Find interval estimation of parameter $\lambda$ with Central limit theorem

Question

Assume that $X_1,X_2, ..., X_n$ is random sample with exponential distribution and with probability density function:

$$f(x)=\left\{ \begin{array}{c} \lambda e^{-\lambda x} \; x>0, \\ 0 \qquad x <0, \end{array} \right. $$

where $\lambda >0$ is unknown parametr. Find interval estimation of parameter $\lambda$ with central limit theorem

May someone give me an idea, how to find estimation of the parameter?

Answer 1

First, it is easy to see that $ E(X_1) = \frac{1}{\lambda} $ and $ Var(X_1) = \frac{1}{\lambda^2}$. From CLT it follows that

$$ \frac{\sqrt{n}\left (\overline{X} -\frac{1}{\lambda}\right)}{\frac{1}{\lambda}} \xrightarrow {d} N(0,1)$$

and equivalently,

$$ \sqrt{n} \left (\lambda \overline{X} - 1\right)\xrightarrow {d} N(0,1).$$

Therefore, the asymptotic $ 100(1-\alpha)\% $ CI of $ \lambda $ is

$$ P \left ( - z_{\frac{\alpha}{2}} \leq \sqrt{n} \left (\lambda \overline{X} - 1\right) \leq z_{\frac{\alpha}{2}} \right) = 1 - \alpha $$

and equivalently,

$$ P \left (\frac{ - z_{\frac{\alpha}{2}} + \sqrt{n}}{\overline{X} \sqrt{n}} \leq \lambda \leq \frac{z_{\frac{\alpha}{2}} + \sqrt{n}}{\overline{X}\sqrt{n}}\right) = 1 - \alpha. $$