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8.17 - Here 0t is the shape parameter of the distribution...

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Unformatted text preview: Here 0t. is the shape parameter of the distribution i.e., as 0t. changes the shape of the distribution changes We know, 1 V = —; 0 ar(X) 4(20H1) Of. :- So Variance is inversely proportional to a. As 01. increases variance decreases and shape of distributions becomes tail and narrow i.e., for large values of a, the distribution is less spread. (b) Here we can compute and see that, 1 “-1: 00:5 and, H _ 1 +1 9 8a+4 4 So, 8a+4:;1 Ila—E 1 1 => : —— gag—2 2 Now 1L -1iX2 ’ 2 _ P3§_1 i :> 61— 1 1 _ 8(ZXEm)—2 2 a“ 1 _1 _ SZXE—zn 2 k‘ (c) Log likelihood of 0t. is, :[a) = n[logT2 OL—ZlogTOL]+[OL—l)illog[xi (1—x,)] 1(a) = Pelogl—‘2OL—22elog1—'OL+[OL—l)|:Zlog xi +Zlog [1—4)] 815(05): 0 Bot 21"2oc'21"oa ’1 S’s—[ma 701},le 379W ”:0 _>(D The solution 66 of equation Will be MLE of on and can be obtained numeric ally. 32 Mot) d :0 (l 303 2120a 2OL"—2l 2061 20L‘ 10L]. OL"-l 06']. 06' _ 2n 2 — 2 U 1"[2 05) F06 32 L[OL) I = —E (a) l M, 2 rearw—[ra'f 21"2oel"2oc"—2(may)2 : )3 — To? F2012 Therefore, Assymptotic variance of MLE 66 = ISA) roe2 r2 of _ [2n|:FCEFOL"—[Fog')2 _ 21’? OLF205"_2[F2OL,)2:|]—1 F201” (e) Likilihood(oa) = [mg] exp|:[cr—l)ilog[x,—X3):| i-l which is one parameter exponential family equivalent to following General form, eXp[d[ Wjexpl: [8)T[x)]; —00 < X < 00 V 8 Bot => 01(06): 22::[fi] SW= 0(a): (01-) TlXJ= 21° Sle’gl -1 N :> Zlog(x,—xf) is sufficient statistic for Oh. s-1 [ Note' By factorization theorem we see directly same result ] ...
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