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hw7_stat210a

# hw7_stat210a - UC Berkeley Department of Statistics STAT...

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Unformatted text preview: UC Berkeley Department of Statistics STAT 210A: Introduction to Mathematical Statistics Problem Set 7 Fall 2006 Issued: Thursday, October 19, 2006 Due: Thursday, October 26, 2006 Problem 7.1 Show that if X n d → X > 0 and X n /Y n p → 1, then Y n d → X . Problem 7.2 Consider i.i.d. samples X i , i = 1 , . . . , n with finite moments up to order four. Consider the sample mean and variance ¯ X n = 1 n n summationdisplay i =1 X i , and s 2 n = 1 n n summationdisplay i =1 ( X i- ¯ X n ) 2 . (a) Prove that bracketleftbigg ¯ X n s 2 n bracketrightbigg- bracketleftbigg μ σ 2 bracketrightbigg d → N parenleftbiggbracketleftbigg bracketrightbigg , bracketleftbigg σ 2 μ 3 μ 3 μ 4- σ 4 bracketrightbiggparenrightbigg . (b) Use your result from (a) to find the asymptotic distribution of s 2 n / ¯ X n . (c) Suppose that the underlying distribution of X i is Poisson( λ ). Show that the asymptotic distribution of s 2 n / ¯ X n is independent of λ . Why might this be useful for testing whether....
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hw7_stat210a - UC Berkeley Department of Statistics STAT...

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