[1] Neyman-Pearson Lemma
\star Consider a joint distribution, f(X_{1},X_{2},...,X_{n};\theta) is unknown. The f would be either f_{0} or f_{1} which will be specified.
\star H_{0}: f=f_{0} vs. H_{1}: f=f_{1}.
\star The level of the test, alpha is specifed.
1) \phi^*(X_{1},...,X_{n})= 1 if \frac {f_{1}(X_{1},...,X_{n})}{f_{0}f(X_{1},...,X_{n})}\geqslant k - outside the critical region!
2) \frac {f_{1}(X_{1},...,X_{n})}{f_{0}f(X_{1},...,X_{n})}< k - inside the critical region.
Example) University of Toronto STA355, Final 2013 Q. 1-(c) Suppose that X_{1}, X_{2},...X_{n} are independent exponential random variables with density f(x;\lambda)=\lambda\cdot\exp(-\lambda\cdot x) for x \geq 0, \lambda > 0
c) H_{0}: \lambda =1 vs. H_{1}: \lambda > 1 By using test statistic T=2\sum_{i=1}^{n}X_{i}. For an alpha level test, for what values of T would reject H_{0}?
\triangleright Solution??!!
[2] Uniformly Most Powerful
\star H_{0}: \lambda \leq \lambda_{0} vs. H_{1}: lambda > \lambda_{0} \star We reject the null hypothesis if \sum_{i=1}^n X_{i}=k' at level alpha.
\star Remark) The UMP test does not exist for H_{0}: \lambda=\lambda_{0} vs. H_{1}: \lambda \neq \lambda_{0}. If there are two rejection regions ( when \lambda_{1} > \lambda_{0} or \lambda_{1} < \lambda_{0}) , then this is not the UMP test case.
[3] Likelihood Ratio Tests
\star H_{0}:\theta \in \Theta_{0} vs.H_{1}:\theta \in \Theta_{1}, where \Theta_{0}+\Theta_{1}=\Theta
\star LR Test \Lambda = \frac{L(\theta \in\Theta)}{L(\theta\in\Theta_{0})}. Reject H_{0} for large values of \Lambda
\star We need to know the distribution of lambda when H_{0} is true.
\Rightarrow We can use a T as our test statistics.
\Rightarrow We can approximate the distirbution of \lambda by another distribution which we may know.
Example) Mathematical Statistics by Keith Knight, Chapter 7-19
Let X_{1}, X_{2},...X_{n} be iid N(\mu,\sigma^2). Both are unknown. We want to test H_{0}: \mu=\mu_{0} vs. H_{1}: \mu \neq \mu_{0}.
\triangleright Solution.
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