Hypotheses Testing-Likelihood Ratio Test (ISS (Statistical Services) Statistics Paper II (Old Subjective Pattern)): Questions 1 - 2 of 2

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Question number: 1

» Hypotheses Testing » Likelihood Ratio Test » ASN Function

Appeared in Year: 2015

Essay Question▾

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Derive the likelihood ratio test for comparing the means of k independent homoscedastic normal populations.

Explanation

Given that there are k independent homoscedastic normal populations that is the variance is same i. e. Equation ; i = 1, 2, …, k. We have to test

Equation

In the X population the sample is = { (x i1, x i2, …, x ini)

The

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Question number: 2

» Hypotheses Testing » Likelihood Ratio Test » ASN Function

Appeared in Year: 2014

Essay Question▾

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x 1 , x 2 , …, x n is a random sample from N (θ, σ 2 ) (σ 2 not specified). Derive likelihood ratio test of testing H 0 : θ=θ 0 against H 1 : θ ≠ θ 0 .

Explanation

Under the given model, the parameter space is

Equation

Under H 0 , the parameter space is

Equation

The likelihood function is

Equation

Under the whole space, the unrestricted MLE is

Equation

Under H 0 , the restricted MLE is

Equation

The statistic is

Equation

Equation

Equation

Equation

The likelihood

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