stat612-notes3 - Statistics 612: Superefficiency,...

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Unformatted text preview: Statistics 612: Superefficiency, Contiguity, LAN, Regularity, Convolution Theorems Moulinath Banerjee December 6, 2006 1 Some concepts from probability Uniform integrability: A sequence of random variables { X n } (with X n defined on ( X n , A n ,P n )) is said to be uniformly integrable if: lim sup n 1 E n ( | X n | 1( | X n | )) = 0 . The following lemma gives a necessary and sufficient condition for uniform integrability. Lemma 1.1 { X n } is uniformly integrable if and only if the following two conditions are satisfied: (i) sup n 1 E n | X n | < . (ii) For any sequence of sets { B n } with B n A n , whenever P n ( B n ) converges to 0, then E P n ( | X n | 1 B n ) converges to 0. The following theorem illustrates the usefulness of uniform integrability; uniform integrability in conjunction with convergence in distribution implies convergence of moments. Theorem 1.1 Suppose that X n L r ( P ) with < r < and X n p X . Then, the following are equivalent. (i) {| X n | r } are uniformly integrable. (ii) X n r X ; in other words: E ( | X n- X | r ) converges to 0. (iii) E | X n | r converges to E | X | r . 1 2 Hodges Superefficient Estimator Let X 1 ,X 2 ,...,X n be i.i.d P from a onedimensional regular parametric model P , for which the conditions of the information inequality hold. If T n is unbiased for estimating q ( ), then, from the information inequality: Var ( T n ) q ( ) 2 nI ( ) . Now, consider a general n consistent estimator S n of q ( ), such that n ( S n- q ( )) d N (0 ,V 2 ( )) . If n is the MLE of , then n ( q ( n )- q ( )) N (0 ,q ( ) 2 /I ( )) . An application of the Skorokhod representation in conjunction with Fatous lemma yields: liminf E [ n ( S n- q ( ))] 2 V 2 ( ) . If S n is unbiased, we have: V 2 ( ) liminf n Var ( S n ) . Now, suppose that lim n Var ( S n ) exists and equals V 2 ( ). This is the case, for example, if the sequence { n ( S n- q ( )) 2 } is uniformly integrable. Then, the information inequality would imply that: V 2 ( ) q ( ) 2 I ( ) . A question that then naturally arises is whether this inequality holds under the usual restrictions on the parametric model alone. So, for usual regular parametric models, is it possible to find n consistent estimators that are asymptotically normal with the variance of the limiting distribution strictly less than the bound from the information inequality? The following example due to Hodges shows that this is indeed the case. Hodges superefficient estimator: Let X 1 ,X 2 ,...,X n be i.i.d. N ( , 1), so that I ( ) = 1, identically. Let | a | < 1 and define: T n = X n 1( | X n | > n- 1 / 4 ) + a X n 1( | X n | n- 1 / 4 ) ....
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This note was uploaded on 04/14/2010 for the course STATS 612 taught by Professor Moulib during the Winter '08 term at University of Michigan.

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stat612-notes3 - Statistics 612: Superefficiency,...

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