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(1, 1) and Y₁,..., Yn Suppose that these two samples are independent. We would like to test Ho:H= H2, HH#U2 i.i.d. using the likelihood ratio test. N(μ2, 1). (a) Prove that the likelihood function can be written as 1 L(μ41,2)= exp (2π)(m+n)/2 m n {− ½Σ(ª¡ − µ1)²} exp{ − ½Σ(v; – 12)²}. i=1 j=1 (b) Prove that the maximum likelihood estimators of μ₁ and μ2 are respectively the sample mean of X and Y, that is, m Â₁ = X = X₁₁₂ == ΣX με m ΣΥ 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 11 ΣX + ΣY _ mX + n m+n 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 Ho:H= H2, HH#U2 i.i.d. using the likelihood ratio test. N(μ2, 1). (a) Prove that the likelihood function can be written as 1 L(μ41,2)= exp (2π)(m+n)/2 m n {− ½Σ(ª¡ − µ1)²} exp{ − ½Σ(v; – 12)²}. i=1 j=1 (b) Prove that the maximum likelihood estimators of μ₁ and μ2 are respectively the sample mean of X and Y, that is, m Â₁ = X = X₁₁₂ == ΣX με m ΣΥ 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 11 ΣX + ΣY _ mX + n m+n m+n

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 19E
See similar textbooks
icon
Related questions
Question
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 Ho:H= H2, HH#U2 i.i.d. using the likelihood ratio test. N(μ2, 1). (a) Prove that the likelihood function can be written as 1 L(μ41,2)= exp (2π)(m+n)/2 m n {− ½Σ(ª¡ − µ1)²} exp{ − ½Σ(v; – 12)²}. i=1 j=1 (b) Prove that the maximum likelihood estimators of μ₁ and μ2 are respectively the sample mean of X and Y, that is, m Â₁ = X = X₁₁₂ == ΣX με m ΣΥ 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 11 ΣX + ΣY _ mX + n m+n m+n
Transcribed Image Text: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 Ho:H= H2, HH#U2 i.i.d. using the likelihood ratio test. N(μ2, 1). (a) Prove that the likelihood function can be written as 1 L(μ41,2)= exp (2π)(m+n)/2 m n {− ½Σ(ª¡ − µ1)²} exp{ − ½Σ(v; – 12)²}. i=1 j=1 (b) Prove that the maximum likelihood estimators of μ₁ and μ2 are respectively the sample mean of X and Y, that is, m Â₁ = X = X₁₁₂ == ΣX με m ΣΥ 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 11 ΣX + ΣY _ mX + n m+n m+n
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