Question

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.

Short answer

The likelihood ratio \(\Lambda\) for testing \(H_{0}: \mu_{1}=\mu_{2}=0\) against all alternatives is \(\Lambda = \exp(n)\). After rewriting \(Λ\), it becomes a function of a statistic \(Z\), specifically \(Λ = \exp(\frac{Z^2}{2})\). The distribution of \(Z\) under the null hypothesis is a standard normal distribution \(N(0,1)\), and under the alternative hypothesis, it is not precisely defined without additional information since it depends on the difference between the means \(\mu_{1}\) and \(\mu_{2}\).

Step-by-step solution

Calculate Likelihood Ratio

The likelihood ratio \(Λ\) is given by \[Λ = \frac{L(\mu_{1}=0, \mu_{2}=0)}{L(\hat{\mu}_{1}, \hat{\mu}_{2})}\] Where \(L(\mu_{1}=0, \mu_{2}=0)\) is the likelihood of the null hypothesis and \(L(\hat{\mu}_{1}, \hat{\mu}_{2}\) is the likelihood of the alternate hypothesis. After inserting the given distributions and doing the needed calculations, this ratio simplifies to \(Λ = \exp(n)\.

Rewrite the Ratio

The next step is to rewrite the likelihood ratio \(Λ\) so that it is a function of the statistic \(Z\). The \(Z\) statistic is defined as \(Z = \frac{\bar{X} - \bar{Y}}{s}\) where \(\bar{X}\) and \(\bar{Y}\) are the means, and \(s\) is the standard deviation. After substitution, the new expression for \(Λ\) will become \(Λ = \exp(\frac{Z^2}{2})\).

Provide the Distribution

The final step is to provide the distribution of \(Z\) under both null and alternative hypotheses. Under \(H_{0}\) the \(Z\) score follows a standard normal distribution, typically denoted as \(N(0,1)\). Contrarily, under any other alternative hypothesis \(H_{1}\), since \(Z\) is a random variable that varies according to the difference of the means \(\mu_{1}\) and \(\mu_{2}\), the distribution cannot be precisely determined without further information.