Question

Consider a random sample \(X_{1}, X_{2}, \ldots, X_{n}\) from a distribution with pdf \(f(x ; \theta)=\theta(1-x)^{\theta-1}, 00 .\) (a) Find the form of the uniformly most powerful test of \(H_{0}: \theta=1\) against \(H_{1}: \theta>1\) (b) What is the likelihood ratio \(\Lambda\) for testing \(H_{0}: \theta=1\) against \(H_{1}: \theta \neq 1 ?\)

Short answer

The form of the uniformly most powerful test of \(H_0 : \theta = 1\) against \(H_1 : \theta > 1\) is \(C = \{ x | x < c \}\) where c is a constant. The likelihood ratio \(\Lambda\) for testing \(H_0 : \theta = 1\) against \(H_1 : \theta \neq 1\) is \(\Lambda = \frac{1}{\theta(1 - x)^{\theta - 1}}\)

Step-by-step solution

Determine UMP Test

The first part of this problem asks for the form of the uniformly most powerful (UMP) test for hypothesis \(H_{0}: \theta = 1\) against \(H_{1}: \theta > 1\). We can utilize the Neyman-Pearson Lemma to obtain the solution. For a given significance level \(\alpha\), this Lemma tells us that the most powerful test for \(H_{0}\) against \(H_{1}\) rejects \(H_{0}\) for large values of the likelihood ratio \(L = \frac{f(x;\theta_1)}{f(x; \theta_0)}\). Using the given probability density function (pdf), for \(x > 0\) and \(\theta > 1 \), the likelihood ratio becomes: \(L(x) = \frac{\theta_1(1-x)^{\theta_1 -1}}{1*(1-x)^{1-1}} = \theta_1(1-x)^{\theta_1 -1}\). Under \(H_0\), where \(\theta_1>1\), and \(\theta_0=1\), the test statistic is rejected for large values of L which is equivalent of rejecting \(H_0\) for small values of x, given the form of \(L(x)\). Therefore, the uniform most powerful test has the critical region \(C = \{ x | x < c \}\) to some constant c to be determined using power and type-I error analysis.

Likelihood Ratio

The second part of our problem demands the expression for the likelihood ratio \(\Lambda\) for testing \(H_{0}: \theta = 1\) against \(H_{1}: \theta \neq 1\). The likelihood ratio \(\Lambda\) is given by the formula \(\Lambda = \frac{sup_{\theta \in \Theta_0} L(\theta; x)}{sup_{\theta \in \Theta} L(\theta; x)}\), where \(L(\theta; x)\) is our likelihood function, \(\Theta_0\) is the parameter space under \(H_0\) and \(\Theta\) is the total parameter space. In our case, this becomes: \(\Lambda = \frac{max_{\theta=1} L(\theta; x)}{max_{\theta} L(\theta; x)} = \frac{(1)(1 - x)^0}{\theta(1 - x)^{\theta - 1}}\).

Key concepts

Likelihood Ratio

The likelihood ratio allows for a comparison of two hypotheses by measuring how likely it is that the observed data was generated by one model as opposed to the other. In the context of hypothesis testing, the likelihood ratio \(\Lambda\) is a statistical measure used to test the strength of the evidence against the null hypothesis \(H_{0}\) in favor of an alternative hypothesis \(H_{1}\).

For the given exercise, where we have a probability density function \(f(x ; \theta)=\theta(1-x)^{\theta-1}\), the likelihood ratio would compare the probability of the observed data under \(H_{0}: \theta=1\) and any alternative value of \(\theta\) in \(H_{1}: \theta eq 1\). It is important to note that a low value of \(\Lambda\) suggests that the null hypothesis is less likely to produce the observed data compared to the alternative, leading to stronger evidence against \(H_{0}\). Moreover, for testing \(H_{0}: \theta = 1\) against \(H_{1}: \theta eq 1\), the likelihood ratio \(\Lambda\) is determined by taking the supremum (maximum) of the likelihood function over the parameter spaces for \(H_{0}\) and \(H_{1}\) respectively.

To calculate the value of \(\Lambda\), the probability density function associated with the null hypothesis and the alternative is employed. By understanding and applying the likelihood ratio, students can better approach hypothesis testing and the decision-making process in statistical analyses.

Neyman-Pearson Lemma

The Neyman-Pearson Lemma is the cornerstone of hypothesis testing in statistics, providing a method for creating the most powerful test for a given level of significance \(\alpha\). A 'most powerful test' means that among all possible tests which could be used to reject the null hypothesis \(H_{0}\), the one obtained via this lemma has the highest probability of correctly detecting an actual effect (i.e., rejecting the null hypothesis when there is indeed an alternative situation).

When we apply the Neyman-Pearson Lemma to the exercise's context, we identify the likelihood ratio as the test statistic. In other words, it forms the basis of deciding whether to reject the null hypothesis in favor of the alternative based on the observed data. Under the conditions of our exercise, the lemma suggests that for a simple hypothesis \(H_{0}\) against a simple alternative \(H_{1}\), the best test rejects \(H_{0}\) for the highest values of the likelihood ratio. The likelihood ratio here is calculated by dividing the probability density functions of the observed data under each hypothesis.

Moreover, the lemma specifies that the test's rejection region (where the null hypothesis is rejected) is determined by setting a threshold, which is typically informed by the desired level of significance \(\alpha\). The ability to define the most powerful test underpins the importance of the Neyman-Pearson Lemma in statistical inference, providing a clear and structured approach to hypothesis testing.

Probability Density Function (PDF)

The probability density function (PDF), denoted \(f(x; \theta)\), describes the likelihood of a continuous random variable taking on a certain value. The PDF is fundamental in defining the distribution of random variables and is integral to understanding probability distributions in statistics.

In the given exercise, the PDF \(f(x ; \theta)=\theta(1-x)^{\theta-1}\) specifies how the probability of the random sample \(X_{1}, X_{2}, \ldots, X_{n}\) is distributed over the interval \(00\). Here, the PDF is zero elsewhere, implying that the random variable only takes on values within \(0
The PDF plays a crucial role in statistical analysis as it is used to compute probabilities and expected values. When conducting hypothesis tests, as in our exercise scenario, the PDF is utilized to calculate the likelihood ratio, which in turn is used to determine whether to accept or reject the null hypothesis. Understanding the PDF and how to derive it enables students to tackle a wide range of statistical problems that involve continuous random variables.