Question

Let \(X_{1}, X_{2}, \ldots, X_{n}\) be a random sample from a \(\Gamma(\alpha, \beta)\) -distribution where \(\alpha\) is known and \(\beta>0\). Determine the likelihood ratio test for \(H_{0}: \beta=\beta_{0}\) against \(H_{1}: \beta \neq \beta_{0}\)

Short answer

The likelihood ratio test statistic, \(\Lambda(X)\), is equal to twice the difference between the log-likelihoods of the predictions made by the two models associated with the null and alternative hypothesis respectively. It follows a chi-square distribution with 1 degree of freedom in this case, and if it is larger than chi-square critical value, one should reject the null hypothesis that \(\beta\) equals \(\beta_0\).

Step-by-step solution

Understanding of Hypothesis

Our null hypothesis \(H_0: \beta=\beta_0\) suggests that the \(\beta\) parameter in the Gamma (alpha, beta) distribution equals a particular value \(\beta_0\). The alternative hypothesis \(H_1: \beta \neq \beta_0\) suggests that the \(\beta\) parameter is different from \(\beta_0\).

Formulate Likelihood Function

The likelihood function for this sample from the Gamma distribution is given by \[L(\beta|X) = \prod_{i=1}^{n} f(x_i|\beta) = \prod_{i=1}^{n} \frac{1}{\Gamma(\alpha)\beta^{\alpha}}x_{i}^{\alpha−1}e^{\frac{−x_{i}}{ \beta}}\]where \(f(x_i|\beta)\) is the density function of a gamma distribution.

Compute the Log-likelihood Function

Take the natural logarithm of the likelihood function to compute the log likelihood function, which simplifies the math and doesn't change the estimation. It will be \[ \ell(\beta|X) = \ln(L(\beta|X)) = -n\alpha \ln(\beta) - \frac{1}{\beta}\sum_{i=1}^{n} x_i\]

Compute Derivative of the Log-likelihood Function

Take the derivative of the log-likelihood function with respect to \(\beta\). Setting the derivative equal to zero and solving for \(\beta\) gives the maximum likelihood estimate \(\hat{\beta}\) of \(\beta_0\) under \(H_1\):\[-\frac{n\alpha}{\beta} + \frac{1}{\beta^2}\sum_{i=1}^{n}x_i = 0\] which gives us \[\hat{\beta} = \frac{1}{n\alpha}\sum_{i=1}^{n}x_i\]

Compute the Likelihood Ratio Test Statistic

Construct the likelihood ratio test (LRT) statistic by substituting \(\hat{\beta}\) and \(\beta_0\) into the log likelihood function and taking the difference:\[ \Lambda(X) = 2[\ell(\hat{\beta}|X) - \ell(\beta_0|X)] \]This \(\Lambda(X)\) follows a chi-square distribution with 1 degree of freedom, under null hypothesis. If the \(\Lambda(X)\) is larger than the tabulated chi-square critical value, we reject the null hypothesis \(H_0\).

Key concepts

Gamma Distribution

The Gamma distribution is a continuous probability distribution often used in statistical models where the data are skewed, typically to the right, and positive-valued. It's characterized by a shape parameter, denoted as \(\alpha\), and a scale parameter, \(\beta\). Understanding the behavior of the Gamma distribution is crucial for various fields, including insurance, finance, and biology because it's adept at modeling waiting times and can describe phenomena such as the amount of rainfall accumulated in a reservoir. For the rates and amount of events, knowing how to work with this distribution helps to predict likelihoods and manage expectations in real-life situations.

In the given exercise, the random sample is drawn from a \(\Gamma(\alpha, \beta)\)-distribution with a known shape parameter \(\alpha\) and an unknown scale parameter \(\beta\) that needs to be estimated. The distinctiveness of the Gamma distribution in this scenario allows us to conduct specific kinds of hypothesis tests, such as the likelihood ratio test, tailored for parameters like \(\beta\).

Hypothesis Testing

Hypothesis testing is a fundamental technique in statistics used to infer whether a certain belief about a population parameter is true based on sample data. It starts by stating two opposing hypotheses—the null hypothesis \(H_0\), representing the status quo, and the alternative hypothesis \(H_1\), signifying a claim that we want to test. The main goal is to determine which hypothesis is supported by the sample data. This process involves calculating a test statistic, which then gets compared to a critical value from a known distribution under the null hypothesis.

In our exercise, the null hypothesis is that the scale parameter \(\beta\) of the Gamma distribution equals a specified value \(\beta_0\), suggesting no effect or difference from the standard belief, while the alternative hypothesis indicates that \(\beta\) is different from \(\beta_0\), implying a deviation from the norm. The likelihood ratio test is the chosen method to address this statistical dubiety.

Maximum Likelihood Estimation

Maximum Likelihood Estimation (MLE) is a method used in statistics to estimate parameters of a model based on observed data. By maximizing the likelihood function, which measures how probable the observed data is for different parameter values, MLE provides a point estimate for the model parameters that would make the observed data most likely. This approach is particularly well-suited for complex models and large datasets.

During our exercise, we employ MLE to find the best estimate for the unknown scale parameter \(\beta\) of the Gamma distribution. We calculate the log-likelihood function of \(\beta\) given the random sample and differentiate it to acquire the maximum likelihood estimate \(\hat{\beta}\). By applying MLE, we are essentially looking for the value of \(\beta\) that makes our sample data the most probable, given the underlying assumptions of a Gamma distribution.

Statistical Inference

Statistical inference involves deducing properties of an underlying probability distribution by analyzing random samples. This is the linchpin for a plethora of statistical analysis, where conclusions about populations are derived from datasets. Central to statistical inference is the concept of drawing reliable conclusions about a population based on a finite amount of data, along with an assessment of the uncertainty inherent in those conclusions.

In statistical inference, parameters of the distribution, such as the mean or variance, can be estimated, and hypotheses about them can be tested. In our exercise, we use statistical inference to test a hypothesis about the scale parameter \(\beta\) of the Gamma distribution using the Likelihood Ratio Test. The process of determining the acceptability of our null hypothesis is contingent on how well the estimated parameter from our sample aligns with the theoretical distribution under \(H_0\). Thus, we combine both hypothesis testing and estimation—two main pillars of statistical inference—to decide whether the evidence from our data is strong enough to reject the null hypothesis.