Let X1, X2, …, Xn and Y1, Y2, …, Yn be independent random samples from two normal distributions

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Let X1, X2, …, Xn and Y1, Y2, …, Yn be independent random...

Let $X_{1}, X_{2}, \ldots, X_{n}$ and $Y_{1}, Y_{2}, \ldots, Y_{n}$ be independent random samples from two normal distributions $N\left(\mu_{1}, \sigma^{2}\right)$ and $N\left(\mu_{2}, \sigma^{2}\right)$, respectively, where $\sigma^{2}$ is the common but unknown variance. (a) Find the likelihood ratio $\Lambda$ for testing $H_{0}: \mu_{1}=\mu_{2}=0$ against all alternatives. (b) Rewrite $\Lambda$ so that it is a function of a statistic $Z$ which has a well-known distribution. (c) Give the distribution of $Z$ under both null and alternative hypotheses.

Let $X_{1}, X_{2}, \ldots, X_{n}$ and $Y_{1}, Y_{2}, \ldots, Y_{n}$ be independent random samples from two normal distributions $N\left(\mu_{1}, \sigma^{2}\right)$ and $N\left(\mu_{2}, \sigma^{2}\right)$, respectively, where $\sigma^{2}$ is the common but unknown variance.
(a) Find the likelihood ratio $\Lambda$ for testing $H_{0}: \mu_{1}=\mu_{2}=0$ against all alternatives.
(b) Rewrite $\Lambda$ so that it is a function of a statistic $Z$ which has a well-known distribution.
(c) Give the distribution of $Z$ under both null and alternative hypotheses.

Key Concepts

Likelihood Ratio Test

The likelihood ratio test (LRT) is a fundamental method in hypothesis testing used to compare two nested models by forming the ratio of their maximum likelihoods. In this framework, one calculates the maximum likelihood under the restrictions imposed by the null hypothesis and compares it to the maximum likelihood when these restrictions are removed. This ratio, often transformed via a logarithm, provides a test statistic whose null distribution can be used to assess whether the data provide sufficient evidence to reject the null hypothesis.

Maximum Likelihood Estimation

Maximum likelihood estimation (MLE) is the process of determining the parameter values that maximize the likelihood function, effectively identifying the most plausible parameters given the observed data. In the context of the LRT, MLE is used to obtain estimates under both the null hypothesis (with restrictions) and the alternative hypothesis (without restrictions), serving as the foundation for the computation of the likelihood ratio.

Sufficient Statistics and Data Reduction

Sufficient statistics are functions of the data that encapsulate all the information needed about a parameter for inference purposes. In many normal model settings, sample means (and related sums) act as sufficient statistics for the population means. By rewriting the likelihood ratio in terms of these statistics, unnecessary complexity is removed, allowing the test statistic to be expressed in a form that follows a recognized probability distribution, thereby simplifying analysis.

Test Statistic Distribution under Null and Alternative

The distribution of the test statistic derived from the likelihood ratio plays a crucial role in hypothesis testing. Under the null hypothesis, the statistic often follows a central distribution (such as a chi-square or normal distribution), while under the alternative it adopts a noncentral form. This property is essential for determining critical regions and power calculations, as it allows for the assessment of how likely the observed data would be under each hypothesis.

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