Let Y1, Y2, …, Yn denote a random sample from a normal...

Let $Y_{1}, Y_{2}, \ldots, Y_{n}$ denote a random sample from a normal distribution with mean $\mu$ (unknown) and variance $\sigma^{2} .$ For testing $H_{0}: \sigma^{2}=\sigma_{0}^{2}$ against $H_{a}: \sigma^{2}>\sigma_{0}^{2},$ show that the likelihood ratio test is equivalent to the $\chi^{2}$ test given in Section $10.9 .$

Key Concepts

Likelihood Ratio Test

The likelihood ratio test (LRT) is a method for comparing the fit of two statistical models, one of which is nested within the other. It involves calculating the ratio of the maximum likelihood under the null hypothesis to the maximum likelihood over the entire parameter space. In the context of testing hypotheses about the variance of a normal distribution, the LRT conveniently transforms into a test statistic that can be evaluated using known distributions, making it a powerful and general approach in hypothesis testing.

Chi-Squared Distribution

The chi-squared distribution is a key probability distribution in statistics that arises naturally in the context of variance estimation and hypothesis testing. When sampling from a normal distribution, the sum of squared deviations (appropriately scaled) follows a chi-squared distribution. This property underpins the chi-squared test for variance, where the test statistic is formulated as a sum of squared standardized observations and compared to critical values from the chi-squared distribution.

Sufficient Statistics in Normal Models

In normal distribution models, particularly when testing for parameters such as variance, the sum of squared deviations from the mean is a sufficient statistic. This statistic captures all necessary information about the variance in the sample, and its distribution under the null hypothesis is pivotal (i.e., independent of the unknown parameters) and known, making it ideal for constructing tests like the likelihood ratio test and the chi-squared test.

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