(c) Let X1, ... , Xn Fo be a random sample of size n from the distribution Fo indexed by the un- known parameter 0 with corresponding density function f(x; 0). Also, we let T be a statistic and fit(x | t; 0) denote the conditional density of X1 given T = t, where t is the observed value of T. It is proposed to estimate f(k; 0), the value of the density f(r; 0) at an arbitrary point : = k by W (t) = fit(k | t), assuming that this conditional density does not depend on 0. (i) Give a condition on T for W(T) not to depend on 0. (ii) Under this condition, show that W(T) is an unbiased estimator of f (k; 0). (iii) Give a further condition on T for W(T) to be also a UMVU estimator of f(k; 0). (iv) For the normal density 6(x; 14, 62) = (27)-1/20-1 exp(-}(x- 14)?/0?} with unknown mean / but known variance o?, show that o(r; I, (1 - -)?) is the UMVU estimator of o(x; /, 0?)