Normal approximation to the hypergeometric distribution in ...

S.N. Lahiri et al. / Journal of Statistical Planning and Inference 137 (2007) 3570 –3590 3573
Then there exists universal constants C1,C2 ∈ (0, ∞) (not depending on r, nr,Mr and Nr) such that:
P
Xr − nrpr
C1
x − (x)
1 +|x|
r
r
2
r(x) exp(−C2x 2 2 r (x)) (2.5)
for all x ∈ R, where r(x) = qrI(x0) + prI(x0).
Theorem 1 is a non-uniform Berry–Esseen theorem for the Hypergeometric distribution. It shows that the error of
Normal approximation to the Hypergeometric distribution dies at a sub-Gaussian rate in the tails. The only condition
needed for the validity of this bound is (2.4). It is easy to check that
r ∈ 1
25 , 1
20
(2.6)
for all r satisfying (2.4). Hence, the bound in (2.5) is available for all r such that r 25.
As pointed out in Section 1, when both the sequences {pr} {r 1} and {fr} {r 1} are bounded away from 0 and 1, the
rate of approximation in Theorem 1 matches the standard rate O(n −1/2
r ) of Normal approximation for the sum of nr
iid random variables with a finite third moment. Although the Hypergeometric random variable Xr can be written as
a sum of nr dependent Bernoulli (pr) variables, the lack of independence of the summands does not affect the rate of
Normal approximation as long as the sequence {pr} r 1 is bounded away from 0 and 1 and {fr} r 1 is bounded away
from 1. On the other hand, if either of the sequences {pr} {r 1} and {fr} {r 1} converge to one of the extreme values 0
and 1, then
r = o(n 1/2
r ) as r →∞
and the rate of Normal approximation to the Hypergeometric distribution is indeed worse than the standard rate O(n −1/2
r )
in such nonstandard cases.
An immediate consequence of Theorem 1 is the following exponential (sub-Gaussian) probability bound on the tails
of Xr.
Theorem 2. Suppose that Xr ∼ Hyp(nr,Mr,Nr), r ∈ N. Then, there exist universal constants C3,C4 ∈ (0, ∞) (not
depending on r, nr,Mr,Nr) such that for all r satisfying (2.4),
Xr − nrpr
P
x
C3
(pr ∧ qr) 3 exp(−C4x 2 [pr ∧ qr] 2 ) for all x>0.
r
3. Numerical results
To gain some insight into the quality of Normal approximation to the Hypergeometric distribution in finite samples
and to compare it with the accuracy in the case of the Binomial distribution, first we consider some joint plots of the
cdfs of normalized Hypergeometric and Binomial random variables against the standard Normal cdf. Figs. 1–5 show
these plots for different values of the parameters n and p for N = 60, 200.
From the figures, it follows that the quality of Normal approximation to the Hypergeometric distribution is comparable
to that for the Binomial distribution for values of f and p close to 0.5, but there is a stark loss of accuracy for high values
of f and p.
Next, to get a quantitative picture of the error of Normal approximation, we conducted a moderately large numerical
study with different values of the population size N and with different values of the parameters p and f. The population
sizes considered were N = 60, 200, 500, 2000. For a given value of N, the set of values of p and f considered was
{0.5, 0.6, 0.7, 0.8, 0.9}. We considered the Kolmogorov distance, i.e., the maximal distance between the cdfs of the
normalized Hypergeometric variable and a standard Normal variable as a measure of accuracy. More specifically, the
measure of accuracy for the Hypergeometric case is defined as
(N,p,f)=
P
X − np
x
− (x)
, (3.1)
where X ∼ Hyp(n, M, N), f = n/N, p = M/N, 2 = Nf (1 − f)p(1 − p) and (·) denotes the cdf of the N(0,1)
distribution.