Dr. Hackney STA Solutions pg 105

Dr. Hackney STA Solutions pg 105 - 7-8 Solutions Manual for...

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Unformatted text preview: 7-8 Solutions Manual for Statistical Inference 7.19 a. L(|y) = i 1 2 2 exp - 1 (y - xi )2 2 2 i 1 2 2 2 = (2 2 )-n/2 exp - (2 2 )-n/2 exp - Yi2 , (y i -2xi yi + 2 x2 ) i i 2 = 2 i xi 2 2 exp - 1 2 2 2 yi + i 2 xi yi i . By Theorem 6.1.2, ( b. i i xi Yi ) is a sufficient statistic for (, 2 ). 2 2 2 2 logL(, 2 |y) = - For a fixed value of 2 , n 1 n log(2) - log 2 - 2 2 2 2 2 yi + xi yi - i x2 . i i logL 1 = 2 Also, xi yi - i 2 set ^ x2 = 0 = i i i xi yi 2 . i xi 2 logL 1 = 2 2 x2 < 0, i i ^ ^ so it is a maximum. Because does not depend on 2 , it is the MLE. And is unbiased because xi E Yi xi xi ^ E = i = i = . 2 2 i xi i xi ^ c. = i ai Yi , where ai = xi / tributed with mean , and ^ Var = i j ^ x2 are constants. By Corollary 4.6.10, is normally disj 2 a2 Var Yi i = i xi 2 j xj 2 = ( x2 2 i = x2 )2 j j i 2 2. i xi 7.20 a. E b. Var Because i i Yi = i xi 1 i xi i E Yi = 1 i xi xi = . i Yi i xi = ( 1 2 i xi ) Var Yi = i i ( 2 n 2 2 = 2 2 = . 2 n x n2 x i xi ) i i x2 - n2 = x i i (xi - x)2 0, x2 n2 . Hence, x i i ^ Var = 2 2 = Var 2 n2 x i xi Yi i xi . ^ (In fact, is BLUE (Best Linear Unbiased Estimator of ), as discussed in Section 11.3.2.) ...
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