1.31.4 kindly help

Let {x;}" , be a random sample of a scalar random variable a with E(x] = p, V(x] = o', E[(x - p) ] = 0, E[(x - 4) ] = T, where all parameters are finite. (a) Define Th = =, where 1=1 Derive the limiting distribution of v47, under the assumption / = 0. (b) Now suppose it is not assumed that a = 0. Derive the limiting distribution of Vn (In - plim In ) Be sure your answer reduces to the result of part (a) when / = 0. (c) Define R. = where is the constrained estimator of o' under the (possibly incorrect) assumption / = 0. Derive the limiting distribution of Vn ( Rn - plim Rn) for arbitrary at and o' > 0. Under what conditions on a and of will this asymptotic distrib ution be the same as in part (b)?Let X = {x1, .... ", } be a random sample from some population of a with E [x] = p and V [x] = 02. Let A, denote an event such that P{An} = 1 - 2, and let the distribution of An be independent of the distribution of r. Now construct the following randomized estimator of : In if An happens, in = 1 n otherwise. (i) Find the bias, variance, and MSE of ,. Show how they behave as n - co. (ii) Is a, a consistent estimator of #? Find the asymptotic distribution of va(/, - /). (iii) Use this distribution to construct an approximately (1-a) x 100% confidence interval for z. Compare this CI with the one obtained by using 2, as an estimator of p