Provide best solution with clear defination of the problem

1. Suppose that VT(4 -2#) - N (0, 1). Find the limiting distribution of 7(1 - cos d). 2. Suppose that T(d - 2x) - N (0, 1). Find the limiting distribution of T'sin . 3. Suppose that TO - x4. Find the limiting distribution of T log , Let the positive random vector (Un, V,)' be such that Vi ( (Vm ) - (" ) ) = N CO). ( as n -+ 00. Find the joint asymptotic distribution of (In Un - In Vm) In Un + In Vn / What is the condition under which In Un - In V,, and In U. + In V,, are asymptotically independent? Let X = {:1,. .., 'n} be a random sample from some population of a with E [x] = p and V [x] = 0. Let An, denote an event such that P{A,,} = 1 - 2, and let the distribution of A, be independent of the distribution of r. Now construct the following randomized estimator of /: In if An happens, otherwise. (i) Find the bias, variance, and MSE of A,. Show how they behave as n - co. (ii) Is ,, a consistent estimator of /? Find the asymptotic distribution of vn(/, - /). (iil) Use this distribution to construct an approximately (1 -a) x 100% confidence interval for p. Compare this CI with the one obtained by using 2, as an estimator of a.Let far,}" , be a random sample of a scalar random variable a with Ex] = p, V(x] = of, E[(x - #) ] = 0, E[(x - p)'] = T, where all parameters are finite. (a) Define T, = =, where Derive the limiting distribution of vn, under the assumption # = 0. (b) Now suppose it is not assumed that # = 0. Derive the limiting distribution of Vi (The - plim In ) Be sure your answer reduces to the result of part (a) when a = 0. (e) Define R. = =, where d'= is the constrained estimator of o' under the (possibly incorrect) assumption / = 0. Derive the limiting distribution of Vn Re - plim R.) for arbitrary a and o' > 0. Under what conditions on / and o' will this asymptotic distrib ution be the same as in part (b)? Consider a positive (r, y) orthant R; and its unit simplex, ie. the line segment r ty = 1, x 2 0, 1 2 0. Take an arbitrary natural number & E N. Imagine a bug starting creeping from the origin (x, y) = (0, 0). Each second the bug goes either in the positive r direction with probability p, or in the positive y direction with probability 1 - p, each time covering distance -. Evidently, this way the bug reaches the simplex in & seconds. Suppose it arrives there at point (re, y). Now let k -+ co, i.e. as if the bug shrinks in size and physical abilities per second. Determine (a) the probability limit of (IA, VA); (b) the rate of convergence; (c) the asymptotic distribution of (Ex, yk)