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.

Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018

18th Edition

ISBN:9780079039897

Author:Carter

Publisher:Carter

Chapter10: Statistics

Section10.1: Measures Of Center

Problem 9PPS

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