Question

Let \(X_{1}, X_{2}, \ldots, X_{n}\) be a random sample from a distribution with pdf \(f(x ; \theta)=\theta \exp \left\\{-|x|^{\theta}\right\\} / 2 \Gamma(1 / \theta),-\infty0 .\) Suppose \(\Omega=\) \(\\{\theta: \theta=1,2\\}\). Consider the hypotheses \(H_{0}: \theta=2\) (a normal distribution) versus \(H_{1}: \theta=1\) (a double exponential distribution). Show that the likelihood ratio test can be based on the statistic \(W=\sum_{i=1}^{n}\left(X_{i}^{2}-\left|X_{i}\right|\right)\).

Short answer

The likelihood ratio test for the given hypotheses is indeed based on the statistic \(W=\sum_{i=1}^{n}\left(X_{i}^{2}-\left|X_{i}\right|\right)\). The likelihood ratio was found to be equivalent to this statistic after simplifying the expression for the likelihood ratio test statistic.

Step-by-step solution

Understand the Hypotheses & Test

The null hypothesis \(H_{0}: \theta=2\) indicates a normal distribution and the alternative hypothesis \(H_{1}: \theta=1\) indicates a double exponential distribution. The likelihood ratio test is commonly used in this type of scenario to choose between these two options. The test compares the likelihood of the observed data under the null hypothesis to the likelihood of the observed data under the alternative hypothesis.

Find the Likelihood Ratio Statistic

To determine if \(W\) is a valid statistic for this test, let's first write down the likelihood functions for both hypotheses. Under \(H_{0}\), the likelihood function is \(\prod_{i=1}^{n}\frac{2 \exp \left\{-x_{i}^{2}\right\}}{2 \Gamma(1 / 2)}\). Under \(H_{1}\), the likelihood function is \(\prod_{i=1}^{n}\frac{1 \exp \left\{-|x_{i}|\right\}}{2 \Gamma(1)}\). Now, the likelihood ratio test statistic is given by \(-2 \log \left(\frac{Likelihood\ under\ H_{0}}{Likelihood\ under\ H_{1}}\right)\), where we ignore constants in the likelihood functions as they don't affect the result of the test.

Simplify the Likelihood Ratio Statistic

So, we have \(-2 \log \left(\frac{Likelihood\ under\ H_{0}}{Likelihood\ under\ H_{1}}\right) = -2 \sum_{i=1}^{n}\left(-x_{i}^{2}+|x_{i}|\right) = \sum_{i=1}^{n}\left(x_{i}^{2}-|x_{i}|\right)\). This is indeed the given statistic \(W\).

Key concepts

Probability Density Function

A Probability Density Function (PDF) is a function that describes the likelihood of a random variable to take on a particular value. For continuous random variables, the PDF is used to specify the probability of the variable falling within a certain range of values, as opposed to taking on any one value. This range probability is found by calculating the area under the PDF’s curve over that interval.

In the given exercise, the PDF for a random variable follows the formula:

• \[f(x ; heta)=\frac{\theta \exp \left\{-|x|^{\theta}\right\}}{2 \Gamma(1 /\theta)} , -\infty < x < \infty\]

The parameter \(\theta\) influences the shape of the PDF. Here, \(\theta\) must be greater than 0, which ensures that the PDF is well-defined. Understanding this PDF is crucial for performing hypothesis testing on the parameter, as it governs the distribution of the data. The ancient Greek letter \(\Gamma\) in the expression represents the gamma function, which generalizes the factorial function for real and complex numbers. This function often appears in probability distributions, normalizing the areas to sum to unity.

Hypothesis Testing

Hypothesis testing is a statistical method used to decide whether there is enough evidence in a sample of data to infer that a certain condition holds for the entire population. The hypotheses are often statements about a parameter of a statistical model, such as the mean of a population.

In the exercise, we have two hypotheses:

• The null hypothesis, \(H_{0}: \theta=2\), which suggests the data follows a distribution akin to a normal distribution.
• The alternative hypothesis, \(H_{1}: \theta=1\), which implies that the data follows a double exponential distribution.

The process begins with assuming the null hypothesis is true and then performing a test to determine if there is significant evidence to reject it in favor of the alternative hypothesis.

Likelihood Ratio Test

The likelihood ratio test checks the plausibility of two competing hypotheses by comparing how well they fit the observed data. If the likelihood of the data under the null hypothesis is significantly lower than under the alternative hypothesis, we reject \(H_{0}\). In the given problem, the test statistic is designed to simplify this comparison.

Double Exponential Distribution

The Double Exponential Distribution, also known as the Laplace distribution, is a continuous probability distribution with two key characteristics:

• It has a peak at the mean where it is most likely to return values, with probabilities decreasing exponentially as we move away from the mean.
• It possesses heavier tails than the normal distribution, meaning that it predicts outliers relatively more often.

The PDF for the Laplace distribution, when the parameter \(\theta = 1\), simplifies to:

• \[f(x; 1) = \frac{1}{2} \exp(-|x|)\]

In hypothesis testing, identifying distributions like the Laplace distribution offers a useful model for data distributions that are "pointy" in the middle and thicker at the tails compared to the normal distribution. This makes it suitable for datasets where outliers are common or where observations cluster heavily around a central value. Understanding the characteristics of the double exponential distribution helps in choosing the correct test and interpreting results when analyzing data that might follow this pattern.