
The first factor of the expression Pr(A) is due to the different orders in which "r - 1 sucesses and x failures" can appear. The second factor is the probability of one of these events; this probability is obtained by the expression: (N1NI-
N N-1 -
N2
-
Nl-r+2) N-r+ N2-x+l_ ) N-r-x+2
1 ,
°
,
N2-1
N-r+lN-r
N~
N2~
(N1 - r + 1)! (N2 - x)!
N!
(N- r-
x + 1)[
( N - z - r + 1)!
(N1 - r -I- 1 ) ] . (N2 - x)]
N! N I ! . N2~
N1 - r + l
/
136
Thus the probability
Pr(X=x)=Pr(A)xPr(B)
for x = 0 , 1 , . . . , N 2 .
T h e m a t h e m a t i c a l e x p e c t a t i o n of this r a n d o m v a r i a b l e is s h o w e d in a r e c u r r e n t way; in t h e first case we consider r = NI:
N2 x (Nl + x-1) E(X I r= N1) = ~ ( Na x
x-----1
N1 for a m i n o r value of r, we h a v e t h e following r e c u r r e n t e x p r e s s i o n
N2 x=O
_
(N)N1x=l
r x
Nl-r l
\ N1J a n d n o w we d e n o t e y = x - 1, a n e w r a n d o m v a r i a b l e w h i c h allow us this e x p r e s s i o n r
~_1(
E ( X ' r ) - ( N)N1
y+
=
Y
N1
r
'
137
a n d now we d e n o t e s = r + 1, a k n o w n c o n s t a n t which verifies r _< N1 - 1 (the case r = N1 has been solved before); t h e n we can write
_
r
=
y
\Nl-s+l
'
but N1 -
NI -s+1'
and then,
r E (X , r) - ( N
~
y+s-1 Y
)( N Nl- y - -s s ) NlN-2 -sy+ l N
N
\
N2
j
7•
-
N1 - r {N2 - z ( x I ~ + 1)}
which is a r e c u r r e n t expression of t h e m a t h e m a t i c a l e x p e c t a t i o n for r = N1 - 1, N1 - 2, ..., 3,2 a n d 1. T h i s m a t h e m a t i c a l exp e c t a t i o n is u n b i a s e d l y e s t i m a t e d by x in each p a r t i c u l a r ease. Thus, we have d e v e l o p e d t h e q u a n t i t y f u n c t i o n of t h e negative h y p e r g e o m e t r i c d i s t r i b u t i o n a n d calculated its m a t h e m a t i c a l exp e c t a t i o n for all t h e values of r.
References C o c h r a n W G (1977) Sampling Techniques, 3rd edition. Wiley, New York
Haldane JBS (1945) On a method of estimating frequencies. 33, 222-225
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