Homework Help is Here – Start Your Trial Now!
Math
Statistics
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
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.8: Probability
Problem 22E
See similar textbooks
icon
Related questions
Question
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
Expert Solution
steps

Step by step

Solved in 2 steps with 2 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Calculus For The Life Sciences
Calculus For The Life Sciences
Calculus
ISBN:
9780321964038
Author:
GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:
Pearson Addison Wesley,
SEE MORE TEXTBOOKS