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1. Suppose we model X₁,..., XnExp(A) as iid RVs with unknown rate X. Suppose that we observe = 21.79 and n = 100. (a) Conduct a uniformly most powerful test at size a = 0.01 for the test of Ho: λ=1/23 vs H₁: λ=1/22 (b) Recall that the MLE of A is given by  = 1/. Using the MLE as the test statistic show that for the two-sided test of Ho: A = 1/23 vs H₁: A1/23 at level a = 0.01 the rejection region is of the form n 72 {x:Σ>cor Σ≤D} i=1 i=1 where C and D are constants to be specified. Hint: Use the hinted result in Assignment 3 Qu(1)(f) to help derive the constants. What is the conclusion in this case? Is the Null rejected? can be written as (for some constant C) (c) Now, using the likelihood ratio test statistic as the test statistic show that the rejection region for the two sided test of Ho: λ=1/23 vs H₁: λ1/23 {* : *exp(-23²) ≤C} (d) For the setting in (c). Derive an expression for -2 log x(x) and evaluate this given the observed data. (e) For the setting in (c). Given that n is large, use Wilk's Theorem and your answer from (d) to conduct the stated hypothesis test at a level a = 0.01.

1. Suppose we model X₁,..., XnExp(A) as iid RVs with unknown rate X. Suppose that we observe = 21.79 and n = 100. (a) Conduct a uniformly most powerful test at size a = 0.01 for the test of Ho: λ=1/23 vs H₁: λ=1/22 (b) Recall that the MLE of A is given by  = 1/. Using the MLE as the test statistic show that for the two-sided test of Ho: A = 1/23 vs H₁: A1/23 at level a = 0.01 the rejection region is of the form n 72 {x:Σ>cor Σ≤D} i=1 i=1 where C and D are constants to be specified. Hint: Use the hinted result in Assignment 3 Qu(1)(f) to help derive the constants. What is the conclusion in this case? Is the Null rejected? can be written as (for some constant C) (c) Now, using the likelihood ratio test statistic as the test statistic show that the rejection region for the two sided test of Ho: λ=1/23 vs H₁: λ1/23 {* : *exp(-23²) ≤C} (d) For the setting in (c). Derive an expression for -2 log x(x) and evaluate this given the observed data. (e) For the setting in (c). Given that n is large, use Wilk's Theorem and your answer from (d) to conduct the stated hypothesis test at a level a = 0.01.

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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1. Suppose we model X₁,…, Xn ~ Exp(A) as iid RVs with unknown rate X. Suppose that we observe = 21.79 and n = 100. (a) Conduct a uniformly most powerful test at size a = 0.01 for the test of Ho: A = 1/23 vs H₁: λ=1/22 (b) Recall that the MLE of A is given by  = 1/. Using the MLE as the test statistic show that for the two-sided test of Ho: λ=1/23 vs H₁: λ1/23 at level a = 0.01 the rejection region is of the form n n {x : Σx₁ ≥C or Σxi ≤ D} i=1 i=1 where C and D are constants to be specified. Hint: Use the hinted result in Assignment 3 Qu(1)(f) to help derive the constants. What is the conclusion in this case? Is the Null rejected? (c) Now, using the likelihood ratio test statistic as the test statistic show that the rejection region for the two sided test of Ho: λ=1/23 vs H₁: λ1/23 can be written as (for some constant C) {2: * Exp(-1/27 ) ≤C} 23 (d) For the setting in (c). Derive an expression for -2 log x(x) and evaluate this given the observed data. (e) For the setting in (c). Given that n is large, use Wilk's Theorem and your answer from (d) to conduct the stated hypothesis test at a level a = 0.01.
Transcribed Image Text:1. Suppose we model X₁,…, Xn ~ Exp(A) as iid RVs with unknown rate X. Suppose that we observe = 21.79 and n = 100. (a) Conduct a uniformly most powerful test at size a = 0.01 for the test of Ho: A = 1/23 vs H₁: λ=1/22 (b) Recall that the MLE of A is given by  = 1/. Using the MLE as the test statistic show that for the two-sided test of Ho: λ=1/23 vs H₁: λ1/23 at level a = 0.01 the rejection region is of the form n n {x : Σx₁ ≥C or Σxi ≤ D} i=1 i=1 where C and D are constants to be specified. Hint: Use the hinted result in Assignment 3 Qu(1)(f) to help derive the constants. What is the conclusion in this case? Is the Null rejected? (c) Now, using the likelihood ratio test statistic as the test statistic show that the rejection region for the two sided test of Ho: λ=1/23 vs H₁: λ1/23 can be written as (for some constant C) {2: * Exp(-1/27 ) ≤C} 23 (d) For the setting in (c). Derive an expression for -2 log x(x) and evaluate this given the observed data. (e) For the setting in (c). Given that n is large, use Wilk's Theorem and your answer from (d) to conduct the stated hypothesis test at a level a = 0.01.
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