Homework Set

# Homework Set - Stat 210B Homework Assignment 1 (due January...

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Stat 210B Homework Assignment 1 (due January 25) 1. Show that if EX n μ and Var X n 0, then X n P -→ μ . 2. Show that if n E | X n - X | r < , then X n as -→ X and X n r -→ X . 3. Suppose X n is uniformly distributed on the set of points { 1 /n, 2 /n, . . . , 1 } . Show that X n d -→ X , where X is Un(0 , 1), where Un denotes the uniform distribution. Does X n P -→ X ? 4. Given densities p n and q n with respect to some measure μ , deFne the likelihood ratio L n ( x ) as L n ( x ) = q n ( x ) /p n ( x ) for p n ( x ) > 0, L n ( x ) = 1 if p n ( x ) = q n ( x ) = 0 and L n ( x ) = otherwise. Show that the likelihood ratio is a uniformly tight sequence. 5. Let X n and X have densities p n and p , respectively, with respect to a measure μ . Show sup B | P ( X n B ) - P ( X B ) | = 1 2 Z | p n - p | dμ, where the supremum ranges over measurable sets B . 6. Let X 1 , . . . , X n be a sample of size n from Be( θ, 1), where θ > 0. Let ¯ X n denote the sample mean. The method-of-moments estimate of θ
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## This note was uploaded on 01/25/2011 for the course STAT 201b taught by Professor Michaeljordan during the Fall '05 term at Berkeley.

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