The Strong Consistency of Regression Quantiles Estimators in Nonlinear Censored Regression Models

전체 글

(1) . Æ

(2) .

(3). . 157. Journal of the Korean Data & Information Science Society 2002, Vol. 13, No. 1, pp. 157 ∼ 164. The Strong Consistency of Regression Quantiles Estimators in Nonlinear Censored Regression Models Seunghoe Choi. 1. Abstract In this paper, we consider the strong consistency of the regression quantiles estimators for the nonlinear regression models when dependent variables are subject to censoring, and provide the sufficient conditions which ensure the strong consistency of proposed estimators of the censored regression models. one example is given to illustrate the application of the main result. Key Words: Nonlinear Censored Regression Model, Regression Quantiles Estimators, Strong Consistency.. 1.. . , , .

(4) .

(5). . , yt = f (xt , θo ) + t ,. t = 1, · · · , n. (1.1).

(6) . .

(7) (x ) (y )

(8) Æ ! "# $% . ).

(9). .

(10) . & '

(11) .

(12)

(13) Æ θ &'

(14) (

(15) % 3 ( )Æ

(16) + ,4

(17) 5 6 .

(18)

(19) Æ% * + , .. /0 1, 2 (robustness) 7$% &'

(20)

(21) 8 9 + ,- ). (Wu(1981), Wang(1995), Procházka(1994), Choi et al.(2001) 4 ). : ; , !

(22) . Jureˇ ckov´ a " # $<= !

(23) " % !% ! > % 7 $).

(24) ". . .

(25)

(26) Æ% $##%89 :; ?@Æ > % ' & A B$ % % &

(27) . & . % . C%

(28) ./#$

(29) ./#$ DE F GH . F I () & . ' , (J '

(30) . 4 ) '

(31) Æ%

(32) # $K.

(33) *+# AB #$ L " ,% 8 9 (. & . # + . 6 . I MN )*+ ,> - +. /

(34) . .

(35) F GK I >

(36) . . )

(37) + #+

(38)

(39) %&% 1( F I () & . 5 6 # + A B L " ,% 8 9 .

(40)

(41). +. O

(42)

(43) .

(44) .

(45) + $ )C P QR '. (. &

(46) &'

(47)

(48) Æ +

(49) ,

(50) .

(51) . & '

(52) .

(53) . . + , " +S T $. * 4 ) ' C%=<, #+

(54) I () & . t. t. o. t. 1 Assistant. Professor, Department of General Studies, Hankuk Aviation University, Koyang, Kyungkido, 412-791, Korea E-mail : [email protected].

(55) . . 158. +

(56) )%,- '

(57) # + . (Nonlinear Censored Regression Model, . NCRM). yt = min{ct , f (xt , θo ) + t },. t = 1, 2, · · · , n. (1.2). +. 0 ). x Γ ⊂ R

(58) % . 8 9

(59) . /U . V8 9 θ = (θ , · · · , θ ). Θ⊂R

(60) % W> ./UV). :;H89 . /% , , f. R × Θ

(61) R . /# $I . , 3 #$ 89 ' L " I > . . $# F G. 23

(62) .. I 10, 2 ) . y # $K*+#C

(63) c

(64)

(65) Æ% . ,. +# A B ) . X /,)% . (1.1)

(66) L " , & '

(67) %. . +. / 6 # + .

(68) & 12H #+ .

(69) . A B / ) , " # $). . 5 B A L " ,%89 &(

(70) )% - , ' Y Z# $( q. t. p. q. 1. o. 1. p. t. t. t. yt∗. . =. f (xt , θo ) + t , if yt < ct (uncensored) if yt ≥ ct (censored) ct ,. > C ,(indicator function) . δt =. 0, if yt < ct 1, if yt ≥ ct. % % & (1.2)

(71) "L, & '

(72) %. . ) . #+

(73) Æ & '

(74) %. ![ \0 +H % & . ] 3^ _0 1(Leaest Powell(1984) '

(75) 4

(76) 05,./# $ LS& '

(77) .

(78) . . /$# ] 3^ _* + 1 3 2 (Least Squares, LS)& '

(79) % 98 ) % - , , Absolute Deviation, LAD)& 1 1 Rn (θ : ) = |yt − min{ct , f (xt , θ)}| 2 n t=1 n. ] 3^ _ 6% θˆ ( ) ]3^_*+123&'

(80) . 6 % % () 2 . Chen Wu(1994).

(81). +# θ

(82) 3 % 4 . * $. 4 ) ' C% % (8 2 9 Choi. .

(83) LAD&'

(84) Kim(1998) # $K

(85) *+# AB L" ,% 8 9 (. &

(86). .

(87) . LAD& '

(88) .

(89) Æ +

(90)

(91)

(92) +O

(93)

(94) % +

(95) ,% % () 2 . 3 ) LAD& '

(96) . 2 3 Æ

(97) + ,4I ` a b c4

(98) . , LS& '

(99) . ' 2 3

(100) Æ+ ,4I 'J H X=Æ( F LS&'

(101)

(102) % 3

(103) Æ + ,4I . + F . # $!.

(104) < = . + F$

(105)

(106)

(107) + $) ). 2 '

(108) 7$) de+ ! <=. & '

(109) . D Eb c4

(110) . + () & . LAD& . . L"

(111) LAD&'

(112) 7$ % ![ \ Koenker Basset(1978). d e+ !

(113).

(114) +.-/6% )%,- , 1 n 2. o. Rn∗ (θ : β) =. 1 ρβ (yt − min{ct , f (xt , θ)}) n t=1 n. (1.3). ] 3^ _ 6% θˆ (β) !(Regression Quantiles, RQ)& '

(115) . 6 % % () 2 . , ρ (·) n. β. ρβ (ω) =. βω, (β − 1)ω,. ω≥0 ω<0. 8 9 0 < β < 1 ).

(116)

(117) . RQ& '

(118) .

(119) Æ

(120) + Powell(1986) Port

(121)

(122) Æ% +, -%&(./01 Jureˇckova Procházka(1994) : ;H 98 Choi Kim(2001). noy(1991)

(123) .

(124) + + ,% % () 2 .

(125).

(126) RQ&'

(127)

(128) Æ

(129) .

(130) . Æ

(131) .

(132). . 159. +

(133) # $K*+#C

(134)

(135) ,. # + BA $# L " ,% 8 9 (. & Æ & '

(136) %. !% !& '

(137) . ^ _

(138). +# (1.2)

(139)

(140) 98 !& '

(141) . Z%98 &'

(142)

(143) Æ +

(144) $)% . * $. 4 ) ' ) . ;:H

(145) Æ +

(146) 5 +%

(147) *+ f C . ) . 2.. Æ. * + +

(148) +#

(149) !& '

(150) .

(151) Æ +

(152) $ )% . * $. 4 ) ' % % () 2 . +

(153) %& g h % # * % !% ) % - , ' g h

(154).

(155) Æ% % () 2 . ft (θ) = f (xt , θ), at (θ) = ft (θ) − ct , bt (θ) = ft (θ) − ft (θo ), ∂ ∂2 ∇ft (θ) = ft (θ) , ∇2 ft (θ) = ft (θ) . ∂θi ∂θi ∂θj (p×1) (p×p). '

(156) θˆ (β)

(157) Æ +

(158) !% . # + . (1.2)

(159) % )% -4 , ) ' I

(160) . RQ& % . ÆA % Θ R

(161) Æ 67/ij (compact) J ' 8 98 Γ. R

(162) ÆF G 8 ) . A : A : % Θ

(163) % θ

(164) [\ ∇f (θ) ∇ f (θ)I 5 k \% 8 9, ∇f (θ). Γ×Θ

(165) . ) . 8 D (θ) = {t : f (θ) = f (θ ), θ = θ }

(166) Æ ^ _9 : d (θ) ; l 6 m BA n

(167) Æ A : , ) I , d(θ)

(168) 34%89 0 < d(θ) < 1 ) . ÆB 3

(169) L", G(x)89, +K , g(x) = G (x). η = G (β)

(170) Æ B :2

(171) . ) < 10 ). e+ d ! Z=8

(172) +.-/6. RQ& '

(173) . β

(174) 5 6 . ) . R ' β

(175) 5 6 RQ& '

(176) . 8 n% . :K F G & ,. /

(177) 5. 6 RQ&'

(178) +,% $# )Ǳ β $) # % - , ' f (x, θ) = θ + f˜(x, (θ , · · · , θ )) +1 * 2 I >

(179) , .-/%,,#$ 9. . +% 0 . : RQ& '

(180) .

(181) Æ ) , > D E8 ) :2 5 +" &(). ![\ 9 n. p. 1. q. 2 3. 2. t. dn (θ) n. n. t. t. o. t. t. o. n. . 1. 1. 2. β. −1. p. 1 ρβ (yt − ηβ − min{ct , ft (θ)}) n t=1 n. Rn (θ : β) =. (2.1). % 6 . : ;? R (θ : β) ]3^_#$ % (1.3)

(182)

(183)

(184). Æ RQ& '

(185) . $# & .# / $ R (θ : β) R (θ : β)#$ % RQ& '

(186) .

(187) Æ:

(188) . , # $ % & " () & . % ) -

(189) , H & (2.1) . . ∗ n. n. n. Qn (θ : β) = Rn (θ : β) − Rn (θo : β).

(190) . . 160.

(191) Æ <. ; + 5 +%89 &().

(192) Æ 2.1

(193) #+ (1.2) I

(194) A, B. $ )% ? Q (θ : β) − E[Q (θ : β)]I 0. /# $ 34 ). ,), Q (θ : β) − E[Q (θ : β)] −→ 0 ) . Æ n. n. n. a.s.. n. Vt (β, θ, θo ) = ρβ (yt − min{ct , ft (θ)} − ηβ ) − ρβ (yt − min{ct , ft (θo )} − ηβ ). % 6 ? Chebyshev J'& . max V ar[Vt (β, θ, θo )]. 1≤t≤n. P {|Qn (θ) − E[Qn (θ)]| > } ≤. n2. ..

(195) ÆW ,./#$

(196) H 2.1 ,% ![ \ . V (β, θ, θ )

(197) Æ ; GF. , 7 $ 'R? * $. %. ). , ρ (·)

(198) Æ =3)

(199) ./#$J'UV t. o. β. ρβ (x + y) ≤ ρβ (x) + ρβ (y). F GK " &(). 56 #> ; + # $) % - , ' .  0,    ρβ (ft (θo ) − ft (θ)), Vt (β, θ, θo ) ≤ ρβ (ft (θo ) − ct ),    ρβ (ct − ft (θo )),. on on on on. Ω1 Ω2 Ω3 Ω4. = {x ∈ Ω : ct < ft (θ), ct < ft (θo )} = {x ∈ Ω : ct > ft (θ), ct > ft (θo )} = {x ∈ Ω : ft (θ) < ct < ft (θo )} = {x ∈ Ω : ft (θo ) < ct < ft (θ)}. + % ' &(). ( tI Ω !

(200) " & Ω

(201) % . F . 3. 4. ρβ (ct − ft (θo )) ≤ ρβ (ft (θ) − ft (θo )), ρβ (ft (θo ) − ct ) ≤ ρβ (ft (θo ) − ft (θ)). o /$# ) . 5 6 τ. Vt (β, θ, θo ) ≤ ρβ (ft (θo ) − ft (θ)) β. 6 %? . = max{β, 1 − β}. Vt (β, θ, θo ) ≤ τβ |ft (θo ) − ft (θ)|. ) . : ;/o$#, ! & Hölder 'J&. /# $'JUV. ¯ Vt (β, θ, θo ) ≤ τβ ∇ft (θ) θ − θo . (2.2). F GK " &(./10 θ¯ = λθ + (1 − λ)θ, 0 ≤ λ ≤ 1 8 9, .. F G? H p/ q 3 - , d e ;/o$# & (2.2) I

(202) A#$J'U V V (β, θ, θ )

(203) Æ ; F G. , " () & . C. ) . : # + (1.2)

(204) %, -<

(205) C . I

(206) A, B.

(207) ) % -I ,

(208) . & 'I % . ÆC o. t. o.

(209) . Æ

(210) .

(211). . 161. : 8 C (θ ) = {t : f (θ ) < c }

(212) Æ ^ _9 : q (θ ) ; ml 6 BA n

(213) Æ , ) , q(θ )

(214) 34%89 0< q(θ ) ≤ 1) I . @# " A3 V (θ ) = ∇f (θ )∇f (θ ). )

(215) < Æ

(216) . B! = @3 " A V (θ )# # $ 3 . 4 ) C :!. C1 qn (θo ) n. n. o. t. o. o. 2. t. n. o. o. n. 1 n. o. t. T t. o. o. o. t∈Cn (θo ).

(217) !

(218)

(219) Æ # $K. * +# C

(220) r.

(221) # +s. " ( & % I >8 9 ! % =< 3 ) " F . . I. % . : # $ST ). * $ C q(θ ) = 1 ./$# I

(222) C

(223) .

(224) R '

(225) 1@ . 4 ) . ' /# $ . 8 9 + .

(226) C % r

(227) " &(

(228) Æ <* ;

(229) . F (q(θ ) = 0). .

(230) .

(231) 8% > &(). 56 I

(232) C

(233) C. : ' # $K. * +# C

(234) r.

(235) " (. &

(236) Æ ;6lm AB

(237) Æ . +

(238) . 4 F G>% . . ) . % ) -

(239) , H +# (1.2)

(240) . θ

(241) . !& '

(242) . Æ

(243) +

(244) 5 +. . ). Æ 2.2

(245) +# (1.2) I

(246) A, B. C $ )% ? !& '

(247) . θˆ (β). . (θ + η , θ , · · · , θ )# $ 34 ). ,), 2. o. o. 1. o. n. 1. β. 2. p. a.s. θˆn (β) −→ (θ1 + ηβ , θ2 , · · · , θp ). ) . Æ RQ& '

(248) θˆ (β)

(249) Æ +

(250) 7$R' ![. \

(251) & ÆÆ

(252) δ > 0

(253) % ) % - , & n. lim. inf. n→∞ θ−θo >δ. {Rn (θ; β) − Rn (θo ; β)} > 0 a.e.. I

(254) , 7$'R10 * $ %).

(255) H2.1

(256) Qn (θ : β) = E[Qn (θ : β)] + o(1). o /$# θ I o. Q(θ : β) ≡ lim E[Qn (θ : β)]. Æ

(257) <^ ; _ & 7$'R. Ω

(258) % t

(259) . V (β, θ, θ )

(260) Æ F (θ) 6% 98 F +%? , . n→∞. i. β(at (θo ) − bt (θ)) −. bt (θ)+ηβ. −∞. t. (at (θo ) − bt (θ)) −. 8 9, I

(261) B %&% %#*

(262) H%?. . ηβ. bt (θ)+ηβ. o. (at (θo ) − bt (θ)) +. ) . A ./$# F. . ti. at (θo ). at (θo ). bt (θ)+ηβ. bt (θ)+ηβ. (at (θo ) + ηβ ) dGt (λ). (λ − at (θo ) − ηβ ) dGt (λ). t4 (θ) 

(263) ∞ at (θo )  (a (θ ) − λ) + η dGt (λ), t o η ηβ β β  at (θo ) (at (θo ) − λ)dGt (λ), ηβ. at (θo ). λ+ bt (θ)+ηβ. . t3 (θ). ηβ ≥ 0 ηβ < 0.

(264) . . 162. 89, F . . t2 (θ)  η +b (θ) t β  (bt (θ) − λ + ηβ )dGt (λ),  

(265) η β ∞ at (θo ) (bt (θ) − λ + ηβ ) + at (θo ) (bt (θ) − at (θo ) + ηβ ) dGt (λ), η β  ∞   b (θ)dGt (λ), ηβ t. at (θo ) > bt (θ) ηβ < at (θo ) < bt (θ) + ηβ at (θo ) < ηβ < bt (θ) + ηβ. ) . ! & b (θ)I 0$7) ? F

(266) ; . + 01 07 $) + 0 F . A. .. / $+ # , " &(.//o$# >? ).. 1 Ω 2 ! "

(267) Ω

(268) % . t

(269) % θ

(270) . E[V (β, θ, θ )]

(271) Æ 1 W t. 2. t. 3. o. ηβ +bt (θ). ∇ft (θ). 8 9, 2 W . dGt (λ) ηβ. ηβ +bt (θ). dGt (λ)∇2 ft (θ) + gt (ηβ + bt (θ))∇ft (θ)∇T ft (θ). ηβ. ) . # tI Ω

(272) " F a (θ ) > b (θ) F $ +% 0 % () 2 . a (θ ) < b (θ) F . . t+ ," &(89,

(273) Æ θ I <^ ; _ & , % . u v<. /$#. % & > /o . /$# >?%%2(). >;B, k (β) = min g (η )6 % ? 2. t. o. t. t. o. t. o. n. ∇2 E[Qn (θo : β)]. 1≤t≤n. 1 n. =. . β. gt (ηβ )∇ft (θo )∇T ft (θo ). t∈Cn (θo ). kn (β) n. ≥. t. . ∇ft (θo )∇T ft (θo ),. t∈Cn (θo ). 8 9, ∇E[Q (θ : β)] = 0o/#$ I

(274) C

(275)

(276) Æ%

(277) Æ θ. Q(θ : β)

(278) Æ <^ ; _ ). ) % -./$# lim E[Q (θ : β)]I N (θ ) ≡ {θ : θ − θ ≥ ζ} ∩ Θ ,

(279) ) < & 7 $ '. I R

(280) A . n. o. o. n. n→∞. ζ. EQn (θ : β) =. o. o. inf. θ ∗ ∈Nζ (θo ). E[Qn (θ∗ : β)]. $ )% θ

(281) Æ 5 k\

(282) 5 + ). :;H 89 t ∈ Ω +? @ 0 < b (θ) < a (θ ) o /$# 2. ηβ +bt (θ). Ft2 (θ) >. t. t. o. (bt (θ) − λ + ηβ )dGt (λ). ηβ. ) . η η η /# o $ ∗ β. β. = gt (ηβ∗ )[Gt (ηβ + bt (θ)) − Gt (ηβ )] ≡ ηt2 > 0.

(283) Æ r

(284) . ) . ( t∈Ω +? @ b (θ) < a (θ ) < 0 . β + bt (θ). at (θo ). Ft3 (θ). ≥. ηaβt (θo ). (λ − at (θo ) − ηβ )dGt (λ),. ηβ +at (θo ). ≡ ηt3 > 0. 3. (λ − at (θo ) − ηβ )dGt (λ),. t. if ηβ < at (θo ) < 0 if at (θo ) < ηβ < 0. t. o.

(285) . Æ

(286) .

(287). . 163. 8 9, t ∈ Ω + ?@ 0 < a (θ ) < b (θ)o/# $ 4. t. o. t. at (θo ). Ft4 (θ) ≥. ) . 5 6 η. t. (at (θo ) − λ)dGt (λ) ≡ ηt4 > 0. ηβ. 6 %?, I

(288) C.

(289)

(290) Æ% . ≡ {ηt2 , ηt3 , ηt4 }. lim. inf. n→∞θ∈Nζ (θo ). E[Qn (θ : β)] ≥ 3q(θ)η. o /$# !&'

(291) θˆ (β) θ

(292) 3 . 4 ) . η= n. o. min. 1≤t≤qn (θo ). ) .. {ηt }. $ * +) G( ) = 0? RQ&'

(293) LAD& '

(294) . . /# $,

(295) H. 2.2. NCRM

(296) . LAD& '

(297) .

(298) Æ +

(299) 5 +%89 &(). % ) - , f !&'

(300)

(301) Æ +

(302)

(303) %. & . . 5 +. % C ) . . 1 # + (1.2)

(304) Æ .-/%,, 1 2. f (xt , θ) = θ1 + θ2 eθ3 xt. 8 9, x L", H(x)#$ I>. L "

(305) D ! +% t 7

(306) /C% (. & /0 12 3 Æ

(307) L ", G(x) I

(308) B $) ) 8 9% . : ;H 98 θ. +

(309). I >0 1

(310) . ) 8 9% . (. I

(311) A,.

(312) Æ θ Θ = [0, a ] × {k} × [0, a ], a , a < ∞

(313) . )98 I

(314) %? C t. 2. 1. o. 3. 1. 3. 2. 1. Vn (θo ). =. 1 ∇ft (θo )∇ftT (θo ) n t∈Cn (θo )  qn (θ) 1.  =  . . q(θ). n 1 n. . n θ3 xt. kxt e. t∈Cn (θo ). dH(x). θ3 x kxe dH(x). 1 n. kxt eθ3 xt. t∈Cn (θo ). . . θ3 xt 2. kxt e.   . . t∈Cn (θo ). kxeθ3 x dH(x) 2 kxeθ3 x dG(x). $ # 3 . 4 ) . DEF α = (α , α ) = (0, 0)

(315) % 1. 2. T. αV (θ)α = q(θ). . α1 + α2 kxeθ3 x. 2. dH(x) > 0. o /$# !&'

(316)

(317) Æ +

(318) $ ). ) .. . 1. Chen, X. R., and Wu, Y. (1994). Consistency of L1 Estimates in Censored Linear Regression Models, Commmnication in Statistics, 23, 1849-1858..

(319) 164. . . 2. Choi, S. H., and Kim, H. K. (1998). Asymptotic Properties of LAD Estimators in Censored Nonlinear Regression Model, Journal of the Korean Statistical Society, 27, 101-112. 3. Choi S. H., Kim H. K., and Park, K. O. (2001). Nolinear regression quantiles estimation, Journal of the Korean Statistical Society, 30, 551-561. 4. Jureˇ ckov´ a, J. and Proch´ azka, B. (1994). Regression quantiles and trimmed least squares estimators in nonlinear regression model, Nonparametric Statistics, 3, 201222. 5. Koenker,R., and Bassett, G. (1978). Regression Quantiles, Econometrica, 46, 33-50. 6. Portnoy, S. (1991). Asymptotic behavior of regression quantiles in non-stationary dependent cases, Journal of Multivariate Analysis, 50, 100-113. 7. Powell, J. L. (1984). Least Absolute Deviation Estimation for the Censored Regression Model, Journal of Econometrics, 25, 303-325. 8. Powell, J. L. (1986). Censored regression quantiles, Journal of Econometrics, 32, 143-155. 9. Wang, J. (1995). Asymptotic normality of L1 -estimators in nonlinear regression, Journal of Multivariate Analysis, 54, 227-238. 10. Wu, C. F. (1981). Asymptotic theory of nonlinear least squares estimation, The Annals of Statistics, 501-513..

(320)