Exercise 24 (#5.51). Let Fn be the empirical distribution based on a random sample of...
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Exercise 24 (#5.51). Let Fn be the empirical distribution based on a random sample of size n from a distribution F on R having Lebesgue density f. Let An(t) be the Lebesgue density of the pth sample quantile Fn(p). Prove that Pull) -- () - 1)- Fop="5(0), where lp = np if np is an integer and lp = 1+ the integer part of np if is not an integer, by (i) using the fact that nFn(t) has a binomial distribution; (ii) using the fact that Fr?(p) = Cnp X(mp) + (1 - Cnp)X(mp+1), is the jth order statistic, mp is the integer part of np, Cnp = 1 if np is an integer, and Cnp = 0) if np is not an integer. where X(j)
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