(c) Under Hoμ₁ = μ₂, let us write µ₁ = µ₂ = µ. Prove that the maximum likelihood estimator of u, assuming Ho is true, is the pooled estimator == Σ "1X: +2=1Y; mX+nY m+n m+n (d) Using part (a), (b) and (c), prove that the likelihood ratio can be written as max=2= L(µ, µ) maxμ₁₂ (μ1, 2) = exp 1 mn 2 (m+n) (x - (e) Using part (d), prove that the critical region of the likelihood ratio test with significance level a is of the form C = {√ mn m+n -
(c) Under Hoμ₁ = μ₂, let us write µ₁ = µ₂ = µ. Prove that the maximum likelihood estimator of u, assuming Ho is true, is the pooled estimator == Σ "1X: +2=1Y; mX+nY m+n m+n (d) Using part (a), (b) and (c), prove that the likelihood ratio can be written as max=2= L(µ, µ) maxμ₁₂ (μ1, 2) = exp 1 mn 2 (m+n) (x - (e) Using part (d), prove that the critical region of the likelihood ratio test with significance level a is of the form C = {√ mn m+n -
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.6: Applications And The Perron-frobenius Theorem
Problem 25EQ
Related questions
Question
Let X1, . . . , Xm i.i.d.∼ N (μ1, 1) and Y1, . . . , Yn i.i.d.∼ N (μ2, 1).
Suppose that these two samples are independent. We would like to test
H0 : μ1 = μ2, H1 : μ1 != μ2 using the likelihood ratio test.
![(c) Under Hoμ₁ = μ₂, let us write µ₁ = µ₂ = µ. Prove that the maximum likelihood estimator of
u, assuming Ho is true, is the pooled estimator
==
Σ "1X: +2=1Y; mX+nY
m+n
m+n
(d) Using part (a), (b) and (c), prove that the likelihood ratio can be written as
max=2=
L(µ, µ)
maxμ₁₂ (μ1, 2)
= exp
1 mn
2 (m+n)
(x
-
(e) Using part (d), prove that the critical region of the likelihood ratio test with significance level a
is of the form
C =
{√
mn
m+n
-](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5fc235f5-c6fd-4e1c-bac8-7d394d2e8123%2F9577ebfd-1045-4be3-85e0-c1e8989304cd%2Fpwaqxey_processed.png&w=3840&q=75)
Transcribed Image Text:(c) Under Hoμ₁ = μ₂, let us write µ₁ = µ₂ = µ. Prove that the maximum likelihood estimator of
u, assuming Ho is true, is the pooled estimator
==
Σ "1X: +2=1Y; mX+nY
m+n
m+n
(d) Using part (a), (b) and (c), prove that the likelihood ratio can be written as
max=2=
L(µ, µ)
maxμ₁₂ (μ1, 2)
= exp
1 mn
2 (m+n)
(x
-
(e) Using part (d), prove that the critical region of the likelihood ratio test with significance level a
is of the form
C =
{√
mn
m+n
-
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