Question

Stat 5644 Spring 2019 Homework 4 Due 3/20 Problem :UMVUE via Rao-Blackwell, Lehmann-Scheffe, and Basu theorems This problem is on the estimation of a reliability function. Let Xi, , x, be IID from N(μ, σ2). Let Φ(-) be the c.d.f. of the standard normal distribution. Assume that σ2-of is known, for now. It is of interest to estimate the reliability number σο for some c, with )-1-(). c is called cut of point in reliability (a) Give, without any proof, a complete and sufficient statistic for μ. (b) To find an UMNUE for Te(μ), we just need to obtain an estimator that is unbiased for Te(a) and that depends only on X. Consider the estimator Is 6(X) an unbiased estimator for Te(H)? Is it a good estimator? Why or why not? (c) Since X is a complete and sufficient statistic, we can apply the Rao-Balckwell theorem to improve δ(2). To that end, following Rao-Balckwell, define 6 (X) E(6(x)|X). (i) Express δ"(X) as a probability (ii) Express X1-X as a linear combination Σ.-1 a,x, and give expression of an-1, 2, ,n. (ii) Find the distribution of X1 -X based on your answer in (ii) and conclude that Xi -X is ancillary, that is its distribution does not contain any information about p (d) Use Basu's theorem to conclude that X and X1 -X are independent. (e) Based on your answers in (c) and (d), show that 6 (x)- (f) Conclude by using the Lehmann-Scheffe theorem that δ"(X) is the UMVUE for Te(μ). (e) Use the law of large numbers along with the Continuous Mapping Theorem to show that 6"(X) is & consistent estimator of Te(u). (h) Assume now that σ2 is unknown, outline steps in obtaining the UMVUE for Te(u).

Answer 1

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