Question

Let \(X_{1}, X_{2}, \ldots, X_{n}\) be a random sample from a distribution with pdf \(f(x ; \theta)=\theta x^{\theta-1}, 00 .\) Show the likelihood has mlr in the statistic \(\prod_{i=1}^{n} X_{i}\). Use this to determine the UMP test for \(H_{0}: \theta=\theta^{\prime}\) against \(H_{1}: \theta<\theta^{\prime}\), for fixed \(\theta^{\prime}>0\).

Short answer

The likelihood of the given pdf is \(L(\theta)= \theta^{n} \prod_{i=1}^{n}x_{i}^{\theta-1}\). The Monotone Likelihood Ratio in \(\prod_{i=1}^{n} X_{i}\) is established as \((\theta' / \theta)^n \prod_{i=1}^{n} x_{i}^{\theta'-\theta}\). Based on this, the UMP test rejects the null hypothesis \(H_0: \theta = \theta'\) if \(\prod_{i=1}^{n} x_{i} < k\), with \(k\) derived from the test's size condition.

Step-by-step solution

Computing Likelihood Function

As given, the pdf is \(f(x ; \theta)=\theta x^{\theta-1}, 0<x<1\) and zero elsewhere. So, we can define likelihood function \(L(\theta)\) as the joint probability function by \(L(\theta)= \prod_{i=1}^{n} f(x_i; \theta)= \theta^{n} \prod_{i=1}^{n}x_{i}^{\theta-1}\).

Showing Monotonic Likelihood Ratio

Next, we need to show that the likelihood ratio is a monotonically increasing function in \(\prod_{i=1}^{n} X_{i}\). The likelihood ratio is given by \( L(\theta') / L(\theta) = (\theta' / \theta)^n [\prod_{i=1}^{n} x_{i}^{\theta'-1}] / [\prod_{i=1}^{n} x_{i}^{\theta-1}] = (\theta' / \theta)^n \prod_{i=1}^{n} x_{i}^{\theta'-\theta}\). For any fixed \(x_1, x_2, ... , x_n\), \(L(\theta') / L(\theta)\) is a decreasing function of \(\theta\). Hence, for any \(0< s < t< 1\), \(L(s) <= L(t)\) gives \(s <= t\). So, the likelihood ratio has a monotonically increasing function in \(\prod_{i=1}^{n} x_{i}\).

Finding UMP Test

The null hypothesis is \(H_0: \theta = \theta'\)and alternative hypothesis is \(H_1: \theta < \theta'\). As we have already shown the likelihood ratio is a monotonically increasing function in \(\prod_{i=1}^{n} x_{i}\), applying Neyman-Pearson Lemma, we get that the uniform most powerful (UMP) test rejects \(H_0\) if \(\prod_{i=1}^{n} x_{i} < k\), where \(k\) is determined from the size condition of the test, \(E_{\theta'} (I(\prod_{i=1}^{n} X_{i} <k)) = \alpha\) where \(\alpha\) is the significance level.