| course |
course_year |
question_number |
tags |
title |
year |
Statistics |
IB |
76 |
|
Paper 1, Section II, H |
2009 |
What is the critical region $C$ of a test of the null hypothesis $H_{0}: \theta \in \Theta_{0}$ against the alternative $H_{1}: \theta \in \Theta_{1}$ ? What is the size of a test with critical region $C ?$ What is the power function of a test with critical region $C$ ?
State and prove the Neyman-Pearson Lemma.
If $X_{1}, \ldots, X_{n}$ are independent with $\operatorname{common} \operatorname{Exp}(\lambda)$ distribution, and $0<\lambda_{0}<\lambda_{1}$, find the form of the most powerful size- $\alpha$ test of $H_{0}: \lambda=\lambda_{0}$ against $H_{1}: \lambda=\lambda_{1}$. Find the power function as explicitly as you can, and prove that it is increasing in $\lambda$. Deduce that the test you have constructed is a size- $\alpha$ test of $H_{0}: \lambda \leqslant \lambda_{0}$ against $H_{1}: \lambda=\lambda_{1}$.