Question

Let \(X\) have the pmf \(f(x ; \theta)=\theta^{x}(1-\theta)^{1-x}, x=0,1\), zero elsewhere. We test the simple hypothesis \(H_{0}: \theta=\frac{1}{4}\) against the alternative composite hypothesis \(H_{1}: \theta<\frac{1}{4}\) by taking a random sample of size 10 and rejecting \(H_{0}: \theta=\frac{1}{4}\) if and only if the observed values \(x_{1}, x_{2}, \ldots, x_{10}\) of the sample observations are such that \(\sum_{1}^{10} x_{i} \leq 1\). Find the power function \(\gamma(\theta), 0<\theta \leq \frac{1}{4}\), of this test.

Short answer

The power function of the test is \(\gamma(\theta) = (1 - \theta)^{10} + 10 * \theta * (1 - \theta)^{9}\). Knowing the power function allows us to evaluate how powerful our test is for different true values of \(\theta\) in the alternative hypothesis region.

Step-by-step solution

Specify the rejection condition

For this test, \(H_{0}: \theta=\frac{1}{4}\) is rejected if and only if the observed values \(x_{1}, x_{2}, \ldots, x_{10}\) of the sample observations are such that \(\sum_{1}^{10} x_{i} \leq 1\). This can be interpreted as having 1 or 0 success in 10 trials.

Calculate the probability under null hypothesis

Under the hypothesis \(H_{0}: \theta=\frac{1}{4}\), the probability of getting 0 or 1 success in 10 trials is calculated using the binomial distribution formula, which is reduced to: \(P_{H_{0}} = (1 - \frac{1}{4})^{10} + 10 * (\frac{1}{4})*(1 - \frac{1}{4})^{9}\)

Calculate the power function

The power function \(\gamma(\theta)\) is the probability to reject \(H_{0}\) when a particular \(\theta\) under \(H_{1}\) is true. If we let \(p_{H_{0}}\) be the probability calculated in Step 2, and using the binomial distribution formula under a general \(\theta<\frac{1}{4}\), we get \(\gamma(\theta) = (1 - \theta)^{10} + 10 * \theta * (1 - \theta)^{9}\)