# 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

### Describe in Detail

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. ; i = 1, 2, …, k. We have to test

In the __X__ population the sample is = { (x _{i1}, x _{i2}, …, x _{ini})

The parameter space is

The parameter space under H _{0} is

The likelihood function is

The unrestricted MLE is

The restricted MLE is

The…

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## Question number: 2

» Hypotheses Testing » Likelihood Ratio Test » ASN Function

Appeared in Year: 2014

### Describe in Detail

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

Under H _{0 }, the parameter space is

The likelihood function is

Under the whole space, the unrestricted MLE is

Under H _{0 }, the restricted MLE is

The statistic is

The likelihood ratio test is reject H _{0 } if λ (__x __) ≤ c is equivalent to

Under H _{0 }

And these two random …

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