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Unformatted text preview: UC Berkeley Department of Statistics STAT 210A: Introduction to Mathematical Statistics Problem Set 8 Fall 2006 Issued: Thursday, October 26, 2006 Due: Thursday, November 2, 2006 Some useful notation: The p th quantile of a continuous random variable with invertible CDF F is given by x p = F =1 ( p ). Given an i.i.d. set of samples X 1 ,...,X n , let X (1) ,...,X ( n ) denote the associated order statistics. Define for any p ∈ (0 , 1), the p th-sample quantile X ( np ) ≡ X ( ⌈ np ⌉ ) , where for any x ∈ R , the notation ⌈ x ⌉ denotes the smallest integer greater than or equal to x . Problem 8.1 (Asymptotics of sample quantiles) Keener (Chapter 10) provides one method for an- alyzing the asymptotic behavior of the sample quantitles X ( np ) . In this problem, we de- velop an alternative approach. We begin by analyzing the uniform case. In particular, let U (1) ,...,U ( n ) be the order statistics of a set of n i.i.d. samples from the Uni[0 , 1] distribu- tion. Let E 1 ,...E n +1 be i.i.d. samples from an exponential distribution with mean 1, and set S k = ∑ k i =1 E i , for k = 1 ,..., ( n + 1)....
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