Problem 4 Suppose X₁ and X₂ are independent with E(X₁) = E(X₂) = μ, Var (X₁) = 1, and Var(X₂) = 2. Your goal is to estimate the unknown mean . Consider the estimator (1) X₁ 4X₂ + 5 5 What is the mean-square error MSE(1) ? û^1 (b) Consider an unbiased estimator of the form: 2 = aX₁ + (1 - a)X2. What value should be chosen for the constant a in order to minimize the mean squared error of μ₂? (c) For the value of a found above, compute the relative efficiency of 2 with respect to 1. (If you are unsure about the value of a you found in part (b), you may leave the answer in terms of a.)
Problem 4 Suppose X₁ and X₂ are independent with E(X₁) = E(X₂) = μ, Var (X₁) = 1, and Var(X₂) = 2. Your goal is to estimate the unknown mean . Consider the estimator (1) X₁ 4X₂ + 5 5 What is the mean-square error MSE(1) ? û^1 (b) Consider an unbiased estimator of the form: 2 = aX₁ + (1 - a)X2. What value should be chosen for the constant a in order to minimize the mean squared error of μ₂? (c) For the value of a found above, compute the relative efficiency of 2 with respect to 1. (If you are unsure about the value of a you found in part (b), you may leave the answer in terms of a.)
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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![Problem 4: Suppose \( X_1 \) and \( X_2 \) are independent with \( E(X_1) = E(X_2) = \mu \), \( \text{Var}(X_1) = 1 \), and \( \text{Var}(X_2) = 2 \). Your goal is to estimate the unknown mean \(\mu\). Consider the estimator:
\[
\hat{\mu}_1 = \frac{X_1}{5} + \frac{4X_2}{5}
\]
(a) What is the mean-square error \( \text{MSE}(\hat{\mu}_1) \)?
(b) Consider an unbiased estimator of the form: \( \hat{\mu}_2 = aX_1 + (1-a)X_2 \). What value should be chosen for the constant \( a \) in order to minimize the mean squared error of \( \hat{\mu}_2 \)?
(c) For the value of \( a \) found above, compute the relative efficiency of \( \hat{\mu}_2 \) with respect to \( \hat{\mu}_1 \). (If you are unsure about the value of \( a \) you found in part (b), you may leave the answer in terms of \( a \).)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff12ec146-5781-453b-8388-2e7cc9921f6c%2F54d7780b-58c0-45ba-af66-b7f40cde4166%2F0xh7ats_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Problem 4: Suppose \( X_1 \) and \( X_2 \) are independent with \( E(X_1) = E(X_2) = \mu \), \( \text{Var}(X_1) = 1 \), and \( \text{Var}(X_2) = 2 \). Your goal is to estimate the unknown mean \(\mu\). Consider the estimator:
\[
\hat{\mu}_1 = \frac{X_1}{5} + \frac{4X_2}{5}
\]
(a) What is the mean-square error \( \text{MSE}(\hat{\mu}_1) \)?
(b) Consider an unbiased estimator of the form: \( \hat{\mu}_2 = aX_1 + (1-a)X_2 \). What value should be chosen for the constant \( a \) in order to minimize the mean squared error of \( \hat{\mu}_2 \)?
(c) For the value of \( a \) found above, compute the relative efficiency of \( \hat{\mu}_2 \) with respect to \( \hat{\mu}_1 \). (If you are unsure about the value of \( a \) you found in part (b), you may leave the answer in terms of \( a \).)
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