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 involves finding the maximum of the log-likelihood function under both the null and alternative hypothesis, then comparing the difference with a chi-square distribution. The decision to accept or reject the null hypothesis is made depending on whether the test statistic equals or exceeds the critical value.

Step-by-step solution

Formulate the Likelihood Function

To begin with, we need to understand what the likelihood function for the given distribution is. The Gamma distribution has the probability density function as: \( f(x ; \alpha, \beta) = \frac{1}{\Gamma(\alpha)\beta^{\alpha}} x^{\alpha -1} e^{-x/\beta} \). So, the likelihood function given a random sample \(X_{1}, X_{2}, \ldots, X_{n}\) is : \( L(\beta; x) = \prod_{i=1}^{n} f(x_{i}; \alpha, \beta) \).

Compute the Log-Likelihood Function

Next, we compute the log-likelihood to simplify our computations. Given the log-likelihood function \(\ell(\beta ; x) = log(L(\beta ; x))\), under the assumption that the sample observations are independent, we have: \(\ell(\beta ; x) = \sum_{i=1}^{n} log(f(x_{i}; \alpha, \beta)) \).

Compute the Maximum Log-Likelihood

We want to compute the maximum of the log-likelihood function under both the null and alternative hypotheses. That is, we first impose the restriction \(H_{0}: \beta = \beta_{0}\) to determine the maximum log-likelihood under the null hypothesis:

Construct the Likelihood Ratio Test Statistic

The next step is to construct the likelihood ratio test statistic. It is given by the following expression: \(LR = 2 * [\ell(\hat{\beta}, x) − \ell(\beta_{0}, x)]\)

Make a Decision

Finally, the decision is made based on the calculated likelihood ratio. If the test statistic equals or exceeds the critical value (found from the Chi-Square distribution table with appropriate degrees of freedom), we reject the null hypothesis, otherwise we fail to reject it.