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
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.8: Probability
Problem 22E
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