i.i.d. 4. (Pooled z-test as likelihood ratio test) Let X₁,..., Xm N(1, 1) and Y₁,..., Yn Suppose that these two samples are independent. We would like to test using the likelihood ratio test. Họ: t1 = #2, H1#U2 (a) Prove that the likelihood function can be written as 1 L(H1, H2)= m i.i.d. (2x)(m+n)/2 °xp { ¯ ½Σ(z; − µ1)²} exp{− ž½¯‹¥; − #2)²}. j= N(μ2, 1). (b) Prove that the maximum likelihood estimators of μ₁ and μ2 are respectively the sample mean of X and Y, that is, m n ₁₁₁ = X = Xi, =Y= m n i=1 j=1 (c) Under Ho₁₂, let us write μ₁ =μ₂ =μ. Prove that the maximum likelihood estimator of μ, assuming Ho is true, is the pooled estimator ΣX + ΣY _ mX + nY m+n m+n (d) Using part (a), (b) and (c), prove that the likelihood ratio can be written as exp 1 mn 2 (m+n) (e) Using part (d), prove that the critical region of the likelihood ratio test with significance level a is of the form C ={√ √ mn |-|≥a/2 m+n

i.i.d. 4. (Pooled z-test as likelihood ratio test) Let X₁,..., Xm N(1, 1) and Y₁,..., Yn Suppose that these two samples are independent. We would like to test using the likelihood ratio test. Họ: t1 = #2, H1#U2 (a) Prove that the likelihood function can be written as 1 L(H1, H2)= m i.i.d. (2x)(m+n)/2 °xp { ¯ ½Σ(z; − µ1)²} exp{− ž½¯‹¥; − #2)²}. j= N(μ2, 1). (b) Prove that the maximum likelihood estimators of μ₁ and μ2 are respectively the sample mean of X and Y, that is, m n ₁₁₁ = X = Xi, =Y= m n i=1 j=1 (c) Under Ho₁₂, let us write μ₁ =μ₂ =μ. Prove that the maximum likelihood estimator of μ, assuming Ho is true, is the pooled estimator ΣX + ΣY _ mX + nY m+n m+n (d) Using part (a), (b) and (c), prove that the likelihood ratio can be written as exp 1 mn 2 (m+n) (e) Using part (d), prove that the critical region of the likelihood ratio test with significance level a is of the form C ={√ √ mn |-|≥a/2 m+n

College Algebra (MindTap Course List)

12th Edition

ISBN:9781305652231

Author:R. David Gustafson, Jeff Hughes

Publisher:R. David Gustafson, Jeff Hughes

Chapter8: Sequences, Series, And Probability

Section8.7: Probability

Problem 39E: Assume that the probability that an airplane engine will fail during a torture test is 12and that...

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