Monrad.410.27 - begin/i Hon ( ’ {Tami/“.54 LM Z {a...

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Unformatted text preview: begin/i Hon ( ’ {Tami/“.54 LM Z {a & (JRHSH M M or— aw {fie/[aging fa {am/fig (m p P lee®§ I M 44% ‘PA 6 66/0 0; 611414 3e} ® a FfiZH/Eaé M fifimfi PMM %fl(‘2}9§ : 66 69} 1s hie mgr fine _M0tWLC(‘€/\ 9&4 H3 m Eefuézfl o M an (gee?) MAN?“ [**8 Pa[%(2\: 0] :1 fix a“ 966). EKQMPLL Z 51 é‘tnowvia/Q (L/{RML Pawn «3393 = (‘28ez(‘(~e>“‘z 1 TM WW A€\2~6 10<8<1§ ‘ WNLg‘flrg This. (rt/13641433: _€/§WAHons Z=Ofl1wUMl Nakga : Hm m flaw WWMJ Maw {.7me § 0: Eéwzx] = :uzzx—ge'G J e>o ivx EM ‘ 0 E V MW wmiflw 0 == MOJ WW) = Nahfiz: Hm we (1m iné'mfefg ma ' MW } m mg (m {N ew& Wm o€€> m we gawk/5 Mal/lg Mae/64 14(0)) 74(7)) ,, a a _. ,6»; 3 LC 5:chst =0) gear am 156: (aw {AW (:90?) -= 0 {3w almosrL all We (ch (a?) , This; MQWIS: 14: Z is a lww/iwm mmflom mfia/{lfi‘é MM (la/w 56W 6% am! 5 (ca/m? Z W a pfihfi ) HLML H423 : 0] =10 (ME/Laaapaf)- P[Q[Z)=O] =15 Cam {ék Owl- £3:ch gfil c @Qw¥£¢¢_9€®‘ v ‘ 4 4 "I I M‘ 03f PWDCS E x 3 gay 93 g Q 6 fl EW::;_MAnH;:W f" 7 s , I 2 s, e ie “A j m “ m __ ___.;_L<J:L ‘5“ 4er 61% 5k MR matamwa , _ Z m m Em eA ,wfim.4zbr_g%fi E cm > ém £0 659 a” hi ' ems? v wt Magi M g :vakckmw: of. . . y' ’ I K? but I” A I” v u~ I 0' ’i // 7.5 Exponential families of Disfhbabons Consider a family f($;6! : 6 E Q) of probability density or mass functions, where Q is the interval set 9 : {6 : 7 < 6 < 6}, where 'y and 6 are known constants (they may be ioo), and where fix; 9) Z { eXPlP(9lK($l + 5(a) + q(6)] a: e 5 O elsewhere, (751) where S is the support of X. In this section we will be concerned with a particular class of the family called the regular exponential class. Definition 7.5.1 (Regular Exponential Class). A pdf of the form (7.5.1) is said to be a member of @ regular exponential " ; of probability density or mass functions if 1. S, the support of X, does not depend upon 6, 2. p(6) is a nontrivial continuous function of 6 E Q, 5’. Finally, (a) ifX is a continuous random variable then each of K’(ac) i 0 and S($) is a continuous function of a: E 8, (b) if X is a discrete random variable then K is a nontrivial function of a: E S. For example, each member of the family {f(:c;6) : 0 < 6 < 00}, where f(:c;6) is N (0, 6), represents a regular case of the exponential class of the continuous type because 1 —a: f($;6) 2 Tree 2/29 1 . exp (—Eémz — logv27r6>, —oo < a: < oo. II On the other hand, consider the uniform density function given by flag) 2 { exp{——log 6} a: 6 (0,6) 0 elsewhere. This can be written in the form (7.5.1) but the support is the interval (0,6) Which depends on 6. Hence, the uniform family is not a regular exponential family. Consider a family { f (:13; 0) : 49 E Q} of probability density or mass functions, Where Q is the interval set 9 = {6 : 7 < 6 < 5}, Where *y and 6 are known constants (they may be too), and Where . _ eXp[p(0)K(w) + 3(93) + 9(9)] 03 E 5 flw’g) — { 0 elsewhere, (7'5'1) Where 8 is the support of X. Let X 1, X 2, . . . ,Xn denote a random sample from a distribution that represents a regular case of the exponential class. The joint pdf or pmf of X1, X2, . . . ,Xn is n exp [p(6) Z + Z + nq(6) 1 Theorem 7.5.2. Let f(a:;6), *y < 9 < 6, be a pdf or pmf of a random variable X whose distribution is a regular case of the exponential class. Then if X1,X2, . . . ,Xn (where n is a fixed positive integer) is a random sample from the distribution of X , n _ the statistic Y1 : ZK is a sufiicient statistic for 6’ and the family {fy1(y1; 0) : 1 7 < 0 < 6} of probability density functions of Y1 is complete. That is, Y1 is a complete sufficient statistic for 6. ...
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This note was uploaded on 06/11/2010 for the course STAT 410 taught by Professor Alexeistepanov during the Spring '08 term at University of Illinois at Urbana–Champaign.

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Monrad.410.27 - begin/i Hon ( ’ {Tami/“.54 LM Z {a...

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