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hw11_stat210a

# hw11_stat210a - UC Berkeley Department of Statistics STAT...

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UC Berkeley Department of Statistics STAT 210A: Introduction to Mathematical Statistics Problem Set 11 Fall 2006 Issued: Thursday, November 30, 2006 Due: Thursday, December 7, 2006 Problem 11.1 Recall that a statistical function h is said to be continuous at F in the sup-norm topology if, for any sequence { G n } such that bardbl G n F bardbl 0, there holds h ( G n ) h ( F ). Are the following statistical functionals continuous in this sense or not? Prove or disprove. (a) for fixed a R , the evaluation h ( F ) = F ( a ). (b) for fixed CDF F 0 with density f 0 , the Cram´ er-von Mises functional given by h ( F ) = integraltext [ F ( t ) F 0 ( t )] 2 f 0 ( t ) dt . (c) the mean functional h ( F ) = E F [ X ]. (d) For a given CDF F , suppose that the inverse CDF F - 1 ( t ) = inf { x R | F ( x ) t } is continuous at α . Consider the quantile functional h α that returns the level α quantile of F : is it continuous or discontinuous? Problem 11.2 Suppose we form an empirical CDF hatwide F n - 1 based on a set of ( n 1) samples x 1 , . . . , x n - 1 . Now consider the new empirical CDF hatwide F n,x formed when the point x is added to { x 1 , . . . , x n - 1 } as the n th sample. The sensitivity function S h,F ( x ) = bracketleftBig h ( hatwide F n,x ) h ( hatwide F n - 1 )

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