- C Let I I D X1 Xn Fe Be A Random Sample Of Size N From The Distribution Fo Indexed By The Un Known Parameter 0 With 1 (84.19 KiB) Viewed 114 times
(c) Let i.i.d. X1, Xn Fe 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; 2) 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(x;0) at an arbitrary point x = 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 0(X; , 02) = (2+)-1/20-/exp{-}(2 – u)?/o²} with unknown mean u but known variance o?, show that $(x; ž, (1 - 5)(2) n is the UMVU estimator of 0(c; j, 02).