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# home2 - Stat 210B Homework Assignment 2(due February 13 1...

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Stat 210B Homework Assignment 2 (due February 13) 1. Let ˆ U n denote the H´ ajek projection of the U-statistic U n : ˆ U n = n X i =1 E [ U n | X i ] - ( n - 1) θ. Show that ˆ U n - U n can itself be written as a U-statistic. 2. Consider the V-statistic V n = 1 /n 2 n i =1 n j =1 h ( x i , x j ) for a symmetric kernel h such that Eh 2 < . Show that V n is asymptotically normal. 3. Consider the kernel h ( x 1 , x 2 ) = I x 1 + x 2 > 0 (where I denotes an indicator function). Evaluate θ = E [ h ( X 1 , X 2 )] for: (a) the mixture F = (1 - ) N (0 , 1) + N ( α, β ) (b) the uniform distribution F = Un( θ - 1 2 , θ + 1 2 ). 4. Given a probability space, consider a sequence of sigma-fields {F n } on that space such that F 1 ⊃ F 2 ⊃ F 3 ⊃ · · · . Consider a sequence of random variables { X n } such that X n is F n -measurable and E | X n | < . The sequence { X n } is called a reverse martingale if E [ X n | F n +1 ] = X n +1 , (wp1, for all n ) . Let the symbol X ( n ) denote the order statistics of the sequence ( X 1 , X 2 , . . . , X n ). Define F n = σ ( X ( n ) , X n +1 , X n +2 , . . . ). (Intuitively, this sigma-field is the information contained in the infinite sequence (
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