Question

Let \(X_{1}, X_{2}, \ldots, X_{10}\) be a random sample of size 10 from a Poisson distribution with parameter \(\theta\). Let \(L(\theta)\) be the joint pdf of \(X_{1}, X_{2}, \ldots, X_{10}\). The problem is to test \(H_{0}: \theta=\frac{1}{2}\) against \(H_{1}: \theta=1\). (a) Show that \(L\left(\frac{1}{2}\right) / L(1) \leq k\) is equivalent to \(y=\sum_{1}^{n} x_{i} \geq c\). (b) In order to make \(\alpha=0.05\), show that \(H_{0}\) is rejected if \(y>9\) and, if \(y=9\), reject \(H_{0}\) with probability \(\frac{1}{2}\) (using some auxiliary random experiment). (c) If the loss function is such that \(\mathcal{L}\left(\frac{1}{2}, \frac{1}{2}\right)=\mathcal{L}(1,1)=0\) and \(\mathcal{L}\left(\frac{1}{2}, 1\right)=1\) and \(\mathcal{L}\left(1, \frac{1}{2}\right)=2\), show that the minimax procedure is to reject \(H_{0}\) if \(y>6\) and, if \(y=6\), reject \(H_{0}\) with probability \(0.08\) (using some auxiliary random experiment).

Short answer

The exercise involves hypothesis testing for Poisson distributions. We reject \(H_{0}\) when \(y > 9\) or with a 0.5 probability when \(y = 9\) for a significance level of 0.05. Under a given loss function, we reject \(H_{0}\) when \(y > 6\) or with a probability of 0.08 when \(y = 6\).

Step-by-step solution

Formulate the hypotheses

The null hypothesis \(H_{0}\) is \(\theta=\frac{1}{2}\). The alternative hypothesis \(H_{1}\) is \(\theta=1\).

Define the likelihood function

We have \(X_{1}, X_{2}, \ldots, X_{10}\) as a random sample from a Poisson distribution with parameter \(\theta\). The likelihood function for a sample of size n from a Poisson distribution is \(L(\theta) = Π_{i=1}^{10} e^{-\theta} \theta^{x_i} / x_i!\), which represents the probability of getting our observed data given that \(\theta\), our hypothesis, is true.

Show \(L\left(\frac{1}{2}\right) / L(1) \leq k\) is equivalent to \(y=\sum_{1}^{n} x_{i} \geq c\)

Divide the likelihood under \(H_{0}\) by the likelihood under \(H_{1}\), then taking the natural logarithm. If the log-likelihood ratio is less than some constant \(k\), we can reject the null hypothesis or equivalently if \(y=\sum_{1}^{10} x_{i} \geq c\).

Find the rules for rejecting \(H_{0}\) at \(\alpha=0.05\)

Using properties of the Poisson distribution, we determine the critical value \(y = 9\). This means we reject the null hypothesis if our test statistic \(y = \sum_{1}^{10} x_{i}\) is greater than 9 or, if \(y = 9\), reject \(H_{0}\) with a probability of \(\frac{1}{2}\). Use the cumulative distribution function (CDF) of Poisson distribution to find the probabilities.

Determine the minimax procedure with a loss function

The loss function \(\mathcal{L}\) is given in the question. The minimax rule minimizes the maximum loss. Thus, rejecting \(H_{0}\) is preferred if \(y > 6\). If \(y = 6\), reject \(H_{0}\) with a probability of 0.08. This is determined by considering probabilities that minimize the maximum of the total risk.