Example) about MLE, Pivot, N-P Lemma

Example) University of Toronto STA355 2013 Final Test Q1.

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 $x \geq 0$, $\lambda > 0$ 

a) Find the MLE of $\lambda$, and find the limiting distribution of $\sqrt{n}(\hat{\lambda_{n}}-\lambda)$?

b) A pivot for $\lambda$ is $2\lambda \sum_{i=1}^{n}X_{i}\sim \chi^2_{2n}$.
Show how you can use this pivot to construct a CI for $\lambda$. 

c) $H_{0}: \lambda =1$ vs. $H_{1}: \lambda > 1$ Suing test statistic $T=2\sum_{i=1}^{n}X_{i}$.
For an alpha level test, for what values of T would reject $H_{0}$?
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Solution 
a) * First the MLE of $\lambda$,
  Likelihood, $L(\lambda)=\prod_{i=1}^{n} f(x_{i};\lambda)= \lambda^n \cdot \exp(-\lambda \cdot \sum_{i=1}^{n}x_{i})$
  Log likelihood, $l(\lambda)= n \log \lambda - \lambda \sum_{i=1}^{n}x_{i}$
  Taking a derivative w.r.t. $\lambda$, $l'(\lambda)= \frac{n}{\lambda} - \sum_{i=1}^{n}=0$
  Therefore, $\hat{\lambda_{n}}= \frac{1}{\bar{X}}$ 
  * Limiting distribution of $\sqrt{n}(\hat{\lambda_{n}}-\lambda)$
  $I(\lambda)=Var(\frac{d}{d\lambda}\log f(x_{i}; \lambda))=Var(\frac{1}{n}-X_{1}) =Var(X_{1})=\frac{1}{\lambda^2}$ 

b) In this question, follow three steps;
1) Find a pivot statistic ($T=h(\theta; X_{i})$), From the question, we know $2\lambda \sum_{i=1}^{n}X_{i}\sim \chi^2_{2n}$.
2) Pick a & b such that $P(a$P(a< \chi_{2n}^2
3) And rearrange the 2nd step w. r. t. a parameter given a distribution from the question. 
  $P(\frac{a}{2\sum X_{i}} < \lambda <\frac{b}{2\sum X_{i}})= 1-2\alpha$%. Therefore, CI for $\lambda$ : $(\frac{a}{2\sum X_{i}}, \frac{b}{2\sum X_{i}})$

c) We need to use the N-P lemma! 
 Under the null hypothesis, if $\lambda_{1}$>1, $\frac{f(x_{1}, x_{2},...,x_{n}; \lambda_{1})}{f(x_{1}, x_{2},...,x_{n}; 1)}$ = $\lambda_{1}^n \cdot \exp((1-\lambda_{1})\sum X_{i})$
 This is a decreasing function of $\sum_{i=1}^nX_{i}$ (as $1-\lambda_{1}$<0 font="">
 Therefore, the most powerful alpha level test reject the null hypothesis when $T < c \cdot \alpha$,
 where $c \cdot \alpha$ is the alpha quantile of $\chi_{2n}^2$.