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41. Let Xi ~ bin(n; , p), for i = I , . . . , m, be independent. Derive a likelihood ratiotest for the hypothesisHo: p1 = p2 = . . . = pmagainst the alternative hypothesis that the p, are not all equal. What is the large-sample distribution of the test statistic?

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9.41 question attached

41. Let Xi ~ bin(n; , p), for i = I , . . . , m, be independent. Derive a likelihood ratio
test for the hypothesis
Ho: p1 = p2 = . . . = pm
against the alternative hypothesis that the p, are not all equal. What is the large-
sample distribution of the test statistic?
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41. Let Xi ~ bin(n; , p), for i = I , . . . , m, be independent. Derive a likelihood ratio test for the hypothesis Ho: p1 = p2 = . . . = pm against the alternative hypothesis that the p, are not all equal. What is the large- sample distribution of the test statistic?

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Step 1

Solution

Given: Let Xi follows bin(

Solution

Given: Let Xi follows bin(ni, pi), for i = 1, 2, …., m, be independent.

We have to derive a likelihood ratio test for the hypothesis H0: p1 = p2 = … = pm against the alternative hypothesis that the pi are not all equal.

ni, pi), for i = 1, 2, …., m, be independent.

We have to derive a likelihood ratio test for the hypothesis H0: p1 = p2 = … = pm against the alternative hypothesis that the pi are not all equal.

Step 2

Step 1:

Since the hypotheses are composite, we have to use the generalized likelihood ratio test

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Step 3

The null hypothesis may specify that p ϵ ω0, where ω0 is a subset of the set of all possible values of p, and H1 may specify that p ϵ ω1, ω1 is disjoint from &omega...

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