STAT 234 Final, Fall 2007

AMERICAN UNIVERSITY OF BEIRUT STAT 234 Final Exam, Jan 28, 2008 Time = 1 hour 1. Let X be a continuous random variable with a pdf f(x, θ) that satisfies regularity conditions. Show that V ar[ ∂lnf(x,θ) ∂θ ] = −E[∂2lnf(x,θ) ∂θ2 ] 2. Let X1, · · · , Xn be a random sample from a Geometric distribution with parameter θ. (a) Find the maximum likelihood estimator, ˆθ, of θ. (b) Show that the statistic(n−1)/(�n i Xi−1) is an unbiased estimator of θ. 3. Let X1, X2, and X3 be a random sample of size n = 3 from an expo- nential distribution with mean θ. i.e. the common pdf is f(x, θ) = (1/θ)e−x/θ, if x > 0.Also, let Y1 < Y2 < Y3 be the corresponding order statistic. (a) Compute E( Median(X1, X2, X3)) = E(Y2) and hence construct an unbiased estimate of θ, say ˜θ. (b) Compare V ar(˜θ) to the Rao-Cramer lower bound. 4. Given a random sample of size n = 3 from a Bernoulli distribution with parameter θ. i.e. f(x, θ) = θX(1 − θ)1−X if X = 0, 1. Suppose you are testing the following hypothesis: H0 : θ = 1/2 vs H1 : θ = 2/3. If α=P(Type I error)=1/8, determine the critical region and β = P(type II error). 1