Let Y 1 < Y 2 < < Y n be the order statistics

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Let Y1 < Y2 < · · · < Yn be the order statistics of a random sample from a N(θ, σ2) distribution, where σ2 is fixed but arbitrary. Then ‾Y = ‾X is a complete sufficient statistic for θ. Consider another estimator T of θ, such as T = (Yi + Yn+1−i)/2, for i = 1, 2, . . . , [n/2], or T could be any weighted average of these latter statistics.
(a) Argue that T − ‾X and ‾X are independent random variables.
(b) Show that Var(T) = Var(‾X) + Var(T − ‾X).
(c) Since we know Var(‾X) = σ2/n, it might be more efficient to estimate Var(T )by estimating the Var(T − ‾X) by Monte Carlo methods rather than doing that with Var(T) directly, because Var(T) ≥ Var(T −‾X). This is often called the Monte Carlo Swindle.

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Introduction To Mathematical Statistics

ISBN: 9780321794710

7th Edition

Authors: Robert V., Joseph W. McKean, Allen T. Craig

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