STAT 234 Final, Fall 1996

F pasueliadieneeteptiamen neat MEWS UNIV shee Lt eRary Oh notais ( er. Scere ee AMERICAN UNIVERSITY OF BEIRUT MATH 234 FINAL EXAM Time = 1 hour 30 minutes February 3, 1997 . Suppose X1, X2,...,.Xn,... is a sequence of random variables (a) Define what is meant by the sequence {X,,} converges in distribu- tion to random variable X. (b) Define what is meant by the sequence {X,,} converges in proba- bility to random variable X. . State clearly the central limit theorem. . Give an example or a situation where the maximum likelihood estima- tor of parameter @ is unbiased and an example or situation of a biased maximum likelihood estimator but efficient. . Let X have a Gamma distribution with parameters a = 2 and 8 > 0. (a) Find the Fisher information I({). (b) If X1, X2,..., Xn is a random sample from this distribution, show that the maximum likelihood estimator of Gis an efficient estima- tor of B. . Let the random variable X have the p.df. f(x;0) = (1/@)e-*/*, 0 < 8 < 00, zero elsewhere. Consider the simple hypothesis Hp : 8 = 2 and the alternative hypothesis H, : 8 = 4. Let X, and X, be a random sample of size 2 from this distribution. Show that the best test of Ho against H, may be carried out by use of the statistic X; + X2.