Question

Let X₁, X_{2, ......} X_n  be a random sample of size n from EXP( $\Theta$ ) distribution , $f(x)= \frac{1}{\Theta } e^{-\frac{x}{\Theta }} ,x>0$ , zero , elsewhere.

Given, mean of distribution $\Theta$ and variances $\Theta^{2}$ and mgf $(1-\Theta t)^{-1}, t<\frac{1}{\Theta }$

a) Show that the mle $\hat{\Theta }$ for $\Theta$ is $\bar{X}$ . Is $\hat{\Theta }$ a consistent estimator for $\Theta$ ?

b)Show that Fisher information $I(\Theta)=\frac{1}{\Theta ^{2}}$ . Is mle of $\hat{\Theta }$ an efficiency estimator for $\Theta$ ? why or why not? Justify your answer.

c) what is the mle estimator of $\Theta ^{2}$ ? Is the mle of $\Theta ^{2}$ a consistent estimator for $\Theta ^{2}$ ?

d) Is mle of $\Theta ^{2}$ an unbiased estimator for $\Theta ^{2}$ ? why or why not? Justify your answer.

Answer 1

i2t 0 h 2 h2 n M LE 1(0)-(02 )-1

s the mle d6 0 hot am un baiass

(ht)-n t (h2) (x) u 2 4 4