Home6 - is said to be locally asympotically normal(LAN if the log likelihood ratio can be written as h T Z n-1 2 h T Λ h o P n(1 where Z n d-→

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Stat 210B Homework Assignment 6 (due April 19) 1. Consider estimating the distribution function P ( X x ) at a Fxed point x based on a sample X 1 , . . . , X n from the distribution of X . A nonparametric estimator is n - 1 i 1( X i x ). If it is known that the true underlying distribution is N ( θ, 1), another possible estimator is Φ( x - ¯ X ). Calculate the relative e±ciency of these estimators. 2. Let N be a kr -dimensional multinomial variable written as a ( k × r ) matrix ( N ij ). Calculate the likelihood ratio statistic for testing the null hypothesis of independence H 0 : p ij = p i · p · j for every i and j . Here the dot denotes summation over all columns and rows, respectively. What is the limit distribution under the null hypothesis? 3. Recall that a family { P n,h : h H }
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Unformatted text preview: is said to be locally asympotically normal (LAN) if the log likelihood ratio can be written as h T Z n-1 2 h T Λ h + o P n, (1) where Z n d-→ N (0 , Λ) under P n, . Suppose that { P n,h : h ∈ H } is LAN. Show that { P n,h 1 } and { P n,h 2 } are mutually contiguous for any h 1 and h 2 . 4. Suppose that { P n,h : h ∈ H } is LAN. Show that under P n,h we have Z n d-→ N (Λ h, Λ). 5. Consider the Hodge supere±cient estimator in the setting in which data X 1 , . . . , X n are sampled i.i.d. from the distribution N ( θ, 1). ²ind a sequence of parameters { θ n } such that the quadratic risk calculated along the sequence grows to inFnity....
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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 University of California, Berkeley.

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