Let X1,...,Xn ∼ iid N(μ,τ) with both μ and τ unknown. Consider estimators of the form δb(x) = bS2, where S2 = ∑ni=1(Xi −X ̄)2. Note that we have already considered estimators of this form with b = 1/n and b = 1/n−1. (a) Calculate the MSE for δb(x) as a function of the choice b. (b) Find the value of b (depending on n) that minimizes the MSE. (c) Is the resulting estimator unbiased? If not, what is the bias?
Let X1,...,Xn ∼ iid N(μ,τ) with both μ and τ unknown. Consider estimators of the form δb(x) = bS2, where S2 = ∑ni=1(Xi −X ̄)2. Note that we have already considered estimators of this form with b = 1/n and b = 1/n−1. (a) Calculate the MSE for δb(x) as a function of the choice b. (b) Find the value of b (depending on n) that minimizes the MSE. (c) Is the resulting estimator unbiased? If not, what is the bias?
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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Let X1,...,Xn ∼ iid N(μ,τ) with both μ and τ unknown. Consider estimators of the form δb(x) = bS2, where S2 = ∑ni=1(Xi −X ̄)2. Note that we have already considered estimators of this form with b = 1/n and b = 1/n−1.
(a) Calculate the MSE for δb(x) as a
(b) Find the value of b (depending on n) that minimizes the MSE.
(c) Is the resulting estimator unbiased? If not, what is the bias?
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