s6 - Stat 5101 Lecture Slides Deck 7 Charles J Geyer School...

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Unformatted text preview: Stat 5101 Lecture Slides Deck 7 Charles J. Geyer School of Statistics University of Minnesota 1 Asymptotic Approximation The last big subject in probability theory is asymptotic approxi- mation, also called asymptotics, also called large sample theory. We have already seen a little bit. • Convergence in probability, • o p and O p notation, • and the Poisson approximation to the binomial distribution are all large sample theory. 2 Convergence in Distribution If X 1 , X 2 , ... is a sequence of random variables, and X is another random variable, then we say X n converges in distribution to X if E { g ( X n ) } → E { g ( X ) } , for all bounded continuous functions g : R → R , and we write X n D-→ X to indicate this. 3 Convergence in Distribution (cont.) The Helley-Bray theorem asserts that the following is an equiv- alent characterization of convergence in distribution. If F n is the DF of X n and F is the DF of X , then X n D-→ X if and only if F n ( x ) → F ( x ) , whenever F is continuous at x. 4 Convergence in Distribution (cont.) The Helley-Bray theorem is too difficult to prove in this course. A simple example shows why convergence F n ( x ) → F ( x ) is not required at jumps of F . Suppose each X n is a constant random variable taking the value x n and X is a constant random variable taking the value x , then X n D-→ X if x n → x because x n → x implies g ( x n ) → g ( x ) whenever g is continuous. 5 Convergence in Distribution (cont.) The DF of X n is F n ( s ) = , s < x n 1 , s ≥ x n and similarly for the DF F of X . We do indeed have F n ( s ) → F ( s ) , s 6 = x but do not necessarily have this convergence for s = x . 6 Convergence in Distribution (cont.) For a particular example where convergence does not occur at s = x , consider the sequence x n = (- 1) n n for which x n → . Then F n ( x ) = F n (0) = , even n 1 , odd n and this sequence does not converge (to anything). 7 Convergence in Distribution (cont.) Suppose X n and X are integer-valued random variables having PMF’s f n and f , respectively, then X n D-→ X if and only if f n ( x ) → f ( x ) , for all integers x Obvious, because there are continuous functions that are nonzero only at one integer. 8 Convergence in Distribution (cont.) A long time ago (slides 36–38, deck 3) we proved the Poisson approximation to the binomial distribution. Now we formalize that as a convergence in distribution result. Suppose X n has the Bin( n,p n ) distribution, X has the Poi( μ ) distribution, and np n → μ. Then we showed (slides 36–38, deck 3) that f n ( x ) → f ( x ) , x ∈ N which we now know implies X n D-→ X. 9 Convergence in Distribution (cont.) Convergence in distribution is about distributions not variables....
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This note was uploaded on 09/13/2011 for the course STA 4184 taught by Professor Staff during the Spring '11 term at University of Central Florida.

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s6 - Stat 5101 Lecture Slides Deck 7 Charles J Geyer School...

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