Question

Let \(X_{1}, X_{2}, \ldots, X_{n}\) and \(Y_{1}, Y_{2}, \ldots, Y_{m}\) be independent random samples from the two normal distributions \(N\left(0, \theta_{1}\right)\) and \(N\left(0, \theta_{2}\right)\). (a) Find the likelihood ratio \(\Lambda\) for testing the composite hypothesis \(H_{0}: \theta_{1}=\theta_{2}\) against the composite alternative \(H_{1}: \theta_{1} \neq \theta_{2}\). (b) This \(\Lambda\) is a function of what \(F\) -statistic that would actually be used in this test?

Short answer

The likelihood ratio \(\Lambda\) for this hypothesis test is a function of the F-statistic, where the F-statistic is derived from the likelihood ratio using the formula \[F = \frac{n*m/2}{n+m-2} \times (1-\Lambda^{2/n})\].

Step-by-step solution

Setting Up the Likelihood Ratio

The likelihood ratio for the hypothesis test is defined as the maximized value of the likelihood function under the null hypothesis \(H_{0}: \theta_{1}=\theta_{2}\) divided by the maximized value of the likelihood function under the alternative hypothesis \(H_{1}: \theta_{1} \neq \theta_{2}\). In this case, the likelihood function for independent variables under a normal distribution is defined as the product of the individual probability density functions.

Express the Likelihood Ratio in terms of Sum of Squares

Transform the likelihood ratio by expressing it in terms of the sum of squares of the individual variables. This is achieved by replacing \(\theta_{1}\) and \(\theta_{2}\) in the formula by their estimates, which are the sum of squared deviations from the mean.

Deriving the F-statistic from the Likelihood Ratio

The relationship between the likelihood ratio (denoted by \(\Lambda\)) and the F-statistic is given by the formula -\[F = \frac{n*m/2}{n+m-2} \times (1-\Lambda^{2/n})\] Here, n and m are the sizes of the respective samples, \(\Lambda\) is the likelihood ratio.