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 )

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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

1 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,, Tsighua Togfag Optical Disc Co, Ltd All rights reserved

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 Tsighua Togfag Optical Disc Co, Ltd All rights reserved

3 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) Tsighua Togfag Optical Disc Co, Ltd All rights reserved

4 ( 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 < , ( B ), Tsighua Togfag Optical Disc Co, Ltd All rights reserved

5 ( ) ( 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) = { } ,, m, Tukey biweight c , 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 PP I) Huber [6 ], Tsighua Togfag Optical Disc Co, Ltd All rights reserved

6 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 ) 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 Tsighua Togfag Optical Disc Co, Ltd All rights reserved

7 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 = / 2, > 0, C ( ) = { F, F, ƒ ( - ) F+ H, H }, = - / 2,> 0, ( ) = sup F C ( 0 ) E F, ( ) = if E F, F C (+ - / 2 ) Tsighua Togfag Optical Disc Co, Ltd All rights reserved

8 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{> 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) Tsighua Togfag Optical Disc Co, Ltd All rights reserved

9 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 ) Tsighua Togfag Optical Disc Co, Ltd All rights reserved > c }

10 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, , 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,) k p (k, 005) k p 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 Tsighua Togfag Optical Disc Co, Ltd All rights reserved

11 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, Bejamii Y Is the t test really coservative whe the paret distributio is log - tailed?jour Ameri Statist Asso, 983, Box G E P No - ormality ad tests o variaces Biometritica, 953, Che X R ad Wu Y H Strog cosistecy of M - estimates i liear models J Mult A aly, 988, 2 (7) Cui H PP,, Davies, L The Asymptotics of Rousseeuws miimum volume ellipsoid estimator A Statist, 992, Davies L Desirable properties, breakdow ad efficiecy i the liear regressio model Stat Prob letters, 994, Dooho D L Breakdow properties of Multivariate locatio estimators Ph D qualifyig paper, Dept of Statistics, Harvard Uiversity, 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, Efro B Studets t - test uder symmetry coditios Jour A mer Statist Assoc, 969, Hampel F R Cotributio to the theory of robust estimatio Ph D dissertatio Uiv Califoria Berkley, Hampel F R A geeral qualitative defiitio of robustess A Math Statist, 97, 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, Hampel F R, Rochetti E M, Rpusseeuw P J, Stahel W A, Robust Statistics The Approach Based o Ifluece Fuctios Wiley, New York, He X, Simpso D G ad Portoy S L Breakdow robustess of tests JA SA, 990, Huber P J Robust estimatio of a locatio parameter A Math Statist, 964, Huber P J A robust versio of the probability ratio test A Math Statist, 965, 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, Huber P J Projectio pursuit A Statist, 985, Kariya T A robustess property of Hotelligs T 2 - test A Statist, 98, , 983, (2) , 987, 2 (3) ; 2 (4) Li Guoyig ad Che Z Projectio pursuit approach torobust dispersio metrices ad pricipal compoets primary theory ad Mote Carlo JA SA, 985, Li Guoyig ad Shi P D Coverget rates of M - estimators for apartly liear model I Statistical Sciece ad Data aaly Tsighua Togfag Optical Disc Co, Ltd All rights reserved

12 44 27 sis eds K Matusita, M L Puri, T Hayakawa, VSP, the Netherlad, 985, Lopuhaa H P Highly efficiet estimators of multivariate locatio with high breakdow poit A Statist, 992, Lopuhaa H P ad Rousseeuw P J Breakdow poitsof affie equivariat estimators A Statist, 99, 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, Pollard D Covergece of Stochastic Processes, Spriger Verlag, New York, Rao C R ad Zhao L C Liear represetatio of M - estimatio liear models The Caadia Joural of Statistics, 992, Reider H A robust asymptotic testig model A Statist, 978, Rousseeuw P J Least media of squares regressio JA SA, 984, 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, Rousseeuw P J ad C Croux The bias of k - step M estimators Statist Prob letter, 994, Rousseeuw P J Ucovetioal features of positive - breakdow estimators Statist ad Prob letters, 994, Stahel W A Breakdow of covariace estimators Ressarch Report, 3, Fachfruppe fur Statistic, ETH, Zurich, Tukey J W A survey of samplig from cotamiated distributios icotributios to Probability ad Statistics edited by I Olki, Stadford Uiversity Press, Stadford, CA, Yohai V J High breakdow poit ad high efficiecy robust estimators for regressio A Statistics, 987, Yohai V J ad Mara R A The maximum bias of robustcovariaces Com m Statist, Theory Methods, 990, Ylvisaker D Test resistace JA SA, 977, Yu D Improved media estimatio for multivariate locatio Techical Report, Istitute of Sys Sci, Academia Siica, 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, 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, Zhag Jia A PP estimator of multivariate locatio Techical Report, Istitute of Applied Mathematics ad Istitute of Systems Sciece, Academia Siica, 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, Zhag Jia The sample breakdow poits of tests J Stat Pla ad If er, 996, Zhag Jia The optimal breakdow M - ad score - tests Statistics, 997, Zhag Jia The sample breakdow properties of someo parametric tests Com m Statist Theory ad Methods, 996, 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, Tsighua Togfag Optical Disc Co, Ltd All rights reserved

13 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 Tsighua Togfag Optical Disc Co, Ltd All rights reserved

Chinese Journal of Applied Probability and Statistics Vol.25 No.4 Aug (,, ;,, ) (,, ) 应用概率统计版权所有, Zhang (2002). λ q(t)

2009 8 Chinese Journal of Applied Probability and Statistics Vol.25 No.4 Aug. 2009,, 541004;,, 100124),, 100190), Zhang 2002). λ qt), Kolmogorov-Smirov, Berk and Jones 1979). λ qt).,,, λ qt),. λ qt) 1,.