Theory of Statistics - George Mason University

we have for fixed k,
1.3 Sequences of Events and of Random Variables 99
n
lim =
n→∞ k
nk−1
,
k!
lim
n→∞ fY (y) = 1
Γ(k) yk−1e −y I[0,∞[(y), (1.202)
that is, the limiting distribution of {nX(k:n)} is gamma with scale parameter
1 and shape parameter k. (Note, of course, k! = kΓ(k).) For finite values
of n, the asymptotic distribution provides better approximations when k/n
is relatively small. When k is large, n must be much larger in order for the
asymptotic distribution to approximate the true distribution closely.
If k = 1, the PDF in equation (1.202) is the exponential distribution, as
shown in Example 1.27. For k → n, however, we must apply a limit similar to
what is done in equation (1.201).
While the min and max of the uniform distribution considered in Example
1.27 are “extreme” values, the more interesting extremes are those from
distributions with infinite support. In the next example, we consider an extreme
value that has no bound. In such a case, in addition to any normalization,
we must do a shift.
Example 1.29 extreme value distribution from an exponential distribution
Let X(n:n) be the largest order statistic from a random sample of size n from
an exponential distribution with PDF e−xIĪR+ (x) and let
We have
Y = X(n:n) − log(n).
lim Pr(Y ≤ y) = e−e−y
n→∞
(1.203)
(Exercise 1.65). The distribution with CDF given in equation (1.203) is called
an extreme value distribution. There are two other classes of “extreme value
distributions”, which we will discuss in Section 1.4.3. The one in this example,
which is the most common one, is called a type 1 extreme value distribution
or a Gumbel distribution.
1.3.8 Asymptotic Expectation
The properties of the asymptotic distribution, such as its mean or variance,
are the asymptotic values of the corresponding properties of Tn. Let {Tn}
d
be a sequence of random variables with E(|Tn|) < ∞ and Tn → T, with
E(|T |) < ∞. Theorem 1.40 (on page 90) tells us that
E(|Tn| k ) → E(|T | k ).
Theory of Statistics c○2000–2013 James E. Gentle