Question

Let \(X_{1}, \ldots, X_{n}\) and \(Y_{1}, \ldots, Y_{m}\) be independent random samples from the distributions \(N\left(\theta_{1}, \theta_{3}\right)\) and \(N\left(\theta_{2}, \theta_{4}\right)\), respectively. (a) Show that the likelihood ratio for testing \(H_{0}: \theta_{1}=\theta_{2}, \theta_{3}=\theta_{4}\) against all alternatives is given by $$ \begin{aligned} &\qquad\left[\sum_{1}^{n}\left(x_{i}-\bar{x}\right)^{2} / n\right]^{n / 2}\left[\sum_{1}^{m}\left(y_{i}-\bar{y}\right)^{2} / m\right]^{m / 2} \\ &\left\\{\left[\sum_{1}^{n}\left(x_{i}-u\right)^{2}+\sum_{1}^{m}\left(y_{i}-u\right)^{2}\right] /(m+n)\right\\}^{(n+m) / 2} \end{aligned} $$ (b) Show that the likelihood ratio test for testing \(H_{0}: \theta_{3}=\theta_{4}, \theta_{1}\) and \(\theta_{2}\) unspecified, against \(H_{1}: \theta_{3} \neq \theta_{4}, \theta_{1}\) and \(\theta_{2}\) unspecified, can be based on the random variable $$ F=\frac{\sum_{1}^{n}\left(X_{i}-\bar{X}\right)^{2} /(n-1)}{\sum_{1}^{m}\left(Y_{i}-\bar{Y}\right)^{2} /(m-1)} $$

Short answer

The likelihood ratio for testing equal means and variances between independent normal samples can be expressed in the given form. And for testing the equality of variances with unspecified means, the likelihood ratio test can be based on the given F-ratio.

Step-by-step solution

Calculate the likelihood function

Using the properties of normal distribution, the likelihood function for each group \(X[1, n]\) and \(Y[1, m]\) under the null hypothesis can be expressed. Then we multiply these two likelihood functions since the two groups are independent.

Define the test statistic and the likelihood ratio

The test statistic for the likelihood ratio test is the ratio of the maximum likelihood under the null hypothesis (maximized over \(\theta_1\) and \(\theta_3\) or \(\theta_2\) and \(\theta_4\)) to the maximum likelihood under the alternative (maximized over all parameters). Simplify this ratio to match the given form.

Show that the likelihood ratio test can be based on the F-ratio

To show this, first define the variance for each group using the given F-ratio. Then derive the likelihood ratio for the null hypothesis where \(\theta_3\) is equal to \(\theta_4\) and alternative hypothesis where those values are not equal, using these variances. Now, by simplifying this ratio and comparing it with the definition of the F-ratio, we will find out that the likelihood ratio test can indeed be based on the given F-ratio.

Key concepts

Independent Random Samples

Understanding the concept of independent random samples is critical to the application of statistics to real-world problems. Basically, this means that we have two or more sets of data, which can be from different populations, and there is no linkage or influence between any of the samples in these sets. Imagine rolling a die and flipping a coin: the outcome of the die doesn't affect the coin, and vice versa; they are independent events.

When it comes to independent random samples in the context of statistics, this principle implies that the samples are collected separately without any overlap or dependence such that the results from one sample do not affect results from another. This is a foundational assumption in many statistical tests, including the likelihood ratio test. These samples often come from different populations, which in our exercise are modeled by different normal distributions, and their independence is vital for the reliability of the likelihood ratio test results.

Normal Distribution

The normal distribution, often symbolized as \(N(\mu, \sigma^2)\), is a continuous probability distribution that is symmetrical around its mean \(\mu\), indicating that data near the mean are more frequent in occurrence than data far from the mean. Its shape is famously bell-curved, and it is characterized by two parameters: the mean \(\mu\), which locates the center of the distribution, and the variance \(\sigma^2\), which measures the spread of the distribution.

In the exercise we're considering, each sample is drawn from a normal distribution, with means \(\theta_1\) and \(\theta_2\), and variances \(\theta_3\) and \(\theta_4\), respectively. Why is this important? Many statistical methods, including the likelihood ratio test, are based on the assumption of normality, because many things we measure in the natural and social sciences are normally distributed, or approximately so. When the data is normally distributed, formulas derived from the normal distribution can be applied to infer about the whole population from the sample.

Maximum Likelihood Estimation

Maximum likelihood estimation (MLE) is a method used in statistics to estimate the parameters of a statistical model. It identifies the values of the model parameters that make the known data 'most likely' to have occurred. How does it work? MLE finds the parameter values that maximize the likelihood function, which measures the probability of the observed data given a set of parameters.

Let's connect this with our exercise: The likelihood ratio test compares the likelihoods of two hypotheses, typically a null hypothesis and an alternative. To do this, we calculate the likelihood of the observed data under the null hypothesis, maximizing over any specified parameters, versus the likelihood under the alternative hypothesis, potentially maximizing over a different set of parameters. By examining the ratio of these two likelihoods, we can see which set of parameters is more likely given the data we have, and decide whether to reject the null hypothesis in favor of the alternative. This method is especially powerful as it accounts for the variability and uncertainty inherent in real-world data collection.