The Exponential distribution is given by f(X; ) = 1 e X/ X >0 and > 0 This is a skewed and continuous

The Exponential distribution is given by f(X; θ) =

1

θ

e

−X/θ X >0 and θ > 0 This is a skewed and continuous distribution defined only over the positive quadrant.

(a) Show that E(X) = θ and var(X) = θ2.

(b) Show thatθ MLE = ¯X .

(c) Show that the method of moments estimator of θ is also ¯X.

(d) Show that ¯X is an unbiased and consistent estimator of θ.

(e) Show that ¯X is sufficient for θ.

(f) Derive the Cram´er-Rao lower bound for any unbiased estimator of θ? Is ¯X MVU for θ?

(g) For n = 20, derive the Uniformly Most Powerful critical region of size α ≤ 0.05 for testing H0; θ = 1 versus H1; θ = 2.

(h) Form the Likelihood Ratio test for testing H0; θ = 1 versus H1; θ = 1. Derive the Wald and LM statistics for testing H0 versus H1. When is the Wald statistic greater than the LM statistic?