\f 1 Let X 1 , X 2 ,....., X n is a random sample from a normal distribution with mean and variance 02 known. Then probabilily density function 1 x i 2 0 1 f xi 2 e , xi , 0, 0 2 By def, Likelihood function, L f x1 x2 ..... xn f x1 f x2 ... f xn 1 e 0 2 1 0 2 1 1 x 1 2 0 n e L 0 2 1 2i e 0 2 xi 0 1 1 2i e 1 n n 2 2 K 1 0 2 2 1 xn 2 0 e 2 n 1 x 2 2 0 2 xi 0 1 I 2 1 n x ln L n ln 2 n ln 0 i 2 i 1 0 n log L 1 n 1 n 1 2 xi 2 xi 2 0 i 1 0 i 1 i 1 0 2 log L n 2 2 0 n x n i 1 i Therefore, Fisher's information, 2 log L n n E 2 2 E Constant Constant 2 0 0 I E 2 Let x 1 n xi n i 1 1 n 1 n 1 n X i E X i ni 1 ni 1 ni 1 E Since E , is an unbiased estimator of . 1 n 1 n 2 2 1 n X i 2 Var X i 2 0 0 n i 1 n i 1 n ni 1 Also Var Var 2 Since population is N , 0 , 2 E X i and Var X i 0 Also assume 1 Cramer Rao's lower bound for the variance of an unbiased estimator of is, 2 I Clearly Var Therefore, 2 1 0 n 02 n 2 I 1 n X i attains Cramer-Rao's lower bound, so it is the minimum variance ni 1 unbiased estimator MVUB . Since is the MVUB, it is an efficient estimator in such class of estimators as it possesses the smallest variance. 3 The Fisher's information for 1 is, 1 2 1 0 I n I 4 By invariance property of MLE, the MLE for 1 1 n n x i 1 i 1 is, 1 Let X 1 , X 2 ,....., X n is a random sample from a normal distribution with mean and variance 02 known. Then probabilily density function f xi 1 e 0 2 1 x i 2 0 2 xi , 0, , By def, Likelihood function, L f x1 x2 ..... xn f x1 f x2 ... f xn 1 0 2 e 1 x 1 2 0 2 1 e 0 2 n 1 n 1 xi 0 n xi 0 2 i1 e 1 L 2 0 2 i1 e 2 1 0 2 log L 1 2 0 2 2 I 1 n x 2 n ln 0 i 2 i 1 0 ln L n ln e 1 x n 2 0 2 n 1 2 0 1 x 2 2 0 2 log L n 2 2 0 n xi i 1 n 1 n 1 n xi 2 xi n 2 0 i 1 i 1 0 i 1 2 Therefore, Fisher's information, n 2 log L n I E E 2 2 E Constant Constant 2 0 0 n 1 2 Let x xi n i 1 1 n 1 1 E Xi E Xi ni 1 ni 1 ni 1 Since E , is an unbiased estimator of . n n 1 n 1 Also Var Var X i 2 ni 1 n n Var X i i 1 2 1 n 2 0 0 n2 i 1 n 2 Since population is N , 0 , 2 E X i and Var X i 0 Also assume 1 Cramer Rao's lower bound for the variance of an unbiased estimator of is, 2 1 0 I n 02 n 2 Clearly Var I Therefore, 2 1 n X i attains Cramer-Rao's lower bound, so it is the minimum variance ni 1 unbiased estimator MVUB . Since is the MVUB, it is an efficient estimator in such class of estimators as it possesses the smallest variance. 3 The Fisher's information for 1 is, 1 2 1 I 0 n I 4 By invariance property of MLE, the MLE for 1 1 n n x i 1 i 1 is, \f 1 Let X i i 1 ~ Exp n Then probabilily density function f x e x , x 0, 0, By def, Likelihood function, L f x1 x2 ..... xn f x1 f x2 ... f xn e x1 e x2 K e xn n e n Xi L n e i 1 n Xi I i 1 n ln L n ln X i i 1 ln L n Xi i 1 n 2 ln L n 2 2 Therefore, Fisher's information, 2 ln L n n E 2 2 E Constant Constant 2 I E 2 Let x 1 n xi n i 1 1 n 1 n 1 1 1 n X i E X i 1 n i 1 n i 1 ni Since E , is not an unbiased estimator of . E 1 1 1 Also Var Var X Var X n n n n i 1 n i 2 i 1 Since population is Exp , E X i Also assume 1 n i 2 i 1 1 n 2 2 1 1 and Var X i 2 Cramer Rao's lower bound for the variance of an unbiased estimator of is, I 2 1 2 n 2 n Since is not an unbiased estimator, we can't compare it for Cramer-Rao's lower bound. n 1 Therefore, X i doesn't attain Cramer-Rao's lower bound, so it is not the minimum ni 1 variance unbiased estimator MVUB . Since is not the MVUB, it is not an efficient estimator. 1 The Fisher's information for is, 3 Let n n I I I I 1 I 1 n 2 1 n 2 2 1 Let X i i 1 ~ Exp n Then probabilily density function f x e x , x 0, 0, By def, Likelihood function, L f x1 x2 ..... xn f x1 f x2 ... f xn e x e x1 e x2 n e L n e n n Xi i 1 n Xi i 1 I n ln L n ln X i i 1 ln L n Xi i 1 n 2 ln L n 2 2 Therefore, Fisher's information, 2 ln L n n I E E 2 2 E Constant Constant 2 2 Let x 1 n xi n i 1 1 n 1 1 1 n 1 n E Xi E Xi n i 1 ni 1 ni 1 Since E , is not an unbiased estimator of . 1 1 1 1 n Also Var Var X Var X n n n n i 1 n i 2 i 1 Since population is Exp , E X i Also assume 1 n 1 i 2 2 i 1 and Var X i 2 1 2 Cramer Rao's lower bound for the variance of an unbiased estimator of is, 2 1 I n 2 n 2 Since is not an unbiased estimator, we can't compare it for Cramer-Rao's lower bound. n 1 Therefore, X i doesn't attain Cramer-Rao's lower bound, so it is not the minimum ni 1 variance unbiased estimator MVUB . Since is not the MVUB, it is not an efficient estimator. 3 Let 1 The Fisher's information for is, n I I 1 I 1 n 1 I I 2 2 n n 2 \f