Question

Let X1, X2, . . . , Xn iid∼ Exponential(β). The probability density function for these random variables is f(x|β) = 1 β e − x β , x > 0; β > 0.

a. Find the maximum likelihood estimator for β.

b. Compare the MLE to the estimator βˆ = (X1 + Xn)/2. Which do you recommend? Explain. You can use the following facts: E(Xi) = β and V ar(Xi) = β 2 , for any i.

c. Construct a 100(1 − α)% C.I. for β using the pivotal quantity 2nβˆMLE /β. You can use the fact that 2nβˆMLE /β ∼ χ 2 2n . (Note that you don’t need the explicit form of the MLE to answer this question.)

d. Explain how 2nβˆMLE /β can be used to perform tests of hypotheses regarding β. (Note that you don’t need the explicit form of the MLE to answer this question.)

Answer 1