INFERENCE STATISTICS

(b) The likelihood function is Page 2 L ( M , 62 ( x(i ) = 1 RT( 62 ) 2 exp ( - M . + M 262 262 xx ) = D L ( M , 62 / sci ) = hoes . g ( 2 xx , zixi , M , 62 ) where hoes = 1 , -Xi, Zixi, M , 62 ) = ( and ) "exp ( - 1 4 ) exp ( - 2 X + 14 Zixi) depends on u and 62 on the sample through EX and Zixi. Henle Z Xi and Ziki are jointly sufficient statistic for (( ) let Yi, 12,.... I'm be another random Sample from the Sampled distribution. Then we have: M. { ) 27 62 - exp - 2 62 14 ) MP - 26 L (yi / M 62 ) ( snoz ) " exp ( - 1 1 ) exp / - to2 2 M 62 - exp 262 A=I exp ( boz Eixe ) exp 26 2 Z = up 2 Xx 262 is independent of ( 1, 63) if Z Xi = - yi equivalently if Exxi = Ex Yi. Therefore Xx, Eiki) is A minimal sufficient statistic for (M, 62)( d ) yes because Exi Zixi ) ER is a sufficient statistic Kage 3 for ( 1, 62) E RT (dimension of the parameter spale s the Same as the dimension of the sufficient statistic spale) ( e) The likelihood function is ? 127 62) exp Exxi 2 62 62 = 0 Log likelihood function is : ( ( 1 62 ) = - m ( R ( 257) + him (63) ) -1. . 2 + 262 262 F= 1 Eixe . Maximum Likelihood Estimate of N is 1 which solves the equation : 0= ( (W) = 1 ( M, 63 ) = D M : 2 = sixi IN =M 62 + - Sixi zo 62 m i Xi 2 1 = 1 Zi = 1+ 2+ ... + m = D MMLE = 6 Lixi m( m+ 1 ) ( 2m + 1 ) * = m (anti ) (am+ 1 ) 6 . Malimum likelihood Estimate of 6 i 6 which solves the equation : 0 = 6 (62) = 26( 1, 62) 862 16= 62 = D = m + 264 ( Ju - NM ) = m D 2 5 2 2 54 2 (ai- i'm ) 2 = D MLE M M i =(f). Second derivative of ( ( 1 62) with respect to M and 6 Page 4 = CCM) = - 2 x 8 1 2 2 6 ( M 62 ) 62 = ( ( 6#). m ( si- AM ) 26 8 Let find Crammer-Rao Lower Bound (CRLB) for Manol 62 . CRLB for M i : Im ( M ) . CRLB for 62 is 2 62 1 2 062 In ( 62 ) . Fisher Information 2 2 M3 . In ( M ) = - E CC( MJ ) = - EL- m( m+1) ( am+ 1) 6 = 62 6. 62 . In ( 62) = - E (L ( 62) ) = -E ( 26 - iM ) 264 2 68Page ! STA 4812 ASSIGNMENT Question , ( a) f (xi /M, 62 ) 62 2 62 = 0 f ( film , 62 ) = + IM G 262 62 where heard = h(xritz . .. . xu ) = 1 C ( B ) = 2162) xxp (- 4 262 WI (B ) = ->o 62 W2 (0 ) = M 62 *, ( 20 ) = = Ci 2 Then f (xill, 53) = hocs. Cco). exp [ w, cost,ce) + W 2 Co) * 2 (20)]. Hence f (xi / M, 62 ) belongs to the exponential family of Mobability density functions.\f