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, σ2), where σ2 is known.
(i) Show that is complete sufficient for μ, and S2 is ancillary. Hence by Basu's Theorem, and S2 are independent.
(ii) Show that this independence carries over even if σ2 is unknown, as knowledge of σ2 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,..., Xn are iid n((μ, σ2) and that we want to compute the variance of the median, M.
(i) Apply Basu's Theorem to show that Var(M) = Var(M - ) + Var(); thus we only have to simulate the Var(M - ) piece of Var(M) (since Var() = σ2/n).
(ii) Show that the swindle estimate is more precise by showing that the variance of M is approximately 2[Var(M)]2/(N - 1) and that of M - is approximately 2[Var(M - )]2/(N - 1), where N is the number of Monte Carlo samples.
c. (i) If X/Y and Y are independent random variables, show that

(ii) Use this result and Basu's Theorem to show that if X1,... ,Xn are iid gamma(α, β), where a is known, then for T = ˆ‘i Xj

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