Question

Let \(X_{1}, \ldots, X_{n}\) denote a random sample from a gamma-type distribution with \(\alpha=2\) and \(\beta=\theta\). Let \(H_{0}: \theta=1\) and \(H_{1}: \theta>1\). (a) Show that there exists a uniformly most powerful test for \(H_{0}\) against \(H_{1}\), determine the statistic \(Y\) upon which the test may be based, and indicate the nature of the best critical region. (b) Find the pdf of the statistic \(Y\) in part (a). If we want a significance level of \(0.05\), write an equation that can be used to determine the critical region. Let \(\gamma(\theta), \theta \geq 1\), be the power function of the test. Express the power function as an integral.

Short answer

The Uniformly Most Powerful test for hypothesis does exist. The test statistic is defined as the sum of the sample \(Y = \sum X_{i}\). Identifying the critical region is complicated as it involves setting up and solving an integral equation, meaning there's no general solution for this test. The power function is expressed as an integral which computes the probability of correctly rejecting \(H_{0}\) when \(\theta > 1\).

Step-by-step solution

Show the existence of a UMP test

For testing \(H_{0}: \theta=1\) against \(H_{1}: \theta>1\) for a gamma distribution, and by Neyman-Pearson lemma, a uniformly most powerful (UMP) test does indeed exist.

Identify the test statistic

The test statistic can be identified as the sum of the sample, i.e., \(Y = \sum X_{i}\). This comes from the sum of iid gamma random variables which also follows a gamma distribution.

Determine the critical region

The critical region is the set of all values of \(Y\) for which we reject \(H_{0}\). Since we're testing \(H_{0}: \theta=1\) against \(H_{1}: \theta>1\) and our test statistic \(Y = \sum X_{i}\) is a sum of gamma-distributed random variables, the best or most powerful critical region will be of the format \(Y \geq k\), where \(k\) is a certain threshold that depends on the significance level of the test.

Find the pdf of Y

The pdf of the sum of gamma distributed random variables follows a gamma distribution. Hence, if we are summing over \(n\) such variables, the gamma distribution parameters will be \(\alpha = 2n\) and \(\beta = 1\) under \(H_{0}\). Therefore, the pdf of \(Y\) is defined as \(f_{Y}(y) = \frac{y^{2n-1} e^{-y}}{(2n-1)!}\) for \(y \geq 0\).

Write the equation for determining the critical region

To find the critical region for a significance level of \(0.05\), we need to solve the following equation for \(k\): \(P_{\theta=1}(Y \geq k) = 0.05\). This implies \(\int_{k}^{\infty} \frac{y^{2n-1} e^{-y}}{(2n-1)!} dy = 0.05\). Solving this would give us \(k\), the threshold for the significance level.

Express the power function as an integral

The power function \(\gamma(\theta)\) for \(\theta \geq 1\) is given by the probability of correctly rejecting \(H_{0}\) when \(\theta > 1\). This is \(\gamma(\theta) = P_{\theta}(Y \geq k) = \int_{k}^{\infty} \frac{\theta^{2n} y^{2n-1} e^{-\theta y}}{(2n-1)!} dy\), where the density function under \(H_{1}\) is a gamma distribution with parameters \(\alpha = 2n\) and \(\beta = \theta\).