Question

If \(X_{1}, X_{2}, \ldots, X_{n}\) is a random sample from a beta distribution with parameters \(\alpha=\beta=\theta>0\), find a best critical region for testing \(H_{0}: \theta=1\) against \(H_{1}: \theta=2\)

Short answer

The best critical region \(C = \{ x: \lambda \le k \}\) is determined by calculating the likelihood ratio and setting up criteria based on the level of significance \(\alpha\). Here, \(k\) is a constant determined to ensure that the size of the critical region is equal to the given level of significance \(\alpha\).

Step-by-step solution

Definition of Beta Distribution

The probability density function of the Beta distribution is given by: \[f(x;\alpha,\beta) = \frac{x^{\alpha-1}(1-x)^{\beta-1}}{B(\alpha,\beta)}\] for \(0 \le x \le 1\) and \(\alpha > 0\), \(\beta > 0\). Where \(B(\alpha,\beta)\) stands for the beta function.

Calculate the Likelihood ratio

The likelihood ratio for two simple hypotheses \(H_{0}: \theta=\theta_{0}\) and \(H_{1}: \theta=\theta_{1}\) is computed as: \[ \lambda = \frac{L(\theta_{0})}{L(\theta_{1})}\] where \(L(\theta)\) is the likelihood function. In this case, \(\theta_{0} = 1\), \(\theta_{1} = 2\), hence \[ \lambda = \frac{L(1)}{L(2)} \] Calculate the likelihood ratios by replacing all \(\theta\) with 1 in \(H_{0}\) and 2 in \(H_{1}\) in the likelihood function derived from the Beta distribution.

Define the Best Critical Region

Define the best critical region (rejection region) as: \[C = \{ x: \lambda \le k \}\] for some constant \(k > 0\). In this step, find the value of \(k\) such that the size of the critical region is equal to the given level of significance \(\alpha\). Typically, \(\alpha\) is set to be 0.05.