Let (X1,...,Xn), n>2, be a random sample from a distribution P on R with EX21 < ∞, ¯X bethesamplemean,X(j) be the jth order statistic, and T =(X(1) + X(n))/2. Consider the estimation of a parameter θ ∈Runder the squared error loss. (i) Show that ¯X is better than T if P = N(θ,σ2), θ ∈R, σ>0. (ii) Show that T is better than ¯X if P is the uniform distribution on the interval (θ − 1 2,θ+ 1 2), θ ∈R. (iii) Find a family P for which neither ¯ Solution. (i) Since ¯ X nor T is better than the other.

Let (X1,...,Xn), n>2, be a random sample from a distribution P on R with EX21 < ∞, ¯X bethesamplemean,X(j) be the jth order statistic, and T =(X(1) + X(n))/2. Consider the estimation of a parameter θ ∈Runder the squared error loss. (i) Show that ¯X is better than T if P = N(θ,σ2), θ ∈R, σ>0. (ii) Show that T is better than ¯X if P is the uniform distribution on the interval (θ − 1 2,θ+ 1 2), θ ∈R. (iii) Find a family P for which neither ¯ Solution. (i) Since ¯ X nor T is better than the other.

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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