27 5 998 0 ADVANCES IN MA THEMA TICS Vol 27,No 5 Oct, 998 (,,00080, ), () M, M ; (2) PP ; (3), t T 2 M x 2, ; M ; ; ; ; MR (99) 62F35 ; 62F03 ; 62F0,,, 60, P J Huber F R Hampel, 80 Huber [ 8 ] Hampel [ 4 ] 80,, ( ),, M, [ 22 ], [23 ] 80 R Fisher J Neyma E S Pearso ( ) (,,,, ),,,? Box [3 ] Tukey [37 ],, Hampel [3 ], 0 % ( ),, Mask,, 996-09 - 02 997-0 - 6 995-2004 Tsighua Togfag Optical Disc Co, Ltd All rights reserved
404 27 2 I), ; II) ; III) ( Hampel [, 2 ] (i) Huber [6 ] B ( ), 0 Φ<, T, F 0, C ( F 0, ) F 0 B ( ) = B (, F 0, T) ƒ sup F C( F 0, ) T ( F) - T ( F 0 ) C ( F 0, ) F 0 B ( ) T B ( ) 0 Φ< T, (ii) Hampel [ ] T (, ), T, F 0, x x, T F 0 I F( x ; F 0, T) ƒ lim 0 T ( ( - ) F 0 + x ) - T ( F 0 ), r 3 = sup x I F( x ; F 0, T), r 3, T ( r 3 = B (0) ƒ 9B ( ) 9 = 0 ) r 3 T, B (iii) Hampel [ ] (, ), B ( ), 3 ƒ 3 ( F 0, T) ƒ if > 0 B (, F 0, T) = +,, Dooho Huber [9 ], Zhag Li [50 ] (a) R X m m m, R (b) A X = ( x, x 2,, x ) m m, m + m A (c) S R, ;,, A, R S R m 995-2004 Tsighua Togfag Optical Disc Co, Ltd All rights reserved
5 405, A V ( X) = i = x i/, x i, A V ( X) ( ), A V ( X) R = /, A = / ( + ), S R = /,, ;,,,,,, 2 2 M,,,,,, Davies [6 ],, Lopuhaa [26 ], / / 2, Hampel [4 ],,,,, Zhag Li [50 ] M M,, Zhag Li [50 ],,,, X = ( x, x 2,, x ) F, R M T ( X) ( x i - T ( X) ) = mi! (2 ) i = T ( F) M, (( x - T ( F) ) - ( x) ) d F( x) = mi! (2 2) T ( X) = sup{ t t T ( X) } M, ( B) ( x) x = 0 - ; x < 0, ; x > 0, ; x ( x) 0 995-2004 Tsighua Togfag Optical Disc Co, Ltd All rights reserved
406 27 ( U) ( x) x = 0 0 ; 0 ; x > 0, ; x ( x) ; = R, x 0 > 0 (0, x 0 ], ( x 0, ) ( C ) ( x), = ( - ) < 0 < ( + ) - Φ Φ 0,, M ( B ) Huber M ( U ) ( C ) M, Zhag Li [50 ] 2 ( U ),, A = / 2, S R = R = g( + ) / 2, g b b 22 ( B ) N = {,2,, }, I k = { i, i 2,, i k }, N, Φ s Φ k, Φ k Φ X = ( x, x 2,, x ), B k A k = sup t A 0 = 0, B 0 = 0, k - ( x i - t), k =,2,,, i = = mi sup - ( x i - t), k =,2,,, I k < N t i I k m = mi k k Ε A - k, r = mi{ k k Ε B - k } a A g Φ a Φ g A + ( [ b ] b ) A = a + a, m Φ S R Φ m +, r Φ R Φ ( r + ), 0 < c < +, x > c ( x) = 0, a = A g, SR = m, R = r i s 23 ( C ),, - ( - S R = R = ) ( S A = mi, ) ; ( ) - ( - ) } ( ) - ( - ), A = +, S R = R = 24, (i) ( B ), 3 ( F, T) = A + A, A = sup T R ( - ( x - ) ) d F( x) (ii) ( U ), x d F( x) < +, 3 ( F, T) = 2 (iii) ( C ),, 3 ( F, T) = 0 ;, 3 ( F, T) - ( - ) ( = mi{, ) ( ) - ( - ) ( ) - ( - ) } Hampel ( [4 ], p 66) M 3 = / 2 24 (i), F, 3 0 < 3 < 2 2-23, ( B ), 995-2004 Tsighua Togfag Optical Disc Co, Ltd All rights reserved
5 407 25 ( ) ( B ), F, S R A 3 26 ( B ), ( - 2 ) 0 0, F = { - (- ) R } F W / 2 ( A - 3 ) / 2 ( S R - 3 ) d ( + A ) 2 sup T ( F) d ( + A ) 3/ 2 sup T ( F) d (( x - ) - ( x - 2 ) ) 2 d F( x) W ( - (- ) ), W ( - (- ) ), F f ( x - ), f ( x), x Ε 0,, T ( F) = { } 2 5 2 6,, m, Tukey biweight c 3 5 6 5, AVS S R ; 3 ( ) = mi{ j Pr { S R Ε } Ε 0 95, Ε j } ; A (, T), A ( M N, T) biweight, Slash ; Tukey biweight M ( x) = - ( - ( x/ c) 2 ) 3 I { x/ c Φ },, I, c Ε 0 c 3 AVS 3 (004) 3 (045) A (, T) A ( SL, T) A ( M N, T) r 3 0 0257 0590 985 748 3 553 5 0328 0043 373 269 428 297 20 0376 0029 24 43 247 23 25 0439 009 644 55 966 8 79 30 0430 002 9 29 749 5 67 35 0447 0008 9 7 638 38 64 40 0458 0005 230 0 576 3 67 45 0466 0004 0 06 54 28 74 50 0472 0003 0 04 52 27 82 55 0476 0002 03 5 27 92 60 0480 000 02 507 28 203 65 0483 000 04 508 29 25 70 0485 000 00 52 3 227 75 0487 0006 008 58 33 239 80 0489 0005 006 526 34 252 2 2 PP I) Huber [6 ], 995-2004 Tsighua Togfag Optical Disc Co, Ltd All rights reserved
408 27 II) Hampel [ ] B r 3, III) Rousseeuw [33 ] Yohai [38 ] Huber Hampel, Davies [7 ], ( ) Rousseeuw Yohai, ( Rousseeuw [32 ] ), Rousseeuw Croux [34 ] k M S k, k Ε 3, S k (50 %), ( Ε 95 % ), ( r 3 2 69) B k ( ) ƒ sup G S k ( ( - ) F 0 +G) k,, ( ), 3,, 3 3 ( h) = if{> 0 sup G T ( ( - ) F 0 +G) Ε h 0 }, h 0, h 0, h Ε h 0 3 ( h) - 3, 3 3 ( h 0 ), 3 3 ( h 0 ),, 3, 90, Huber [6 ], ( 80), Yohai Maroa [39 ], 80, Stahel [36 ] Dooho [8 ],, / 2, 2,, Li Che[24 ],, / 2,, Maroa [28 ],, / 2, Stahel - Dooho X EC (, ρ ), A A = ρ -, A X,, A, A A X Zhag [46 ],, ( - ) < 0 < ( + ), X = ( x, x 2,, x ) M T ( X) ( x i - T ( X) ) = 0 i =, ( i = x i - t) t M T ( X) t X = ( x, x 2,, x ) p F 0 (- ), PP, sup a = t, ( i = a ( x i - t) ) ( z) = Sg ( z) ( ), g 995-2004 Tsighua Togfag Optical Disc Co, Ltd All rights reserved
5 409 arg if t sup a = g Sg ( a ( x i - t) ), i = arg if ρ sup a = Sg ( a ( x j - x i ) ) i < j a ρ a - F 0 - (3/ 4) ), ρ, F 0 g arg if t sup a = ^S, a X = ( a x, a x,) ( a ( x j - t) ) i = ^S ( a ) X) ) g arg if ρ sup a = i = ( a x i - ^( a ) a ρ - ), a ^, R, PP (i), Fisher ; (ii) 3 / 3 ; (iii) O p ( ) ; (iv), Yu [4 ] Cui [5 ] Zhag [45 ] PP 3 3 (i) Huber [7 ] x, x 2,, x i i d H 0 F = F 0, H F = F i F i ( i = 0,), P P 0 = < Sup F P0 E FΦ a, Huber if F P E F (ii) Reider [3 ] { F }, H 0 = 0, H = 0 + - / 2, > 0, C ( ) = { F, F, ƒ ( - ) F+ H, H }, = - / 2,> 0, ( ) = sup F C ( 0 ) E F, ( ) = if E F, F C (+ - / 2 ) 0 995-2004 Tsighua Togfag Optical Disc Co, Ltd All rights reserved
40 27 lim ( ) Φ, Reider lim ( ) (iii) Rousseeuw Rochetti (Hampel [4 ]), (iv) Ylvisaker [40 ], He [5 ] Zhag [47-49 ] Y = ( y, y 2,, y k ), Z = ( z, z 2,, z l ) Y Z = ( y, y 2,, y k ; z, z 2,, z l ) Ylvisaker [40 ] R = mi{ m 0 Φ m Φ, if Z R - m sup ( Z Y) = }, Y R m A = mi{ m 0 Φ m Φ, sup if ( Z Y) = 0} Z R - m Y R m { F } T ( F) H 0 0, H 0, T 0 = { T ( F) 0 }, G = { }, He [5 ] T ( 3 T) = if{> 0 T 0 { T ( ( - ) F+G) G G} <} ; 3 3 ( T) = if{> 0 0 0 T ( F) { T ( ( - ) F 0 +G) G G}} Zhag [47-49 ] 3 2, Huber, Rieder ( ) ( ) (Zhag [47 ] ),,, t Rieder Rousseeuw Rohchetti, Ylvisaker [40 ], He [5 ], Zhag [47-49 ] X = ( x, x 2,, x ), ( X) X k = ( x, x 2,, x k ) X k 3 X SA ( X) = mi{ m 0 Φ m Φ, sup( X - m Y) = }, Y R m S R ( X) = mi{ m 0 Φ m Φ, if ( X - m Y) = 0 } Y R m SA ( X) S R ( X) 995-2004 Tsighua Togfag Optical Disc Co, Ltd All rights reserved
5 4 (C) Z = ( z, z 2, Z 3 ( Z) = ( Z 3 ) X (C), X 3 X, P( S R ( X) Ε v ( X) = ) = P( S R ( X 3 ) Ε v ( X 3 ) = ), P( SA ( X) Ε v ( X) = 0) = P( SA ( X 3 ) Ε v ( X 3 ) = 0), z ), Z v > 0, P( S R ( X) Ε v ( X) = ) P( SA ( X) Ε v ( X) = 0) ( X) t T 2, X = ( x, x 2,, x ) F F gx H 0 A ( F), gx = = 0, H A ( F) ( X) = gx > c, 0, S R ( X) = SA ( X) = 0, gx Φ c,, ; 0, gx > c,, 2 t H 0 H, S ( X) = ( X) = gx S ( X),, T ( X) > c, i - x i, A ( F) 0 ( i = x i - gx) 2 / ( - ), T ( X) = C, Ε 2 S R ( X) = 0, ; } } 0, T ( X) Φ c,, ; SA ( X) = m, - m = mi{ - 2 Ε m Ε 0 - m - T ( X - m) + m ( - ) > c ( - m) } 3 X = ( x, x 2,, x ) p F, A ( F) F gx = i = x i, V ( X) = Hotellig T 2 H 0 A ( F) = 0 A ( F) 0, c, Ε 2 S R ( X) = i = ( x i - gx) ( x i - gx), T ( X) = gx V ( X) - gx ( X) =, T ( X) > c, 0, ; 0, T ( X) Φ c,, ; SA ( X) = m, - m = mi{ - 2 Ε m Ε 0 - m - T ( X - m) + m ( - ) ( - m ) 995-2004 Tsighua Togfag Optical Disc Co, Ltd All rights reserved > c }
42 27 t T 2 3 2, ( c ) / 2 = t - - ( - / 2),0 < <, t - - ( - / 2 ) - t - / 2 F, SA ( X) (i) S R ( X) 0, a s, ( ) (ii) F, A ( F) = 0, d max{ ( - ( - a/ 2 ) ) 2 - N (0,) 2 g, 0 } ( ) p ( k, a) = lim P ( SA ( X) = k ( X) = 0 ), a = 0 05, 3 3 32 3, C = D ( p, ) + o(), D ( p, ) p 2 p -, F (i) S R 0, a s ; (ii) F A ( F) = 0,, d ( k, p, ) ƒ lim + P ( SA ( X) = SA d max{d ( p, ) - 2 pg, 0 } Ε k ( X) = 0 ), 3 2 d ( k, p, ) - P( 2 p Φ D ( p,) - k + ) = 0 05, k p, d ( k, p,) 3 2 3 3 2 k p (k, 005) 00443 2 00873 3 0939 4 06748 k p 2 4 6 0 8 2 09659 09734 0977 0982 09852 3 09096 09345 09454 09567 09669 4 08467 08784 09022 09250 09444 5 06637 0799 08443 08448 0973 6 044 06905 07689 08347 08849 7 00000 05476 06734 07736 08467 8 00000 0378 05574 07009 08023 Mote Carlo t T 2 SA ( X), 3 3 2, t T 2, t T 2 ; p T 2,, t, t (Efro [0 ], Bejamii [2 ] ) Bejamii [2 ] t, t, Rousseeuw Croux [34 ],,, Bejamii,, Kariya [2 ] t T 2 X,,, He [5 ] t, t Rieder [3 ] t 995-2004 Tsighua Togfag Optical Disc Co, Ltd All rights reserved
5 43 T 2, t T 2, Zhag [47-49 ], ( I) M, ( (> 0) ), ( ), Huber M ( II) 2, Kolmgorou, Wilcoxo Zhag [47-49 ] Bai Z D, Rao C R ad Wu Y M - estimatio of multivariate liear regressio parameters uder a covex discrepacy fuctio Statist Si ica, 99, 2, 237-254 2 Bejamii Y Is the t test really coservative whe the paret distributio is log - tailed?jour Ameri Statist Asso, 983, 78 645-654 3 Box G E P No - ormality ad tests o variaces Biometritica, 953, 40 38-335 4 Che X R ad Wu Y H Strog cosistecy of M - estimates i liear models J Mult A aly, 988, 2 (7) 6-30 5 Cui H PP,, 993 6 Davies, L The Asymptotics of Rousseeuws miimum volume ellipsoid estimator A Statist, 992, 20 828-843 7 Davies L Desirable properties, breakdow ad efficiecy i the liear regressio model Stat Prob letters, 994, 9 36-370 8 Dooho D L Breakdow properties of Multivariate locatio estimators Ph D qualifyig paper, Dept of Statistics, Harvard Uiversity, 982 9 Dooho D L ad Huber P J The Notio of Breakdow Poit i A Festchrift for E L Lehma, eds P J Bickel, et al, Belmot, CA, 983, 57-84 0 Efro B Studets t - test uder symmetry coditios Jour A mer Statist Assoc, 969, 63 278-302 Hampel F R Cotributio to the theory of robust estimatio Ph D dissertatio Uiv Califoria Berkley, 968 2 Hampel F R A geeral qualitative defiitio of robustess A Math Statist, 97, 42 887-896 3 Hampel F R Rejectio rules ad robust estimates of locatio A aalysis of some Mote Carlo results I Trasatios of the Seveth Pargue Coferece ad of the Europea Meetig of Statisticias 977, 87-94 4 Hampel F R, Rochetti E M, Rpusseeuw P J, Stahel W A, Robust Statistics The Approach Based o Ifluece Fuctios Wiley, New York, 986 5 He X, Simpso D G ad Portoy S L Breakdow robustess of tests JA SA, 990, 85 446-452 6 Huber P J Robust estimatio of a locatio parameter A Math Statist, 964, 35 73-0 7 Huber P J A robust versio of the probability ratio test A Math Statist, 965, 36 753-758 8 Huber P J Robust Statistics Wiley, New York, 98 9 Huber P J Fiite sample breakdow poit of M - ad P - estimators A Ststist, 984, 2 9-26 20 Huber P J Projectio pursuit A Statist, 985, 3 435-475 2 Kariya T A robustess property of Hotelligs T 2 - test A Statist, 98, 9 20-23 22, 983, (2) 4-47 23, 987, 2 (3) 369-384 ; 2 (4) 493-50 24 Li Guoyig ad Che Z Projectio pursuit approach torobust dispersio metrices ad pricipal compoets primary theory ad Mote Carlo JA SA, 985, 80 759-766 25 Li Guoyig ad Shi P D Coverget rates of M - estimators for apartly liear model I Statistical Sciece ad Data aaly2 995-2004 Tsighua Togfag Optical Disc Co, Ltd All rights reserved
44 27 sis eds K Matusita, M L Puri, T Hayakawa, VSP, the Netherlad, 985, 45-446 26 Lopuhaa H P Highly efficiet estimators of multivariate locatio with high breakdow poit A Statist, 992, 20 398-43 27 Lopuhaa H P ad Rousseeuw P J Breakdow poitsof affie equivariat estimators A Statist, 99, 9 229-248 28 Maroa R A, Stshel W A ad Yohai V J Bias - robus testimators of multivariate scatter based oprojectio it J M ultiv A a, 992, 42 4-6 29 Pollard D Covergece of Stochastic Processes, Spriger Verlag, New York, 984 30 Rao C R ad Zhao L C Liear represetatio of M - estimatio liear models The Caadia Joural of Statistics, 992, 20 359-368 3 Reider H A robust asymptotic testig model A Statist, 978, 6 080-094 32 Rousseeuw P J Least media of squares regressio JA SA, 984, 79 87-880 33 Rousseeuw P J Multivariate estimatio with high breakdow poit I Mathematical Statistics ad Applicatio Vol B, Edited by Grossma W, et al The Netherlads, 985, 283-297 34 Rousseeuw P J ad C Croux The bias of k - step M estimators Statist Prob letter, 994, 20 4-420 35 Rousseeuw P J Ucovetioal features of positive - breakdow estimators Statist ad Prob letters, 994, 9 47-43 36 Stahel W A Breakdow of covariace estimators Ressarch Report, 3, Fachfruppe fur Statistic, ETH, Zurich, 98 37 Tukey J W A survey of samplig from cotamiated distributios icotributios to Probability ad Statistics edited by I Olki, Stadford Uiversity Press, Stadford, CA, 960 38 Yohai V J High breakdow poit ad high efficiecy robust estimators for regressio A Statistics, 987, 5 642-656 39 Yohai V J ad Mara R A The maximum bias of robustcovariaces Com m Statist, Theory Methods, 990, 9 3925-3933 40 Ylvisaker D Test resistace JA SA, 977, 72 55-556 4 Yu D Improved media estimatio for multivariate locatio Techical Report, Istitute of Sys Sci, Academia Siica, 992 42 Zhag Jia Asymptotic theories for the robust PP estimatorsof the pricipal compoets ad dispersio matrix III, Bootstra pcofidece sets, bootstrap tests System Sciece ad mathematicalscieces, 99, 4 289-30 43 Zhag Jia Breakdow properties of two projectio based statistics To appear i Sakhya, 998, No 3 44 Zhag Jia Limit theorems of radom - sample - size empirical process To appear i Chiese A Math A, also B, 998 45 Zhag Jia A PP estimator of multivariate locatio Techical Report, Istitute of Applied Mathematics ad Istitute of Systems Sciece, Academia Siica, 993 46 Zhag Jia A ew kid of affie equivariat estimators for Multivariate locatio ad dispersio based o projectio pursuit Techical Report, Istitute of Applied Mathematics ad Istitute of Systems Sciece, Academia Siica, 993 47 Zhag Jia The sample breakdow poits of tests J Stat Pla ad If er, 996, 52 6 - - 8 48 Zhag Jia The optimal breakdow M - ad score - tests Statistics, 997, 30 47 - - 68 49 Zhag Jia The sample breakdow properties of someo parametric tests Com m Statist Theory ad Methods, 996, 25 85 - - 98 50 Zhag Jia ad Li Guoyig Breakdow properties of locatio M - estimators To appear i A Stat, 998, No 3 5 Zhag Jia, Zhu L ad Cheg Pig Asymptotic theories for the robust PP estimators of the pricipal compoets ad disper2 sio matrix ( I), Techical Report, Istitute of Systems Sciece, Academia Siica, 988 995-2004 Tsighua Togfag Optical Disc Co, Ltd All rights reserved
5 45 Some Recet Developmets o Robust Estimatio ad Tests Zhag Jia Li Guoyig ( Istit ute of Systems Sciece, Academia Siica, Beijig, 00080, P R Chia) Abstract This paper gives a veviw of some ew developmets o robust estimatio ad tests i recet years, icludig ( ) breakdow properties of M - estimators for oe - dimessioal locatio, i particular, t he asymptotic properties of sample breakdow poit s of redescedig M - estimators ; (2) PP type estimators for multivariate locatio ad dispersio mat rices ad t heir properties ; (3) the breakdow poits of statistical tests, ad breakdow properties of t - test, T 2 - test, M - test, score test ad X 2 - test etc Also, basic cocepts of robustess ad mai theory of the above three topics are briefly itro2 duced Key words sample breakdow ; poit ; redescedig M - estimator ; locatio ; dispersio mat rics ; robust test ; asymptotic properties 995-2004 Tsighua Togfag Optical Disc Co, Ltd All rights reserved