Hypothesis Testing - N-P Lemma, UMP, LR Test

Reference: Mathematical Statistics by Keith Knight, Chapter 7

[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.