Consider a random sample of size n from a distribution with

Consider a random sample of size n from a distribution with CDF F(x) = 1 - 1/x if x >= 1, and zero otherwise. (a) Derive the CDF of the smallest order statistic, X(1). (b) Find the limiting distribution of X(1). (c) Find the limiting distribution of X^n(1).

Solution

a) F(1)(x) = P(X(1) < x) = 1 ? (1 ? F(x))^n

= 1- (1/x)^n for x>1

b) lim n?? F(1)(x) = lim n?? 1 ? (1 ? F(x))^n = 0 if x<=1

1 if x>1

that is, a step function with single step of size 1 at x = 1. Hence the limiting random variable X is a discrete variable with P [X = 1] = 1 that is, the limiting distribution is degenerate at 1. Again, the limiting function is not a cdf as it not right continuous, but this does not affect out conclusion, as the limit function is not continuous at 1.

c) Now if Un = X_n^ n , we have from first principles that for u > 1

FUn (u) = P [Un ? u] = P [X_n^ n ? u] = P [X_n ? u ^1/n] = 1 ? (1 /u^1/n )ⁿ= 1 ? 1 /u

which is a valid cdf, but which does not depend on n. Hence the limiting distribution of Un is precisely FU (u) = 1 ? 1/ u u > 1