Question

Let \(X_{1}, X_{2}, \ldots, X_{n}\) be a random sample from a distribution with pdf \(f(x ; \theta)=\) \(\theta x^{\theta-1}, 0

Short answer

a) The complete and sufficient statistic for \(\theta\) is \(T(X)=\prod_{i=1}^{n}X_{i}\). b) The test is performed by comparing the cumulative likelihood ratios for each observation to the boundaries given by \(\alpha_{a}\) and \(\beta_{a}\).

Step-by-step solution

Find the Joint Distribution

Based on the given probability density function, let's find the joint pdf of \(X_{1}, X_{2}, \ldots, X_{n}\), which is \(f(x_{1}, x_{2}, \ldots, x_{n}; \theta)=\theta^{n} (x_{1}x_{2}\ldots x_{n})^{\theta-1}\) for \(0<x_{i}<1\), and zero elsewhere.

Determine the Sufficient Statistic

The likelihood function given by the joint pdf is a function of \(\theta\) and \(\prod_{i=1}^{n}x_{i}\). According to the factorization theorem, the statistic \(T(X)=\prod_{i=1}^{n}X_{i}\) is a sufficient statistic for \(\theta\). Then, we use the completeness property of the exponential family and observe that our **pdf** belongs to the exponential family. Therefore, \(T(X)\) is also complete.

Sequential Probability Ratio Test

The likelihood ratio for a single observation x under \(H_{0}: \theta=2\) and \(H_{1}: \theta=3\) will be \(\Lambda(x) = \frac{3x^{2}}{2x}\). Then, we perform the sequential probability ration test for all the \(n\) observations, which involves comparing the ratios of the cumulative likelihoods to the given boundaries \(\alpha_{a}\) and \(\beta_{a}\), and make our decision.