Chapter 7

The incidence of a rare disease seems to be increasing. In successive years the numbers of new cases have been \(y_{1}, \ldots, y_{n}\). These may be assumed to be independent observations from Poisson distributions with means \(\lambda \theta, \ldots, \lambda \theta^{n}\). Show that there is a family of tests each of which, for any given value of \(\lambda\), is a uniformly most powerful test of its size for testing \(\theta=1\) against \(\theta>1\).

A random sample \(Y_{1}, \ldots, Y_{n}\) is available from the Type I Pareto distribution $$ F(y ; \psi)= \begin{cases}1-y^{-\psi}, & y \geq 1 \\ 0, & y<1\end{cases} $$ Find the likelihood ratio statistic to test that \(\psi=\psi_{0}\) against \(\psi=\psi_{1}\), where \(\psi_{0}, \psi_{1}\) are known, and show how to calculate a P-value when \(\psi_{0}>\psi_{1}\). How does your answer change if the distribution is $$ F(y ; \psi, \lambda)= \begin{cases}1-(y / \lambda)^{-\psi}, & y \geq \lambda \\\ 0, & y<\lambda\end{cases} $$ with \(\lambda>0\) unspecified?

Independent random samples \(Y_{i 1}, \ldots, Y_{i n_{i}}\), where \(n_{i} \geq 2\), are drawn from each of \(k\) normal distributions with means \(\mu_{1}, \ldots, \mu_{k}\) and common unknown variance \(\sigma^{2}\). Derive the likelihood ratio statistic \(W_{\mathrm{p}}\) for the null hypothesis that the \(\mu_{i}\) all equal an unknown \(\mu\), and show that it is a monotone function of $$ R=\frac{\sum_{i=1}^{k} n_{i}\left(\bar{Y}_{i \cdot}-\bar{Y}_{. .}\right)^{2}}{\sum_{i=1}^{k} \sum_{j=1}^{n_{i}}\left(Y_{i j}-\bar{Y}_{i}\right)^{2}} $$ where \(\bar{Y}_{i}=n_{i}^{-1} \sum_{j} Y_{i j}\) and \(\bar{Y}_{. .}=\left(\sum n_{i}\right)^{-1} \sum_{i, j} Y_{i j}\). What is the null distribution of \(R ?\)

Let \(X_{1}, \ldots, X_{m}\) and \(Y_{1}, \ldots, Y_{n}\) be independent random samples from continuous distributions \(F_{X}\) and \(F_{Y}\). We wish to test the hypothesis \(H_{0}\) that \(F_{X}=F_{Y}\). Define indicator variables \(I_{i j}=I\left(X_{i}

Below are diastolic blood pressures \((\mathrm{mm} \mathrm{Hg})\) of ten patients before and after treatment for high blood pressure. Test the hypothesis that the treatment has no effect on blood pressure using a Wilcoxon signed-rank test, (a) using the exact significance level and (b) using a normal approximation. Discuss briefly. \(\begin{array}{llrrrrrrrrr}\text { Before } & 94 & 105 & 101 & 106 & 118 & 107 & 96 & 102 & 114 & 95 \\ \text { After } & 96 & 96 & 95 & 103 & 105 & 111 & 86 & 90 & 107 & 84\end{array}\)

A source at location \(x=0\) pollutes the environment. Are cases of a rare disease \(\mathcal{D}\) later observed at positions \(x_{1}, \ldots, x_{n}\) linked to the source? Cases of another rare disease \(\mathcal{D}^{\prime}\) known to be unrelated to the pollutant but with the same susceptible population as \(\mathcal{D}\) are observed at \(x_{1}^{\prime}, \ldots, x_{m}^{\prime} .\) If the probabilities of contracting \(\mathcal{D}\) and \(\mathcal{D}^{\prime}\) are respectively \(\psi(x)\) and \(\psi^{\prime}\), and the population of susceptible individuals has density \(\lambda(x)\), show that the probability of \(\mathcal{D}\) at \(x\), given that \(\mathcal{D}\) or \(\mathcal{D}^{\prime}\) occurs there, is $$ \pi(x)=\frac{\psi(x) \lambda(x)}{\psi(x) \lambda(x)+\psi^{\prime} \lambda(x)} $$ Deduce that the probability of the observed configuration of diseased persons, conditional on their positions, is $$ \prod_{j=1}^{n} \pi\left(x_{j}\right) \prod_{i=1}^{m}\left\\{1-\pi\left(x_{i}^{\prime}\right)\right\\} $$ The null hypothesis that \(\mathcal{D}\) is unrelated to the pollutant asserts that \(\psi(x)\) is independent of \(x\). Show that in this case the unknown parameters may be eliminated by conditioning on having observed \(n\) cases of \(\mathcal{D}\) out of a total \(n+m\) cases. Deduce that the null probability of the observed pattern is \(\left({ }_{n}^{n+m}\right)^{-1}\). If \(T\) is a statistic designed to detect decline of \(\psi(x)\) with \(x\), explain how permutation of case labels \(\mathcal{D}, \mathcal{D}^{\prime}\) may be used to obtain a significance level \(p_{\text {obs }}\). Such a test is typically only conducted after a suspicious pattern of cases of \(\mathcal{D}\) has been observed. How will this influence \(p_{\text {obs }}\) ?