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8.7 In this problem we refer to Example 8.6. Suppose the random variables (X₁, Y₁), (X2, Y2),..., (Xn, Yn) are all independent and that for each i we have X, Y, distributed as N(μi, 0²). a) Set v = and express the likelihood function in terms of the parameters H1,Hn, and v. b) Show that the log likelihood is 1 2TT c) Calculate the MLEs for the parameters ₁,..., V. 1 (μ₁,₂ μln, v) = 2n log - +nlogu−, (Xi-H) + (Y - H) 2 y i=1

8.7 In this problem we refer to Example 8.6. Suppose the random variables (X₁, Y₁), (X2, Y2),..., (Xn, Yn) are all independent and that for each i we have X, Y, distributed as N(μi, 0²). a) Set v = and express the likelihood function in terms of the parameters H1,Hn, and v. b) Show that the log likelihood is 1 2TT c) Calculate the MLEs for the parameters ₁,..., V. 1 (μ₁,₂ μln, v) = 2n log - +nlogu−, (Xi-H) + (Y - H) 2 y i=1

MATLAB: An Introduction with Applications
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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In this problem we refer to Example 8.6. Suppose the random variables (X₁, Y₁), (X2, Y2),..., (Xn, Yn) are all independent and that for each i we have X₁, Yį distributed as N(μi, 0²). a) Set v = and express the likelihood function in terms of the parameters 1,,n, and v. b) Show that the log likelihood is n (Xì - Hi) + (Yi - Hi) 2 1 (μ₁,..., fln, v) = 2n log. + n logv - v 1 √2π i=1 c) Calculate the MLEs for the parameters 1,...,fln, V.
Transcribed Image Text:In this problem we refer to Example 8.6. Suppose the random variables (X₁, Y₁), (X2, Y2),..., (Xn, Yn) are all independent and that for each i we have X₁, Yį distributed as N(μi, 0²). a) Set v = and express the likelihood function in terms of the parameters 1,,n, and v. b) Show that the log likelihood is n (Xì - Hi) + (Yi - Hi) 2 1 (μ₁,..., fln, v) = 2n log. + n logv - v 1 √2π i=1 c) Calculate the MLEs for the parameters 1,...,fln, V.
EXAMPLE 8.6 The Neyman-Scott Paradox Here is an example of a distribution where the MLE is not consistent. You will be asked to prove most of the claims here in Problem 8.7. Suppose random variables (X₁, Y₁), (X2, Y2),..., (Xn, Yn) are all independent and that for each i we have X₁, Y₁ distributed as N(i, 02). Note that the variables have means indexed by i but they all have the same variance. In this case, we can form the log-likelihood function and show that the MLEs for the parameters are 22 = = Xi + Yi 2 n 1 - n (Xi - Yi)² 4 These are in fact the unique solutions of the likelihood equation. Because of the last expression, it is easy to see that 22 2 and therefore the MLE is not consistent (it is biased). (8.4) (8.5)
Transcribed Image Text:EXAMPLE 8.6 The Neyman-Scott Paradox Here is an example of a distribution where the MLE is not consistent. You will be asked to prove most of the claims here in Problem 8.7. Suppose random variables (X₁, Y₁), (X2, Y2),..., (Xn, Yn) are all independent and that for each i we have X₁, Y₁ distributed as N(i, 02). Note that the variables have means indexed by i but they all have the same variance. In this case, we can form the log-likelihood function and show that the MLEs for the parameters are 22 = = Xi + Yi 2 n 1 - n (Xi - Yi)² 4 These are in fact the unique solutions of the likelihood equation. Because of the last expression, it is easy to see that 22 2 and therefore the MLE is not consistent (it is biased). (8.4) (8.5)
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