Question

Let the independent normal random variables \(Y_{1}, Y_{2}, \ldots, Y_{n}\) have, respectively, the probability density functions \(N\left(\mu, \gamma^{2} x_{i}^{2}\right), i=1,2, \ldots, n\), where the given \(x_{1}, x_{2}, \ldots, x_{n}\) are not all equal and no one of which is zero. Discuss the test of the hypothesis \(H_{0}: \gamma=1, \mu\) unspecified, against all alternatives \(H_{1}: \gamma \neq 1, \mu\) unspecified.

Short answer

To solve this, it's necessary to formulate the likelihood ratio for the given variables \(Y_{1}, Y_{2}, \ldots, Y_{n}\) and compare this with a critical value. This value should be known from a chi-square distribution. This comparison decides whether to accept or reject the null hypothesis \(H_{0}\). The answer varies with data of \(Y_{i}\)'s, critical value and \(\gamma\).

Step-by-step solution

Formulate Likelihood Ratio

Recall that the likelihood ratio for hypotheses \(H_{0}\) and \(H_{1}\) can be defined as: \( \Lambda = \frac{sup_{\mu,\gamma: \gamma=1 } L(Y|\mu,\gamma)}{sup_{\mu,\gamma: \gamma \neq 1} L(Y|\mu,\gamma)} \) Where \( L(Y|\mu,\gamma) \) is the likelihood function of parameters \(\mu\) and \(\gamma\) given the data Y (Y consists of variables \(Y_{1}, \dots, Y_{n}\)). Considering that the variables are independent normal random variables, and applying log to simplify calculation, we result in using next step for calculation.

Calculate Likelihood Function

For independent normal variables, likelihood function can be expressed as - Total log likelihood = \( \sum_{i=1}^{n} log( L(Y_{i}|\mu,\gamma) ) = -\frac{n}{2} log(2\pi) - \sum_{i=1}^{n} log( (\gamma^{2} x_{i}^{2}) ) - \frac{1}{2} \sum_{i=1}^{n} ( \frac{Y_{i} - \mu}{\gamma x_{i}} )^{2} \) Calculate this for \(\gamma=1\) and for general \(\gamma\).

Find Critical Value

-2 log (\( \Lambda \)) will approximately follows a chi-square distribution with 1 degree of freedom under the null hypothesis. Here, the critical value can be found from a chi-square distribution. If our calculated value is larger than the critical value, we reject the null hypothesis.

Interpret Result

Based on the result obtained in Step 3, conclude whether to accept or reject the null hypothesis. If the null hypothesis is rejected, it means the data provides enough evidence to support the claim that \(\gamma\) is not 1. If the null hypothesis is not rejected, it means there isn't enough evidence in the data to support the claim that \(\gamma\) is different from 1.

Key concepts

Independent Normal Random Variables

Understanding independent normal random variables is pivotal when dealing with many statistical models and tests. These variables, often noted as \(Y_{1}, Y_{2}, ..., Y_{n}\), follow a normal distribution with a specific mean \(\mu\) and variance, which in this case is \(\gamma^{2} x_{i}^{2}\) for each variable. An essential property is their independence, meaning that the value of one variable does not influence or provide any information about the value of another. This independence is crucial when calculating joint probability distributions or when deriving properties like the likelihood function in hypothesis testing.

When handling such variables, we assume each has its own normal distribution curve, characterized by the parameters mean \(\mu\) and variance \(\gamma^{2}x_{i}^{2}\), where \(x_{i}\) are known constants. Visualization can be very helpful: imagine a series of bell-shaped curves representing the normal distribution of each variable; despite sharing a common factor \(\gamma\), their distribution can vary due to \(x_{i}\).

Hypothesis Testing

In the world of statistics, hypothesis testing plays a critical role. It's a method used to determine if there is enough evidence in a sample of data to infer that a certain condition holds for the entire population. In the exercise, we look at testing the null hypothesis \(H_{0}: \gamma=1\), against an alternative hypothesis \(H_{1}: \gamma eq 1\), regarding the variable \(\gamma\) in the probability density functions of our independent normal random variables.

The hypothesis \(H_{0}\) is a statement of no effect or no difference, while \(H_{1}\) suggests a deviation from the norm. We don't specify \(\mu\) because, in this test, we're solely interested in whether \(\gamma\) equals 1 or not. The key is to use sample data to calculate a test statistic, which then helps us decide whether to reject or not to reject the null hypothesis. If the statistic falls into a certain critical region or beyond a critical value, determined by the chosen significance level, we reject \(H_{0}\).

Chi-Square Distribution

With roots in the concept of variance, the chi-square distribution is fundamental to various statistical tests, notably the Likelihood Ratio Test. It is a family of distributions that are defined by degrees of freedom and typically arise when dealing with the squared differences between expected and observed frequencies, or the sum of squared independent standard normal random variables.

When we perform a likelihood ratio test like the one in the given exercise, we end up with a test statistic following a chi-square distribution, provided the null hypothesis is true. Here, \( -2\log(\Lambda) \) approximately follows a chi-square distribution with degrees of freedom usually equal to the number of restrictions imposed by the null hypothesis, which, in this case, is 1. The chi-square distribution table or computational tools enable us to find the critical value that aids in deciding whether to reject or accept \(H_{0}\).

Statistical Inference

Statistical inference is the process by which we generalize observations from sample data to larger populations. It enables us to make predictions, estimate unknown parameters, and test hypotheses. The exercise presents a situation where we are inferring properties about the population parameter \(\gamma\), based on sample observations \(Y_{1}, ..., Y_{n}\).

Two main branches encompassed by statistical inference are estimation of parameters, which provides us with point estimates and confidence intervals, and hypothesis testing, which involves making decisions about population parameters based on sample statistics. In the context of our exercise, statistical inference tools are applied to assess whether the parameter \(\gamma\) equals 1, a process involving estimation (of the likelihood function), decision-making based on a critical value (chi-square distribution), and interpreting results with respect to rejection or acceptance of a hypothesis.