Hello,Can someone please show how to solve this problem? Thank you! 5. Suppose that the observed data X~ f(x; 0). Then the likelihood function is L(0) = f(x;0).An intuitive test for the simple hypothesis Ho: 0 = 0o and H₁ : 0 = 0₁, where and 0₁ aregiven parameters, is the likelihood ratio test, which rejects Ho whenL(01)/L(00) ≥c,for some constant c to be determined so that type I error is a. The test is also called theNeyman-Person test and its equivalent statistic X(x) = log(L(01)/L(00)) is called the log-likelihood ratio statistic.(a) Suppose X₁,..., Xn ~ii.d. N(μ,0²) where σ is known (for simplicity). Consider thetesting problemHoμ0 vs. H₁: = μ1where μ₁ > 0. Derive the log-likelihood ratio statistic and shows that it is a monotonicfunction of sample mean X.(b) If X₁,Xn are a random sample from Bernoulli (p), derive the log-likelihood ratio"statistic forHo p = po vs. H₁: p= P₁where p₁>po and show that it is a monotonic function of p= n−¹(X₁ ++ Xn), thesample proportion.

Given the formula of log-likelihood statistic for testing the hypotheses H0 : θ=θ0 Vs H1 : θ=θ1 as…

5. Suppose that the observed data X~ f(x; 0). Then the likelihood function is L(0) = f(x; 0). An intuitive test for the simple hypothesis Ho: 0 = 0o and H₁ : 0 = 0₁, where 00 and 0₁ are given parameters, is the likelihood ratio test, which rejects Ho when L(01)/L(00) ≥c, for some constant c to be determined so that type I error is a. The test is also called the Neyman-Person test and its equivalent statistic A(x) = log(L(01)/L(00)) is called the log- likelihood ratio statistic. (a) Suppose X₁, Xn ~ii.d. N(μ,02) where o is known (for simplicity). Consider the testing problem Ho = 0 vs. H₁: = μ₁ where μ₁0. Derive the log-likelihood ratio statistic and shows that it is a monotonic function of sample mean X. (b) If X₁, Xn are a random sample from Bernoulli (p), derive the log-likelihood ratio statistic for Ho: p= po vs. H₁: p= P₁ where p₁>po and show that it is a monotonic function of p= n¹(X₁+...+Xn), the sample proportion.

5. Suppose that the observed data X~ f(x; 0). Then the likelihood function is L(0) = f(x; 0). An intuitive test for the simple hypothesis Ho: 0 = 0o and H₁ : 0 = 0₁, where 00 and 0₁ are given parameters, is the likelihood ratio test, which rejects Ho when L(01)/L(00) ≥c, for some constant c to be determined so that type I error is a. The test is also called the Neyman-Person test and its equivalent statistic A(x) = log(L(01)/L(00)) is called the log- likelihood ratio statistic. (a) Suppose X₁, Xn ~ii.d. N(μ,02) where o is known (for simplicity). Consider the testing problem Ho = 0 vs. H₁: = μ₁ where μ₁0. Derive the log-likelihood ratio statistic and shows that it is a monotonic function of sample mean X. (b) If X₁, Xn are a random sample from Bernoulli (p), derive the log-likelihood ratio statistic for Ho: p= po vs. H₁: p= P₁ where p₁>po and show that it is a monotonic function of p= n¹(X₁+...+Xn), the sample proportion.

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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