kniga 7 - Probability and Statistics 1 - Sheynin, Oscar

Let us calculate now, by means of formulas (13) and (13bis), the conditional expectationof h given (1). We have, respectively,h * =Γ [( n + 2)/ 2],S Γ [( n + 1) / 2]Γ(n / 2)h ** =. (15; 15bis)S Γ [( n −1) / 2]It is easy to determine thath * = h **[1 + (1/n)]. (16)We see that for large values of n the difference between h * and h **is not large but that itcan be very considerable otherwise.4. Limit Theorems. Here, we will establish that, under some natural assumptions and asufficiently large number of observations n (and in some cases, for Problem 1, for smallvalues of n as well), it is possible to apply the approximate formulas 1 (a|x 1 ; x 2 ; …; x n ) ~ (h 2 (h|x 1 ; x 2 ; …; x n ) ~ (sn / π )exp[– nh 2 (a – x ) 2 ], (17)2 / π )exp[– 2S 2 (h – h ) 2 ], (18) 3 (a; h|x 1 ; x 2 ; …; x n ) ~ (S 1 h1 2n/)exp[– n h 21(a – x ) 2 – 2S 1 (h – h1) 2 ] (19)whereh = (1/S) n / 2 , h1= (1/S 1 ) ( n −1) / 2 . (20, 21)Introducing = hn (a – x ), = S2(h – h ), 1 = h 1 n (a – x ), 1 = S 1 2(h – h 1)(22 – 25)whose conditional densities, given (1), are, respectively, 1 (|x 1 ; x 2 ; …; x n ) = (1/hn) 1 (a|x 1 ; x 2 ; …; x n ), (26) 2 (|x 1 ; x 2 ; …; x n ) = (1/S2) 2 (h|x 1 ; x 2 ; …; x n ), (27) 3 ( 1 ; 1 |x 1 ; x 2 ; …; x n ) = (1/S 1 h 1we may rewrite formulas (17) – (19) as2 n ) 3 (a; h|x 1 ; x 2 ; …; x n ), (28) 1 (|x 1 ; x 2 ; …; x n ) ~ (1/)exp (– 2 ), (29) 2 (|x 1 ; x 2 ; …; x n ) ~ (1/)exp (– 2 ), (30)2 3 ( 1 ; 1 |x 1 ; x 2 ; …; x n ) ~ (1/)exp (– 1 – 2 ). (31)When considering Problem 1 the following limit theorem justifies the applicability of theapproximate formula (29), – or, which is the same, of its equivalent, the formula (17).Theorem 1. If the prior density 1 (a) has a bounded first derivative and 1 ( x ) 0, then,uniformly with respect to , 1 (|x 1 ; x 2 ; …; x n ) = (1/) exp (– 2 ){1 + O[1/(hn)](1 + | |)}. (32)