Normal approximation to the hypergeometric distribution in ...

S.N. Lahiri et al. / Journal of Statistical Planning and Inference 137 (2007) 3570 –3590 3579
For notational simplicity, we shall drop the suffix r from notation, except when it is important to highlight the
dependence on r. Thus, we write n, M, N for nr,Mr,Nr, respectively, and set p = M/N, q = 1 − p and f = n/N.
We shall use C to denote a generic positive constant that does not depend on r. Unless otherwise stated, limits in order
symbols are taken by letting r →∞.
For proving the result, we shall frequently make use of Stirling’s approximation (cf. Feller, 1971)
m!= √ 2e −m+m m m+1/2
where the error term m admits the bound
1
12m + 1 m 1
12m
for all m ∈ N, (4.2)
for all m ∈ N.
Also note that for g(y) = log y, y ∈ (0, ∞), the kth derivative of g is given by g (k) (y) = ((−1) k−1 (k − 1)!)/y k ,
y ∈ (0, ∞), k ∈ N. Hence, for any k ∈ N and ∈ (0, 1),
|g (k) (k − 1)!
(1 + x) |
(1 − ) k
for all 0|x| < . (4.3)
For Lemma 1 below, let X ∼ Hyp(n; M,N) for a given set of integers n, M, N ∈ N with 1n(N − 1),
1M (N − 1). Let
xk,n =
k − np
√ npq
and ak,n =
xk,n
(1 − f) √ , 0k n, (4.4)
npq
where f = n/N, p = M/N and q = 1 − p. Lemma 1 gives a basic approximation to Hypergeometric probabilities
solely under condition (4.5) stated below.
Lemma 1. Suppose that X ∼ Hyp(n; M,N) for a given set of integers n, M, N ∈ N such that
0