Question

Let \(X_{1}, X_{2}, \ldots, X_{n}\) be a random sample from the beta distribution with \(\alpha=\beta=\theta\) and \(\Omega=\\{\theta: \theta=1,2\\}\). Show that the likelihood ratio test statistic \(\Lambda\) for testing \(H_{0}: \theta=1\) versus \(H_{1}: \theta=2\) is a function of the statistic \(W=\) \(\sum_{i=1}^{n} \log X_{i}+\sum_{i=1}^{n} \log \left(1-X_{i}\right)\).

Short answer

-2 log-likelihood ratio or the Likelihood Ratio Test Statistic \(\Lambda\), for testing \(H_0: \theta=1\) versus \(H_1: \theta=2\), is a function of the statistic \(W = \sum_{i=1}^{n} [\log(x_i) + \log(1 - x_i)]\).

Step-by-step solution

Understanding the Hypotheses

The null hypothesis is \(H_{0}: \theta=1\) and the alternative hypothesis is \(H_{1}: \theta=2\). These hypotheses indicate that we will be testing whether \( \theta \) is 1 or 2.

Understanding the Beta distribution

The Beta distribution is given as \(f(x|\alpha, \beta) = \frac{x^{\alpha - 1}(1 - x)^{\beta - 1}}{B(\alpha, \beta)}\), where \( B(\alpha, \beta) \) is the Beta function and \( \alpha \) and \( \beta \) are the shape parameters.

Derive the likelihood function

For beta distribution, the likelihood function is the product of all individual probability density functions: \(L(\theta) = \prod_{i=1}^{n}f(x_i|\theta, \theta) = \frac{1}{(B(\theta, \theta))^n}\prod_{i=1}^{n}x_i^{\theta - 1}(1 - x_i)^{\theta - 1}\).

Computing the Log-Likelihood Functions

The log-likelihood function for \( H_0 \) and \( H_1 \) are given by \(\log(L(1)) = n(\log(B(1,1)) + \sum_{i=1}^{n} (1 - 1)\log x_i + (1 - 1)\log(1 - x_i))\) and \( \log(L(2)) = n(\log(B(2,2)) + \sum_{i=1}^{n} (2 - 1)\log(x_i) + (2 - 1)\log (1 - x_i) ),\) respectively.

Deriving the Likelihood Ratio Test Statistic

-2 log-likelihood ratio can be formulated as \( -2\log\Lambda = -2[\log(L(1)) - \log(L(2))] = -2[\log(B(1,1)) - \log(B(2,2)) + \sum_{i=1}^{n} (\log(x_i) + \log(1 - x_i))] \) which is a function of the statistic W.