05 统计学基础--点估计(二)

主要介绍了点估计中的极大似然估计的相关内容,似然函数的定义、极大似然估计的求法,从评价估计量优良性的角度出发,判断极大似然估计的无偏性、有效性和相合性。
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1. Lec5: : O( ) Ü•² September 29, 2009 §1 4Œq, O ˜!Úó9½Â 4Œq,{´3ëê©Ùx|Üe~^ ëê O•{. k˜ëê©ÙxF = {Fθ , θ ∈ Θ},d?Θ•ëê˜m. -X = (X1 , · · · , Xn )•lF ¥Ä {ü‘Å , f (x, θ) = f (x1 , . . . , xn , θ)• X VǼê. = oN©Ù•ëY.ž,f (x, θ)L« X —ݼê; oN©Ù• lÑ/ž,f (x, θ)• X VÇ©Ù, =f (x, θ) = Pθ (X1 = x1 , · · · , Xn = xn ). ÄkÚ\q, ¼ê Vg. ½Â 1. f (x, θ) = f (x1 , · · · , xn , θ)• X = (X1 , · · · , Xn ) VǼê. x ½ž, rf (x, θ)w ¤θ ¼ê, ¡•q,¼ê(Likelihood function),P• L(θ, x) = f (x, θ), θ ∈ Θ, x∈X (1.1) d?Θ•ëê˜m, X • ˜m. ¡log L(θ, x)•éêq,¼ê, P•l(θ, x). 51. q,¼êÚVǼê´Ó˜Lˆª(1.1), L«ü«ØÓ¹¿. rθ ½, òÙw¤ ½Â3 ˜mX þ ¼êž, ¡•VǼê; rx ½, òÙw¤½Â3ëê˜mΘþ ¼êž, ¡•q,¼ê. ù´ü‡ØÓ Vg. •)º4Œq, n, •Äe ˜‡{ü ¢~. ~1. -fpkNõç¥Úù¥. b½®•§‚ '~´1 : 3, Ø• ´ç¥õ„´ù¥ õ. •Ò´`Äј‡ç¥ Vǽö´1/4½ö´3/4. XJk˜£/l-f¥Än‡¥, ‡ ŠâÄ êâ, `²Ä ç¥ VÇ´1/4, „´3/4. ) òd¯K^ÚO .5Lã. -Xi L«1igÄ¥ (J,= 1 ÄÑ•ç¥ Xi = 0 ÄK. PzgÄ ¥Ä ç¥ VÇ•θ,d?θ • ŒU ü‡Šθ1 = 1/4, θ2 = 3/4ƒ˜. PX = n Xi ,KX ∼ b(n, θ).½= ©ÙxF = {Fθ1 , Fθ2 }, Ù¥Fθ1 •b(n, θ1 ), Fθ2 •b(n, θ2 ).‡Šâ i=1 Ä (JéθŠÑ O, =θ Š•1/4„´3/4? ½ 5goNFθ1 „´Fθ2 ? 1

2. w,, X ‰½ž, q,¼ê• n x L(θ, x) = θ (1 − θ)n−x , x = 0, 1, 2, · · · , n. x •{üO, n = 3. x = 0, 1, 2, 3žq,¼ê ŠXeL: x 0 1 2 3 L(θ1 , x) 27/64 27/64 9/64 1/64 L(θ2 , x) 1/64 9/64 27/64 27/64 Œ„ x = 0, 1ž, L(θ1 , x) > L(θ2 , x), x = 2, 3ž, L(θ2 , x) > L(θ1 , x). 3 Ïd·‚ Ñ(Ø: * Šx = i=1 Š•0, 1 ž@• xi 5goNFθ1 ,= ëêθ OŠ•θˆ1 = 1/4; x = 2, 3ž@• 5goNFθ2 , = θ OŠ•θˆ2 = 3/4. òd~ .zXe: e X = (X1 , · · · , Xn )•loNF = {Fθ , θ ∈ Θ}¥Ä {ü ‘Å , Ù¥Θ = {θ1 , θ2 }. d= d/`©ÙxF ¥•kü‡oNFθ1 , Fθ2 . ˜ ¼ x,XÛ^4Œq,•{¦Ñýëêθ OŠQ? þ~L²e L(θ1 , x) > L(θ2 , x), K·‚–•u@• X 5goNFθ1 (=ýëêθ•θ1 ) nd'@• X5goNFθ2 (= ýëêθ•θ2 ) nd•¿© . ½ö`, ýëêθ•θ1 /q,50•Œ . ù , ·‚g,r /q,50•Œ(=wå5•”) @‡ŠŠ•ýëêθ OŠ. ù ´/4Œq,0˜c d5. •˜„, e X ©ÙxF = {Fθ , θ ∈ Θ}, ëê˜mΘ•R1 k•f8½Ã•f8. x‰½ž, eθˆ ¦q,¼êL(θˆ∗ , x) •q,¼ê 8Ü{L(θ, x), ˜ƒ θ ∈ Θ}¥•Œö, = ∗ ëêθ ýŠ•θˆ∗ /q,50'θ Θ ¥Ù§Š /q,50•Œ, K /q,50•Œ θˆ∗ Š•θ OŠ, ù˜•{ ëêθ O,¡•/4Œq, O0. òù˜†*Ž{^êÆ Šó5£ã, Xe½Â: ½Â 2. X = (X1 , · · · , Xn )´lëê©ÙxF = {Fθ , θ ∈ Θ}¥Ä {ü‘Å , L(θ, x)´q,¼ê, e•3ÚOþθˆ∗ (X) = θˆ∗ (X1 , · · · , Xn ), ÷v^‡ L θˆ∗ (x), x = sup L(θ, x), x∈X, (1.2) θ∈Θ ½ d/¦ l θˆ∗ (x), x = sup l(θ, x), x∈X, (1.3) θ∈Θ K¡θˆ∗ (X)•θ 4Œq, O (Maximum Likelihood Estimation,{P•MLE). e– ¼ê ´g(θ),K½Âg θˆ∗ (x) •g(θ) MLE. 4Œq, O´R.A. Fisher31912c ˜‘óŠ¥JÑ5 . 3 ©Ùù‡AÏœ¹ e, ù•{ŒJˆ Gauss319-VÐ'u• ¦{ óŠ. Fisher 531922cóŠ¥, c Ù1925cuL 5Theory of Statistical Estimation6˜©¥éù˜ OŠ NõïÄ. Ïdù ‡•{A8õuR.A. Fisher. 2

3. !4Œq, O ¦{9~ ¼ ëêθ 4Œq, Oke ü«•{: 1. ^‡È©¥¦4Š •{ eq,¼êL(θ, x)´θ ëYŒ‡¼ê, K·‚Œ^‡È©¥¦4Š: •{ ¦θ MLE. é¦L(θ, x)ˆ •Œžθ Š. duL(θ, x)Úlog L(θ, x) = l(θ, x)äkƒÓ 4Š:, ·‚Œ ^l(θ, x)5“OL(θ, x). Ï•L(θ, x)˜„´eZ‡¼ê ¦È, l(θ, x) •eZ‡¼êƒÚ ´u?n. θ = (θ1 , · · · , θk )•ëê•þ(AO k = 1, θ•ëê). el(θ, x) 4ŒŠ3ëê˜mΘ S:?( š>.:)ˆ , Kd:7•q,•§| ∂l(θ, x) = 0, i = 1, 2, · · · , k ∂θi ). Ïd¦4Œq, OÄk¦q,•§ )θ. ˆ d)´Ä˜½´θ MLE Q? θˆ÷vq, •§, •´MLE 7‡^‡, š¿©^‡. ˜„•k÷ve ^‡: (i) q,¼ê 4ŒŠ 3ëê˜mΘS܈ , (ii) q,•§•k•˜), Kq,•§ƒ)θˆ7•θ MLE. Ïd·‚¦Ñq,•§(½•§|) ) , ‡ y§•θ MLE, kž¿š´¯. é ©Ùx´•êx |Ü, kš~÷¿ (J, QãXe: X = (X1 , · · · , Xn )•l,oN¥Ä {ü‘Å , X1 ©Ù••êx, = k f (x, θ) = C(θ) exp θi Ti (x) h(x), θ∈Θ i=1 d?Θ•g,ëê˜m, Θ0 •ΘƒS:8, ùžX éÜ—Ý• k n L(θ, x) = C n (θ) exp θi Ti (xj ) h(x), i=1 j=1 n Ù¥h(x) = h(xi ).éþª éê i=1 k n l(θ, x) = log L(θ, x) = n log C(θ) + θi Ti (xj ) i=1 j=1 ½n 1. eé?Û X = (X1 , · · · , Xn ),•§| n n ∂C(θ) =− Ti (Xj ), i = 1, 2, · · · , k C(θ) ∂θi j=1 3Θ0 Sk), K)7•˜, …•θ MLE. y²g´ dug,ëê˜m•˜à8, e•§|kü‡ØÓ ), θ0 Úθ1 , K E¼ê H(t) = l(tθ0 + (1 − t)θ1 ), 0 ≤ t ≤ 1. 3

4.duθ0 Úθ1 Ñ´•§∂l(θ)/∂θ = 0 ), ÏdH (0) = H (1) = 0, ¤±•30 < t0 < 1, ¦ H (t0 ) = 0. P θ˜ = t0 θ0 + (1 − t0 )θ1 K H (t0 ) = (θ0 − θ1 ) Q(t0 )(θ0 − θ1 ) = 0 2 Ù¥Q(θ) = ∂ ∂θ∂θ logC(θ) T = −V ar(T1 (X), · · · , Tk (X)) < 0, é?¿ ΘS:θ, þª†dA5gñ. Ïdθ0 = θ1 . ,˜•¡, eθ0 •˜‡), ^θ1 L«Θ¥?¿˜:, Kc¡ E ¼êH(t)÷vH (1) = 0, H (t) < 0. ÏdH (t) > 0, 0 ≤ t < 1. ¤±H(1) > H(0), =l(θ0 ) > l(θ1 ), θ0 •l 4Œ:. Ïde ©Ù••êx, •‡q,•§)áug,ëê˜m S:8(ù:éN´ y), KÙ)7•θ MLE. ~„ ©Ùx, X ‘©Ùx!Poisson©Ùx! ©Ùx!Gamma ©Ùx Ñ´•êx, ½n3.3.1 ^‡ ¤á. Ïdq,•§ ), Ò´k'ëê MLE. 2. l½ÂÑu eq,¼êL(θ, x)Ø´θ Œ‡¼ê,AO q,¼êL(θ, x)Ø´θ ëY¼êž, ·‚Ø Uæ^þã•{, 7L† l½Â3.3.2Ñu ¦ëêθ 4Œq, O. e¡·‚©O^þãü«•{• ˜ ~f. ~2. X = (X1 , · · · , Xn )´lü:©Ùx{b(1, p) : 0 < p < 1}¥Ä {ü , ¦pÚg(p) = p(1 − p) MLE. ) q,¼ê• n n xi n− xi L(p, x) = p i=1 (1 − p) i=1 , k n n l(p, x) = log L(p, x) = xi log p + n − Xi log(1 − p) i=1 i=1 éêq,•§• n n ∂l(p, x) 1 1 = xi − n− xi = 0, ∂p p i=1 1−p i=1 ) n 1 ¯ pˆ∗ = Xi = X. n i=1 duü:©Ùx••êx, pˆ = X ¯ •p ∗ MLE. §†~3.2.2¥‰ Ñ p Ý OþƒÓ. U½ÂŒ•g(p) = p(1 − p) MLE• ¯ − X). gˆ∗ (X) = pˆ∗ (1 − pˆ∗ ) = X(1 ¯ ~3. X = (X1 , · · · , Xn )´lPoisson©Ùx{P (λ) : λ > 0}¥Ä {ü , ¦λÚg(λ) = e−λ MLE. 4

5. ) q,¼ê,= X = (X1 , · · · , Xn ) ©Ù• n xi λ i=1 e−nλ L(λ, x) = P (X1 = x1 , · · · , Xn = xn ) = , λ > 0. x1 ! · · · xn ! éêq,¼ê• n n l(λ, x) = xi log λ − nλ − log xi ! . i=1 i=1 déêq,•§ n ∂l(λ, x) 1 = xi − n = 0, ∂λ λ i=1 ) n λ ¯= 1 ˆ∗ = X Xi . n i=1 duPoisson©Ùx´•êx, ˆ∗ = X λ ¯ •λ MLE, §†λ Ý OþƒÓ. qd½ÂŒ•g(λ) = e −λ MLE• ¯ gˆ∗ (X) = e−X . ~4. X = (X1 , · · · , Xn )´l ©Ùx{N (a, σ 2 ) : σ 2 > 0, −∞ < µ < ∞}¥Ä {ü , ¦a, σ Úg(θ) = a/σ 2 2 MLE, d?θ = (a, σ 2 ). ) X = (X1 , · · · , Xn ) ©Ù• n n 1 1 f (x, θ) = √ exp − (xi − a)2 . 2πσ 2σ 2 i=1 éêq,¼ê• n n n 1 l(θ, x) = log f (x, θ) = − log 2π − log σ 2 − 2 (Xi − a)2 . 2 2 2σ i=1 déêq,•§| n ∂l(θ, x) 1 = 2 (Xi − a) = 0, ∂a σ i=1 n ∂l(θ, x) n 1 =− 2 + 4 (Xi − a)2 = 0, ∂σ 2 2σ 2σ i=1 ) n ¯ 1 ¯ 2 = Sn2 . ˆ∗ = X, a ˆ∗2 = σ (Xi − X) n i=1 du ©Ùx••êx, ∗¯ Úσ ˆ =X a ˆ∗2 = Sn2 ©O´aÚσ 2 MLE,§‚•©O´aÚσ 2 Ý Oþ. cö´a à O, öØ´σ 2 à O. Œ„4Œq, Oؘ½äkà 5. qd½ÂŒ•g(θ) = a/σ 2 MLE• ¯ Sn2 . gˆ∗ (X) = X 5

6.~5. ‡ Æ·Ñle •ê©ÙEP (λ),Ù—Ý¼ê• λe−λx x>0 f (x, λ) = 0 x ≤ 0. X1 , · · · , Xn ©OL« ÉÁ n‡ ‡Æ·. dužm •›, Á ¢Sþ•?1 kr (r ≤ n)‡ ‡” žÒÊŽ , ±X(1) ≤ X(2) ≤ · · · ≤ X(r) Pùr‡ ‡ Æ·. =, · ‚•* X1 , · · · , Xn cr‡gSÚOþX(1) , X(2) , · · · , X(r) . Äuùcr‡gSÚO þ, ¦λÚg(λ) = 1/λ MLE. ) •Qã•B, Pti = x(i) , i = 1, 2, · · · , r,Kkt1 ≤ t2 ≤ · · · ≤ tr . •(½q,¼ ê, ·‚‡• þã* (J VÇ. ˜‡ ¬3[ti , ti + ∆ti ]S” VÇ•f (ti )∆ti , i = −λtr n−r 1, 2, · · · , r.Ù{n − r‡ ¬Æ·‡Ltr VÇ• e ,¤±þã(JÑy VÇCq• k(λe−λt1 ∆t1 )(λe−λt2 ∆t2 ) · · · (λe−λtr ∆tr )[e−λtr ]n−r , Ù¥k•,˜~ê. ј‡~êÏfé¦MLEÃK•. q,¼ê• r L λ, x(1) , · · · , x(r) = kλr exp −λ x(i) + (n − r)x(r) , i=1 Ùéêq,¼ê• r l λ, x(1) , · · · , x(r) = r log λ − λ x(i) + (n − r)x(r) . i=1 éλ¦ , q,•§• r ∂l r = − x(i) + (n − r)x(r) = 0, ∂λ λ i=1 ) r ˆ∗ = r λ X(i) + (n − r)X(r) . i=1 duq,¼êL λ, x(1) , · · · , x(r) ´•êx /ª, ˆ ∗ •λ λ MLE. d½ÂŒ•g(λ) = 1/λ MLE• r gˆ∗ (X(1) , · · · , X(r) ) = X(i) + (n − r)X(r) r. i=1 ~¥¤ã ¬Æ·Á ?1 1r‡ ¬” žÒªŽ, ù«Á ½ê —Á . ,˜«•ª´, k½e˜‡žmT > 0, Á ?1T žÁ ÒªŽ, ù«Á ‰½ž — Á . ~6. X = (X1 , · · · , Xn )´lþ!©Ùx{U (0, θ) : θ > 0}¥Ä {ü . (1) ¦θ MLE θˆ ; (2) `²θˆ ´Ä•θ à ∗ ∗ O. eØ,, Š· ? ¼ θ à Oθˆ1 ; (3) Áòθˆ1∗ †θ ∗ Ý O?1' , w=˜‡k ? (4)y²θ 4Œq, Oθˆ∗ = X(n) ´θ fƒÜ O. ) (1) X = (X1 , · · · , Xn ) ©Ù• 1 θn 0 < x1 , · · · , xn < θ f (x, θ) = (1.4) 0 Ù ¦. 6

7.Ï•þ!©ÙU (0, θ) | 8•6uθ,q,¼êL(θ, x) = f (x, θ)Š•θ ¼êØ´ëY¼ê, ÏdØU^é,¼ê¦‡û •{ ¦θ MLE. •UlMLE ½ÂÑu5?Ø. •¦L(θ, x)ˆ 4Œ, d(1.4)ªŒ„, A¦©1þ θ ¦ŒU/ , θqØU , ¦q,¼êC•0 . ù‡.•Ò 3 θˆ∗ = max(X1 , · · · , Xn ) = X(n) . Ïdθˆ∗ = X(n) Ò´θ MLE. (2) •¦E X(n) ,Ò‡ŽÑT = X(n) —ݼê, ´¦T —ݼê ntn−1 θn 0≤t≤θ g(t, θ) = (1.5) 0 Ù §, k θ ntn n E(θˆ∗ ) = E(T ) = n dt = θ, 0 θ n+1 θˆ∗ = X(n) Ø´θ à O. w„ n+1 θˆ1∗ = X(n) n •θ à O. (3) θ Ý Oθˆ1 = 2X ¯ ´θ à O. du D(θˆ1∗ ) = θ2 [n(n + 2)], D(θˆ1 ) = θ2 3n, ¤±3n ≥ 2žθˆ1∗ 'θˆ1 k . 3n = 1žθˆ1∗ = θˆ1 ,=ùü‡ O´ƒÓ . (4) ®•T —ݼêd(1.5)‰Ñ, k P (|θˆ∗ − θ| ≥ ε) = 1 − P (|θˆ∗ − θ| < ε) = 1 − P (θ − ε < T < θ + ε) θ ntn−1 1 ε n =1− dt = 1 − n θn − (θ − ε)n = 1 − . θ−ε θn θ θ Ïdk n lim P (|θˆ∗ − θ| ≥ ε) = lim 1 − ε/θ = 0, n→∞ n→∞ •θˆ∗ = X(n) •θ fƒÜ O. ~7. X = (X1 , · · · , Xn )´lþ!©Ùx{U (θ, θ + 1) : −∞ < θ < +∞}¥Ä {ü ,¦θ MLE. ) ‰½ x ž, θ q,¼ê• 1 θ ≤ x(1) ≤ x(n) ≤ θ + 1 L(θ, x) = 0 Ù§, ùž, q,¼ê• 1Ú0ü‡Š, •‡θ ≤ x(1) Úθ + 1 ≥ x(n) ÑŒ¦Lˆ 4Œ. θ MLEØ Ž˜‡, X θˆ1∗ (X) = X(1) , θˆ2∗ (X) = X(n) − 1 Ñ´θ MLE.¯¢þé?‰ 0 < λ < 1, θˆ∗ (X) = λθˆ1∗ (X) + (1 − λ)θˆ2∗ (X) = λX(1) + (1 − λ) X(n) − 1 Ñ´θ MLE, •θ MLEkáõ‡. 7

8. k ~8. k ‡¯‡A1 , A2 , · · · , Ak ¤ ¯‡+, ¯‡Ai u) VÇ•pi , i = 1, · · · , k… pi = i=1 1.òÁ Õá-Eng, ±Xi PAi u) gê, i = 1, 2, · · · , k,KX = (X1 , · · · , Xk ) Ñlõ‘ ©ÙM (n, p1 , · · · , pk ).¦p1 , · · · , pk MLE. ) Pp = (p1 , · · · , pk ).‰½ xž, p q,¼ê• k−1 xk n! n! xk−1 L(p, x) = px1 1 · · · pxkk = px1 · · · pk−1 1− pi . x1 ! · · · xk ! x1 ! · · · xk ! 1 i=1 éêq,¼ê• k k−1 k−1 l(p, x) = log n! − log xi ! + xi log pi + xk log 1 − pi . (1.6) i=1 i=1 i=1 épi ¦ ê, q,•§| ∂l(p, x) xi xk = − = 0, i = 1, 2, · · · , k − 1. ∂pi pi pk e- x1 x2 xk = = ··· = = λ, p1 p2 pk Kk xi = λpi , i = 1, 2, · · · , k. (1.7) òùk‡ ªü>©Oƒ\ k k n= xi = λ pi = λ. i=1 i=1 Ïdd(1.7)Œ•pi MLEXe: pˆ∗i = Xi /n, i = 1, 2, · · · , k. ~9. X1 , · · · , Xn •lXe©Ù¥Ä {ü §¦θ MLE. 1 1 f (x) = [θx (1 − θ)2−x + θ2−x (1 − θ)x ], x = 0, 1, 2; θ ∈ (0, ) x!(2 − x)! 2 ): dK •f (x)•lÑ.§Ù©ÙÆ• X 0 1 2 1 2 2 1 P 2 [(1 − θ) + θ ] 2θ(1 − θ) 2 [(1 − θ)2 + θ2 ] e† ld©ÙÑu§KØU θ MLE wªLˆ"•d§·‚-#ëêz§Pη = 2θ(1− θ). KdK •0 < η < 1/2"K X 0 1 2 1 1 P 2 (1 − η) η 2 (1 − η) 2Pni = #{X1 , · · · , Xn ¥ ui ‡ê}, i = 0, 1, 2, K q,¼ê• 1 1 1 L(η) = ( (1 − η))n0 η n1 ( (1 − η))n2 = ( (1 − η))n−n1 η n1 2 2 2 8

9.¦)¿5¿η e.= η MLE• n1 1 ηˆ = max{ , } n 2 √ 1− 1−2η 2dθ = 2 θ MLE• √ 1− 1 − 2ˆ η θˆ = 2 ~10. loN X 0 1 2 3 P θ/2 θ 3θ/2 1 − 3θ Ä ˜‡{ü X1 , · · · , X10 * Š•(0, 3, 1, 1, 0, 2, 0, 0, 3, 0)§ (1) ¦θ Ý OþθM Ú4Œq, OþθˆL§¿¦Ñ OŠ" ˆ (2) þã Oþ´Ä•Ã ºeØ´§žŠ? . (3) ' ? ü‡ Oþ§•Ñ@‡•k . 3k ¯K¥, p1 , · · · , pk Ñ´,˜ ëêθ1 , · · · , θr (r ≤ k) ¼ê, ùž K(1.6)¥ †θ1 , · · · , θr Ã' Ü© , éêq,¼ê• k l(θ1 , · · · , θr ; x) = xi log pi (θ1 , · · · , θr ) i=1 éθi ¦ ê q,•§| ∂l = 0, i = 1, 2, · · · , r. (1.8) ∂θi eq,•§ƒ)θˆ1∗ , · · · , θˆr∗ •θ1 , · · · , θr MLE, Kp1 , · · · , pk MLE©O´pˆ1 (θˆ1∗ , · · · , θˆr∗ ) · · · , pˆk (θˆ1∗ , · · · , θˆr∗ ). kžq,•§|(1.8)Ãwª), Ò•U^ꊕ{. n!4Œq, O 5Ÿ∗ c¡~f¥®•Ñ4Œq, Oؘ½´Ã . –uƒÜ5¯K, vkÝ O@o {ü. 4Œq, O ƒÜ5¯K, ÚåNõÚOÆö , , † y3ÑØU`®”.)û . 1946cCramer3˜ ^‡e, y² q,•§k˜Š´ëêθ fƒÜ O. duq,• § ŠØ˜½´4Œq, O, ù‡(J„´vk)ûq, O ƒÜ5¯K. † 1949c, WaldâÄgy² 4Œq, O rƒÜ5, ¤‡¦ ^‡éE,. d k˜ ÆöUY ?1ïÄ, F"3 ^‡ey²ƒÜ5. ù (J, ØB3d˜˜[ã. •3‡~, `²4Œq, OŒ±ØƒÜ, k, ÖöŒ wë•©z[2] P81 ‡~. e¡50 4Œq, O A^5Ÿ: 1. 4Œq, O†¿©ÚOþ X = (X1 , · · · , Xn )•goN{f (x, θ), θ ∈ Θ} ¥ÄÑ {ü‘Å , T = T (X1 , · · · , Xn )´ ëêθ ¿©ÚOþ, XJθ 4Œq, O•3, K§7•T ¼ê. dÏf©)½nŒ• X VǼê, ½=q,¼êŒL• n L(θ, x) = f (xi , θ) = g(T (x), θ)h(x). i=1 9

10.Ïd¦sup L(θ, x)ˆ þ(.ƒ:θˆ∗ , =•¦sup g(T (x), θ)ˆ þ(.ƒ:, §7•T (x) ¼ θ θ ê. d5Ÿ`²θ 4Œq, Oθˆ∗ = θˆ∗ (X1 , · · · , Xn ) ŒL•T (X) ¼ê=θˆ∗ = θˆ∗ (T (X1 , · · · , Xn )). X~3.3.2—~3.3.8¥ 4Œq, O •¿©ÚOþ ¼ê. 2. 4Œq, O†k O X = (X1 , · · · , Xn )•g©Ùx{f (x, θ), θ ∈ Θ} Ä {ü‘Å , eg(θ) k O gˆ(X) = gˆ(X1 , · · · , Xn )•3, Kg(θ) MLE gˆ (X) = gˆ (X1 , · · · , Xn ) 7†gˆ(X)-Ü, ∗ ∗ =gˆ∗ (X) = gˆ(X). ù˜5Ÿ y²˜3§3.5,=0 k O Vgƒ , Ùy²òd~3.5.11‰Ñ. 3. ƒÜìC 5 ·‚••Äëêθ•˜‘ œ/. F = {fθ (x), θ ∈ Θ} •˜VǼêx, Θ = (a, b)•R1 þ m«m. f (x, θ)÷ve ^‡: (1) 阃θ ∈ Θ, ê ∂ log fθ (x) ∂ 2 log fθ (x) ∂ 3 log fθ (x) , , ∂θ ∂θ2 ∂θ3 •3. (2) •3½Âu¢¶þ ¼êF1 (x)!F2 (x)ÚH(x), ¦é˜ƒθ ∈ ΘÚx ∈ R1 k ∂fθ (x) ∂ 2 fθ (x) ∂ 3 fθ (x) < F1 (x), < F2 (x), ≤ H(x), ∂θ ∂θ2 ∂θ3 Ù¥ ∞ ∞ Fi (x)dx < ∞, i = 1, 2; H(x)fθ (x)dx < M, θ ∈ Θ, −∞ −∞ ùpM †θÃ'. (3) 阃θ ∈ Θk 2 ∞ 2 ∂ log fθ (x) ∂ logfθ (x) 0 < I(θ) = E = fθ (x)dx < ∞. ∂θ −∞ ∂θ 'u4Œq, O ƒÜìC (CAN)5, ke (J: ½n 2. X = (X1 , · · · , Xn )•g÷vþã^‡(1)-(3) oN¥Ä {ü‘Å , … éêq,•§ n ∂ log fθ (xi ) =0 i=1 ∂θ k•˜Šθˆ = θ(X ˆ 1 , · · · , Xn ), θ0 •ýŠ, Kθˆ•θ0 CAN O. = √ L 1 n(θˆ − θ0 ) −→ N 0, , I(θ0 ) …θˆ•θ0 fƒÜ O. 10

11. n y²g´ ·‚kØ+^‡, PLn (θ) = 1 n i=1 log fθ (xi ), θ ∈ Θ, dŒêÆ• n 1 Ln (θ) = log fθ (xi ) −→ Eθ0 logfθ (X) = L(θ), n → ∞. n i=1 Ù¥L(θ) = Eθ0 log fθ (X) = log fθ (x)fθ0 (x)dx. w,L(θ)† Ã', …´•θ0 •Ù•ŒŠ :. ù´du L(θ) ≤ L(θ0 ), ∀θ ∈ Θ. … Ò¤á …= P (fθ (X) = fθ0 (X)) = 1. ¯¢þ, dué?¿ t > 0Ñklog t ≤ t − 1, Ïd fθ (X) L(θ) − L(θ0 ) = Eθ0 log fθ (X) − Eθ0 log fθ0 (X) = Eθ0 log fθ0 (X) fθ (X) ≤ Eθ0 − 1 = 0. fθ0 (X) Ïd, duθˆ•Ln (θ) •ŒŠ:, Ln (θ)Âñ L(θ), θ0 •L(θ) •ŒŠ:, ÏdθˆÂñ θ0 , =θˆ´θ0 ƒÜ O. ∂Ln (θ) éìC 5, Pl(θ) = ∂θ , dTaylorÐmª ˆ = l(θ0 ) + (θˆ − θ0 )l (θ) 0 = l(θ) ˜ Ù¥θ˜ ∈ [θ, ˆ θ0 ]. ¤± √ √ − nl(θ0 ) ˆ n(θ − θ0 ) = . ˜ l (θ) 5¿ d¥%4•½nk n √ 1 ∂ log fθ (Xi ) L 1 − nl(θ0 ) = − √ −→ N (0, ) n i=1 ∂θ θ=θ0 I(θ0 ) Ù¥I(θ) = E[ ∂ log∂θ fθ (X) 2 ] . , , dŒêÆ•é¤k θ ∈ Θk n 1 ∂ 2 log fθ (Xi ) ∂ 2 log fθ (Xi ) l (θ) = −→ E = −I(θ) n i=1 ∂θ2 ∂θ2 m> ª´Ï• 2 ∂ 2 log fθ (Xi ) f f = − ∂θ2 f f ±9Ef = 0. Ïd,(Üθˆ → θ0 ±9θ˜ ∈ [θ, ˆ θ0 ] • ˜ −→ −I(θ0 ). l (θ) l dSlutsky½n y. ~11. X1 , · · · , Xn ´ g oNN (µ, σ 2 ) . y²~3.3.4‰Ñ aÚσ 2 MLE©O äkìC 5. 11

12. y (1) w, ©Ù÷v½n3.3.2 ^‡(1)-(3). Ïdd~3.3.4®•µ MLE´µ ˆ = ¯ du X. 1 (x − µ)2 f (x, µ) = √ exp − , 2πσ 2σ 2 k 1 1 (x − µ)2 log f (x, µ) = − log 2π − log σ 2 − . 2 2 2σ 2 ¤± ∂ log f (x, µ) 2 x−µ 2 1 I1 (θ) = E =E = , ∂µ σ2 σ2 l d½n3.3.2Œ• √ L µ − µ) −→ N (0, σ 2 ), n(ˆ =µˆCqÑlN (µ, σ 2 /n). (2) d~3.3.4Œ•σ 2 MLE´σ ˆ 2 = Sn2 . du ∂ log f (x, σ 2 ) 2 1 (x − µ)2 2 1 I2 (θ) = E =E − + = , ∂σ 2 2σ 2 2σ 4 2σ 4 d½n3.3.2Œ• √ L σ 2 − σ 2 ) −→ N (0, 2σ 4 ), n(ˆ =σˆ 2 CqÑl ©ÙN (σ 2 , 2σ 4 /n). 12