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i.i.d. 4. (Pooled z-test as likelihood ratio test) Let X1,..., Xm N(μ1, 1) and Y₁,..., Yn Suppose that these two samples are independent. We would like to test Hoμ1=2, H₁₁ μ2 using the likelihood ratio test. i.i.d. N(μ2, 1). (a) Prove that the likelihood function can be written as 1 n L(μ1, H2) = exp (2π)(m+n)/2 {Σ (xi - με U; — 12)²}. Σ - m}{ - Σω (b) Prove that the maximum likelihood estimators of μ₁ and μ₂ are respectively the sample mean of X and Y, that is, μ = = X = = m 1 n 1 Xi, Â=Y == m ΣYj. i=1 n j=1

i.i.d. 4. (Pooled z-test as likelihood ratio test) Let X1,..., Xm N(μ1, 1) and Y₁,..., Yn Suppose that these two samples are independent. We would like to test Hoμ1=2, H₁₁ μ2 using the likelihood ratio test. i.i.d. N(μ2, 1). (a) Prove that the likelihood function can be written as 1 n L(μ1, H2) = exp (2π)(m+n)/2 {Σ (xi - με U; — 12)²}. Σ - m}{ - Σω (b) Prove that the maximum likelihood estimators of μ₁ and μ₂ are respectively the sample mean of X and Y, that is, μ = = X = = m 1 n 1 Xi, Â=Y == m ΣYj. i=1 n j=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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i.i.d. 4. (Pooled z-test as likelihood ratio test) Let X1,..., Xm N(μ1, 1) and Y₁,..., Yn Suppose that these two samples are independent. We would like to test Hoμ1=2, H₁₁ μ2 using the likelihood ratio test. i.i.d. N(μ2, 1). (a) Prove that the likelihood function can be written as 1 n L(μ1, H2) = exp (2π)(m+n)/2 {Σ (xi - με U; — 12)²}. Σ - m}{ - Σω (b) Prove that the maximum likelihood estimators of μ₁ and μ₂ are respectively the sample mean of X and Y, that is, μ = = X = = m 1 n 1 Xi, Â=Y == m ΣYj. i=1 n j=1
Transcribed Image Text:i.i.d. 4. (Pooled z-test as likelihood ratio test) Let X1,..., Xm N(μ1, 1) and Y₁,..., Yn Suppose that these two samples are independent. We would like to test Hoμ1=2, H₁₁ μ2 using the likelihood ratio test. i.i.d. N(μ2, 1). (a) Prove that the likelihood function can be written as 1 n L(μ1, H2) = exp (2π)(m+n)/2 {Σ (xi - με U; — 12)²}. Σ - m}{ - Σω (b) Prove that the maximum likelihood estimators of μ₁ and μ₂ are respectively the sample mean of X and Y, that is, μ = = X = = m 1 n 1 Xi, Â=Y == m ΣYj. i=1 n j=1
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