Page 336 - Elements of Distribution Theory

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11

Approximation of Probability Distributions

11.1 Introduction
Consider a random vector Y, with a known distribution, and suppose that the distribution of
f (Y)is needed, for some given real-valued function f (·). In Chapter 7, general approaches
to determining the distribution of f (Y) were discussed. However, in many cases, carrying
out the methods described in Chapter 7 is impractical or impossible. In these cases, an
alternative approach is to use an asymptotic approximation to the distribution of the statistic
under consideration. This approach allows us to approximate distributional quantities, such
as probabilities or moments, in cases in which exact computation is not possible. In addition,
the approximations, in contrast to exact results, take a few basic forms and, hence, they give
insight into the structure of distribution theory. Asymptotic approximations also play a
fundamental role in statistics.
Such approximations are based on the concept of convergence in distribution. Let
X 1 , X 2 ,... denote a sequence of real-valued random variables and let F n denote the dis-
tribution function of X n , n = 1, 2,.... Let X denote a real-valued random variable with
distribution function F.If
lim F n (x) = F(x)
n→∞
for each x at which F is continuous, then the sequence X 1 , X 2 ,... is said to converge in
distribution to X as n →∞, written

D
X n → X as n →∞.

In this case, probabilities regarding X n may be approximated using probabilities based
on X; that is, the limiting distribution function F may then be used as an approximation
D
to the distribution functions in the sequence F 1 , F 2 ,.... The property that X n → X as
n →∞ is simply the property that the approximation error decreases to 0 as n increases to
∞. The distribution of X is sometimes called the asymptotic distribution of the sequence
X 1 , X 2 ,....

Example 11.1 (Sequence of Bernoulli random variables). For each n = 1, 2,..., let X n
denote a random variable such that

Pr(X n = 1) = 1 − Pr(X n = 0) = θ n ;

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