6.31 Boos and Hughes-Oliver (1998) detail a number of instances where application of Basu's Theorem can simplify calculations. Here are a few. (a) Let X1,..., Xn be iid n(µ, o²), where o² is known. (i) Show that X is complete sufficient for , and S2 is ancillary. Hence by Basu's Theorem, X and S? are independent. (ii) Show that this independence carries over even if o is unknown, as knowledge of o? has no bearing on the distributions. (Compare this proof to the more involved Theorem 5.3.1(a).) (b) A Monte Carlo swindle is a technique for improving variance estimates. Suppose that X1,..., X, are iid n(u, o²) and that we want to compute the variance of the median, M. (i) Apply Basu's Theorem to show that Var(M) = Var(M-X)+Var(X); thus we only have to simulate the Var(M-X) piece of Var(M) (since Var(X) = o²/n).
6.31 Boos and Hughes-Oliver (1998) detail a number of instances where application of Basu's Theorem can simplify calculations. Here are a few. (a) Let X1,..., Xn be iid n(µ, o²), where o² is known. (i) Show that X is complete sufficient for , and S2 is ancillary. Hence by Basu's Theorem, X and S? are independent. (ii) Show that this independence carries over even if o is unknown, as knowledge of o? has no bearing on the distributions. (Compare this proof to the more involved Theorem 5.3.1(a).) (b) A Monte Carlo swindle is a technique for improving variance estimates. Suppose that X1,..., X, are iid n(u, o²) and that we want to compute the variance of the median, M. (i) Apply Basu's Theorem to show that Var(M) = Var(M-X)+Var(X); thus we only have to simulate the Var(M-X) piece of Var(M) (since Var(X) = o²/n).
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter2: The Integers
Section2.3: Divisibility
Problem 30E: Let be as described in the proof of Theorem. Give a specific example of a positive element of .
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