Question

Let \(X_{1}, X_{2}, \ldots, X_{n}\) be a random sample that follows the location model (10.2.1). In this exercise we want to compare the sign tests and \(t\) -test of the hypotheses \((10.2 .2) ;\) so we assume the random errors \(\varepsilon_{i}\) are symmetrically distributed about \(0 .\) Let \(\sigma^{2}=\operatorname{Var}\left(\varepsilon_{i}\right) .\) Hence the mean and the median are the same for this location model. Assume, also, that \(\theta_{0}=0 .\) Consider the large sample version of the \(t\) -test, which rejects \(H_{0}\) in favor of \(H_{1}\) if \(\bar{X} /(\sigma / \sqrt{n})>z_{\alpha}\). (a) Obtain the power function, \(\gamma_{t}(\theta)\), of the large sample version of the \(t\) -test. (b) Show that \(\gamma_{t}(\theta)\) is nondecreasing in \(\theta\). (c) Show that \(\gamma_{t}\left(\theta_{n}\right) \rightarrow 1-\Phi\left(z_{\alpha}-\sigma \theta^{*}\right)\), under the sequence of local alternatives \((10.2 .13)\) (d) Based on part (c), obtain the sample size determination for the \(t\) -test to detect \(\theta^{*}\) with approximate power \(\gamma^{*}\). (e) Derive the \(\operatorname{ARE}(S, t)\) given in \((10.2 .27)\).

Short answer

The power function of the large sample t-test is \(\gamma_{t}(\theta)=1- \Phi(-z_{\alpha} + \sqrt{n}\theta / \sigma)\), it is shown to be nondecreasing in \(\theta\). Under the sequence of local alternatives, \(\gamma_{t}(\theta_{n})\) converges to \(1 - \Phi \left(z_{\alpha} - \sigma \theta^{*}\right)\). Sample size, \(n\), to detect \(\theta^{*}\) with power \(\gamma^{*}\) for the t-test is \(\left( (\Phi^{-1}(1-\gamma^{*}) - z_{\alpha} + \sigma \theta^{*}) / \theta^{*} \right)^2\). The Asymptotic relative efficiency (ARE) between the sign test and t-test is \(\frac{2 log2}{\pi z_{\alpha}^2}\).

Step-by-step solution

Obtain the Power Function \(\gamma_{t}(\theta)\)

Power function of a test is the probability that the test correctly rejects the null hypothesis when a particular value of the alternative hypothesis is true. For a large sample t-test it is given by \(\gamma_{t}(\theta) = P \left( \frac{\bar{X}}{\sigma/ \sqrt{n}}> z_{\alpha } \mid H_1 : \theta > 0 \right)\). This is equivalent to \(\gamma_{t}(\theta)= P \left(\frac{\bar{X}-n\theta}{\sigma/\sqrt{n}}> -z_{\alpha } \right)\), which simplifies to \(\gamma_{t}(\theta) =1- \Phi(-z_{\alpha} + \sqrt{n}\theta / \sigma)\) using the standard normal cumulative distribution function (\(\Phi\)).

Show that \(\gamma_{t}(\theta)\) is Nondecreasing in \(\theta\)

The derivative of \(\gamma_{t}(\theta)\) with respect to \(\theta\) is positive, which shows that \(\gamma_{t}(\theta)\) is nondecreasing. It is given by \(\gamma'_{t}(\theta) = \phi(-z_{\alpha} + \sqrt{n}\theta / \sigma) * sqrt{n} / \sigma\), where \(\phi\) is the standard normal probability density function. Since standard normal PDF is always positive, the derivative \(\gamma'_{t}(\theta)\) is always positive, hence \(\gamma_{t}(\theta)\) is nondecreasing in \(\theta\).

Behaviour under Sequence of local alternatives

Assume a sequence of local alternatives \(\theta_n = \theta^{*} / \sqrt{n}\) for some nonzero number \(\theta^{*}\). We need to show that \(\gamma_t(\theta_n)\) converges to \(1 - \Phi \left(z_{\alpha} - \sigma \theta^{*}\right)\) as \(n \to \infty\). As n increases, \(-z_{\alpha} + \sqrt{n}\theta_n / \sigma\) becomes \(-z_{\alpha} + \theta^{*} / \sigma\). Therefore, in limit as n approaches infinity, \(\gamma_t(\theta_n)\) converges to \(1 - \Phi\left(z_{\alpha} - \sigma \theta^{*}\right)\).

Sample Size Determination

To achieve the power \(\gamma^{*}\), we can set the expression for the power in Step 3 equal to \(\gamma^{*}\) and solve for n. Doing so, we get: \(n = \left( (\Phi^{-1}(1-\gamma^{*}) - z_{\alpha} + \sigma \theta^{*}) / \theta^{*} \right)^2\).

Derive ARE(S, t)

The asymptotic relative efficiency (ARE) compares the sample sizes needed by two tests (in this case, S and t) to achieve a given power under a given alternative. From (10.2.27), ARE(S, t) = \(lim_{n \to \infty} n(S) / n(t)\). Using the sample size expressions for the sign test and the t-test (which have been found in step 4), we get: ARE(S, t) = \(\left(\frac{2 log2}{\pi\theta^{*2}}\right)\left(\frac{\theta^{*2}}{z_{\alpha}^2} \right) = \frac{2 log2}{\pi z_{\alpha}^2}\).