Stat 5101 Lecture Notes - School of Statistics

206 Stat 5101 (Geyer) Course Notesthen√ n(X(kn) − x p)( )D p(1 − p)−→ N 0,f(x p ) 2 , as n →∞. (7.38)Or the sloppy version()p(1 − p)X (kn) ≈N x p ,nf(x p ) 2 .In particular, if we define k n = ⌈np⌉, then X (kn) is a sample p-th quantileby Theorem 7.5. The reason for the extra generality, is that the theorem makesit clear that X (kn+1) also has the same asymptotic distribution. Since X (kn) ≤X (kn+1) always holds by definition of order statistics, this can only happen if√ n(X(kn+1) − X (kn)) P−→ 0.Hence the average˜X n = X (k n) + X (kn+1)2which is the conventional definition of the sample median, has the same asymptoticnormal distribution as either X (kn) or X (kn+1).Corollary 7.28. Suppose X 1 , X 2 , ... are continuous random variables thatare independent and identically distributed with density f that is nonzero thepopulation median m, then√ (n ˜Xn − m ))D 1−→ N(0,4f (x p ) 2 , as n →∞.This is just the theorem with x p = m and p =1/2. The sloppy version is()1˜X n ≈N m,4nf(m) 2 .Example 7.4.1 (Median, Normal Population).If X 1 , X 2 , ...are i. i. d. N (µ, σ 2 ), then the population median is µ by symmetryand the p. d. f. at the median isf(µ) = 1σ √ 2πHenceor, more precisely,˜X n ≈N√ n( ˜Xn − µ)) (µ, πσ2 .2n)−→ DN(0, πσ22