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 the ratio of the joint likelihood functions under the null hypothesis and alternative hypothesis. The ration \( \Lambda \) can be rewritten in terms of a statistic \( Z \), where \( Z = -\frac{n}{2\sigma^2}(\bar{X}^2 + \bar{Y}^2) + n(\frac{\bar{X}^2}{S_X^2} + \frac{\bar{Y}^2}{S_Y^2}) \). \( Z \) has a chi-square distribution (\( Z \sim \chi^2(2) \)) under the alternative hypothesis, and \( Z = 0 \) under null hypothesis.

Step-by-step solution

Part (a): Compute the Likelihood Ratio

First, calculate the joint likelihood function under both the null and alternative hypotheses. Under null hypothesis, \( \mu_{1} = \mu_{2} = 0 \). So, the joint likelihood function of \( X = (X_1, X_2, ..., X_n) \) and \( Y = (Y_1, Y_2, ..., Y_n) \) is \[ L_{0} = \frac{1}{(2\pi \sigma^2)^n} exp\left(-\frac{1}{2\sigma^2}\sum_{i=1}^{n}(x_i^2 + y_i^2)\right)\]. Under alternative hypothesis \( H_1: \mu_1 \neq \mu_2 \), joint likelihood function is \[ L_{1} = \frac{1}{(2\pi \sigma^2)^n} exp\left(-\frac{1}{2\sigma^2}\sum_{i=1}^{n}((x_i - \mu_1)^2 + (y_i - \mu_2)^2)\right)\]. Next, we can calculate the likelihood ratio \( \Lambda \) which is equal to \( \Lambda = \frac{ L_{0} }{ L_{1} }\).

Part (b): Rewrite \(\Lambda\) as a Function of the Statistic Z

Rewrite the likelihood ratio in terms of statistic Z. Notice that maximizing the likelihood function is the same as maximizing the logarithm of the likelihood function, because log function is monotonically increasing. We can call the maximum log-likelihood ratio as Z which equals \[ Z = -\frac{n}{2\sigma^2}(\bar{X}^2 + \bar{Y}^2) + n(\frac{\bar{X}^2}{S_X^2} + \frac{\bar{Y}^2}{S_Y^2})\] where \( S_X^2, S_Y^2 \) are the sample variances for X and Y respectively, and \( \bar{X}, \bar{Y} \) are sample means for X and Y.

Part (c): Determine the Distribution of Z

The distribution of \( Z \) under both null and alternative hypotheses is to be determined. Under null hypothesis (\(H_0\)), \( Z = 0 \). Under alternative hypothesis (\(H_1\)), Z follows a chi-square distribution \( \chi^2 \) with degrees of freedom equal to the number of parameters in the alternative hypothesis which equals the difference in the number of parameters between the alternative and null hypotheses, in this case 2, i.e. \( Z \sim \chi^2(2) \).