Unformatted text preview: 6.} By the Factorization Theorem, [X1 in sufﬁcient because we can write the pdf 01' X as fX(xI52) = $UEX2/2a2m mac {elite/26' _ SUXHV‘?) _ gig“
50‘)
6.2 By the Factorization Theorem, T(X)...  mag/if m gumdgnt because we can Write the joint
pdf of X1" ..., Xn as
rel, New) ills"”5 10,, “0, our ._. e‘" 1a,”, (Ten) if:  w
ammo) “‘3 Notice, we use the fact that i > 9, and the fact that all xis > i8 if and only if miﬁg/i) > 0 3" 1 6.3 Let 1(1): minixi. Then We can write the joint pdf as
a, a .1.. ~( xi'~#)/¢ eP/V / ‘
 1 . {(x110°°)xni ”3 ) iImIlanl It”, m)(x’1iw)  ("7—)1—Exl vim1))
112;) r 3(‘(])! mi 1 #10.) Thus, by the Factorization Theorem, (X0), 2:11)“ a sufﬁcient stetistic for (11,6). 6,5 The sampie density is given by . ‘ =2";
19] {(xi IE) =Hm1§li§I(i(5~l) < xi__ < i(9+1)) I
.iWI i: ll (21—9 n(iI:]1~i1)I(min¥ 2 ~{9»1))I(max§g 9+1) so (min'Xi/ i, max Xi/i) is sufﬁcient for 9. 6.6 The sample density is given by “ 1
n ..  ‘lﬁ 1‘ a . “Sign?
{on}... .=,x,,la,m Iii—"”5"" X1 1 e X: "’r_"“'""( mama) (gig) e . i: _. By the Factorization Theorem, (1—1:: Xi' £1_1Xi)m sufficient for (a, ﬁ). 11 0""‘1
7.10 a) rode): ._iI_j[1 53:59!“ IE0 ,3 (m): _.§%( ) “Ml 1M, (3:01)) I[0 0°)(x(1))“ .. L(a,ﬁlx).
By the Factorization Theorem, (HXi,X(n)) are sufﬁcient.
b) For any ﬁxed a, L(c¢,ﬂ Ix) z 0 if ,6! < 3:01), and L(a,ﬁ ﬁx) a. decreasing function of 19 if [3 2
x01). Thus, X01) is the MLE of [3. For the MLE of (I calculate 32—116ng 5—8 [nloga— nalogﬂ +(awUlOSHXi1: :37 "" n 3°85 +103 HIE: Set the derivative equal to zero and use 3 r... Xm) to obtain .
1
3 ._ n
"‘ was: 1:01) ioglIXi [HE (1°3x(n) ‘°3 19]“  The second derivative is —n/ar2 < 6, so this' 13 the MLE. 6) x01) = 25.0,1eg Hxi a: Slog xi = 43.95 2:» 3 = 25.9, 5; ~_— 12_59_ ...
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 Spring '07
 SENTURK,DAMLA
 Statistics

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