Stat611hw2-win09 - Statistics 611 Homework 2(1(i Let G n be a sequence of random functions defined on R p with continuous sample paths i.e for each

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Unformatted text preview: Statistics 611: Homework 2 (1) (i) Let G n be a sequence of random functions defined on R p with continuous sample paths: i.e. for each ω , G ( ω ) n ( t ) is continuous in t . Suppose that g is a fixed function on R p . All functions are taken to be real-valued. Assume that (i) g is maximized uniquely at t ? , (ii) For every compact subset K of R p , k G n- g k ∞ ,K ≡ sup t ∈ K | G n ( x )- g ( x ) |→ p 0. (iii) Let { T n } be a sequence of random variables maximizing G n such that T n is tight i.e. bounded in probability. Show that T n converges in probability to t ? . (ii) Consider now, a sequence of random functions G n on C ( K ) where K is a compact subset of R , converging to g ∈ C ( K ). Suppose that { T n } solves G n ( T n ) = 0 and that g ( t ) = 0 for a unique t ∈ K . Show that T n → p t . (2) The Glivenko-Cantelli Theorem. Let X 1 ,X 2 ,..., be an i.i.d. sequence of random variables with common distribution F on R . Let F n denote the empirical c.d.f. of the first n observations X 1 ,X 2 ,...,X n . Thus, F n ( x ) = n- 1 ∑ n i =1 1( X i ≤ x ). We seek to show that sup x | F n ( x )- F ( x ) |→ a.s 0. To this end, fix an arbitrary > 0, and choose points-∞ = x < x 1 ≤ x 2 ≤ x 3 ≤ ... ≤ x k- 1 < x k = ∞ , such that (a) k >...
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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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Stat611hw2-win09 - Statistics 611 Homework 2(1(i Let G n be a sequence of random functions defined on R p with continuous sample paths i.e for each

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