Mathematics 341/686 Assignment 2 Due Oct 26, 2015 in Class 1. Let X1 , . . . , Xn be a random sample from N (, 1). Assume a uniform (0, ) prior distribution for . (a) Find the posterior distribution of . (b) Find the Bayesian estimator of . (c) Find the probability that is between 0.2 and 0.5. 2. Let X1 , X2 , . . . , Xn be a random sample from a distribution with pdf f (x; ) = x (1 ), x = 0, 1, 2, . . ., zero elsewhere, where 0 1. (a) Find the mle, , of . (b) Find the mle of the mean. (c) Find the Cramer- Rao Low bound for the variance of unbiased estimator of 1-. 3. Let X have a gamma distribution with parameter and = > 0. (a) Suppose = 4 and = > 0, i. nd the Fisher information I(). ii. If X1 , X2 , . . . , Xn is a random sample from this distribution, show that the mle of is an ecient estimator of . iii. What is the asymptotic distribution of n( )? (You do not have to prove this result, just use the properties of mle). 4. Suppose that both and are unknown and we are interested in obtaining the mle of and . (a) What are the score functions? Is there an analytical solution for the score functions? 1 (b) If we use Newton method to solve the score functions, i. What is the iterative equation? ii. Using the iterative equation, state the algorithms in steps. 5. (For MA 641)Under the regularity conditions, suppose g() is a continuous function of which is dierentiable at such that g () = 0. Then n(g() g()) N (0, g ()2 ) I() in distribution as n goes to innity. 6. Let X be mean of a random sample of size n from a N (, 2 ) distribution, < < , 2 > 0. Assume that 2 is known. (a) Show that X is the mle of . 2 (b) Show that = X 2 n is an unbiased estimator of 2 and nd its eciency. (c) Using -method, nd the approximated expected value and variance of . 7. (Optional) Prove that X, the mean of a random sample of size n from a distribution that is N (, 2 ), < < , is, for every known 2 > 0, an ecient estimator of . 8. (Optional) Let X be N (0, ), 0 < < . (a) Find the Fisher information I(). (b) If X1 , X2 , . . . , Xn is a random sample from this distribution, show that the mle of is an ecient estimator of . (c) What is the asymptotic distribution of n( )? 9. (Optional) Let S 2 be the sample variance of a random sample of size n > 1 from N (, ), 0 < < , where is known. We know E(S 2 ) = . (a) What is the eciency of S 2 ? (b) Under these conditions, what is the mle of ? (c) What is the asymptotic distribution of n( )? 2