Question

Could you help me to solve this question below? Thank you so much!

Problem 1 Suppose i iid N(?, 3) and let ???(n + 1)-? ? n=1 Xi be an estimator of ? I. Derive the distribution of ?? 2. Derive the MSE of 4. Find a MVUE of ?, denote it ??, and compare its MSE to that of ??. 5. Does the relative efficiency Effnn) approach 1 as n?

Answer 1

X~N(mu, 3) mu_n = sigma^n_i = 1 xi (i) As x_1, x_2 .....x_n iid normal, hence, sigma^n_i = 1 xi /n + 1 is also a normal random variable sigma^n_i = 1 xi/n + 1 = nx/n + 1 x~N(mu, 3/n) nX/n + 1 ~N(mu/n + 1, n_0^2/(n + 1)^2 (2) MSE (mu_n) = var (mu_0) + (Basis(mu_n))^2 = n sigma^2/(n + 1)^2 + (mu-n mu/n + 1)^2 = n sigma^2 + mu^2/(n + 1)^2 = 3n + mu^2/(n + 1)^2 (3) minimal sufficient statistic for mu is given by y mu_n = nx/n + 1 Hence mu_n is a function of minimal sufficient statistic. \r\n (4) log f(x/mu_1) = -n/2 log 6 pi-sigma^n_i = 1 (xi-mu)^2/sigma^2 = (0) = E[-partial differential^2/partial differential mu^2 log (x(mu)] = E[n/sigma^2] = n/sigma^2 CRLB = sigma^2/n x is a MVUE of mu MSF (x) = var (x) + (B as (x))^2 = sigma^2/n plusminus 3/n if we compare between 3/n & 3n + mu^2/(n + 1)^2 its difficult to which is larger as its not same for all n