A C T A U N I V E R S I T A T I S L O D Z I E N S I S F O L IA CŁCONOMICA AB, 1985
w ła d y o ła w M i lo * , Z b ig n ie w W a s ile w s k i* *
SOME PROBLEMS OF ROBUST ESTIMATION IN THE CASE OF U N EA R MODELS PART I s CHARACTERIZATION OF ROBUSTNESS
1. In t r o d u c t io n
We l i m i t ou r a t t e n t i o n to the ro b u s tn e s s of p o in t e s tim a t io n methods f o r the f o llo w in g l i n e a r models
t m 0 - ( * n ,,kt 8, Y - X(S ♦ S , k0 - k, n0 - n, <Py - J r y ( X ( i , d 2 I ) ) , ¿Triu, » ( a 0**, a , y - xp + s , k0 - k, n0 - n, $>y - ¿ ^ C x p , n ) ) ,
w here t
^nxk _ th e 9#t o i r o a i nxk m a t r ic e s , Q
S ■ ( ‘U ,* F ,ł>) - a p r o b a b i l i t y space w it h th e co m p le te measure
1 T - B o r e l- t f - f ie ld o f s u b s e ts o f s e t U ,1i. - an elem en t a r y s e t o f e v e n t s , k o » r a n k ( X ) , n o • rank f> (Y ), ... X € Rn . Jb (Y ) - d is p e r s io n o f random v e c t o r Y , i . e . t - e x p e c t a t io n o p e r a t o r , “ f " t)P (X f y if - "n - d lm e n s lo n a l random v e c t o r has n - d lm e n s lo n e l norm al d i s t r i b u t i o n w it h 8 (Y ) • X(3 and & (Y ) ■ A*.
The c o n c e p t o f r o b u s tn e s s was in tro d u c e d in t o s t a t i s t i c s by B o x , A n d e r s o n [ 5 ] and i t was co n cern ed w it h the robu- ł s tn e s s o f t e s t s . A more fo rm a l p r e s e n t a t io n o f th e r o b u s tn e s s
* D r . , L e c t u r e r a t the I n s t i t u t e o f E c o n o m e tric s and S t a t i s t i c s , U n i v e r s i t y o f Ł ó d ź .
* * S e n i o r A s s is t a n t a t th e I n s t i t u t e o f E c o n o m e tric s and S t a t i s t i c s , U n i v e r s i t y o f Łódź.
ic łjn a waa In tro d u c e d by H u b e r [ 9 , 10, 111, H a m p e l [ &, 7 ] , B i c k e 1 [
4
] , B e r g e r [3
] , B a r t o - s z y ń s k i , P l e s z c z y r t s k o [ * ] • Z i e l i ń s k i [20
] . H u b e r , C a r o l [ l 2 ] , and R o us-3
e o u w [ 1 8 ] . i2. flob ustnyjsoj I n t u i t i v e and H®urlat^lc^hean£njj|
The* broad and i n t u i t i v e meaning o f r o b u s tn e s s sh o u ld be ch a r a c t e r i z e d moro f o r m a lly in o r d e r to a c h io v e g r e a t e r p r e c i s i o n . P o s in g the f o llo w in g q u e s tio n s sh o u ld h e lp in i t .
Q l ) What l a r o b u s t ?
Q2) A g a in s t what som ething i s r o b u s t ? ( a g a in s t what k in d o f changes ( d e v i a t i o n , p e r t u r b a t i o n , d is tu r b a n c e s « t r a n s f o r m a t io n s ) t h i3 sûm ething i s r o b u s t ) ?
Q3) In what p r o p e r t ie s o f t h i s som ething th e r o b u s tn e s s i s m a n ife s t e d ? ( w it h r e s p e c t to w h ich p r o p e r t ie s th e r o b u s tn e s s i s m a n if e s t e d ? )
Q4) W ith r e s p e c t to w h ich q u a l i t a t i v e m easures o f p e rfo rm a n ce t h i s som ething I s r o b u s t ?
Q5) How to measure r o b u s t n e s s ?
Q6) What do we lo o s e In r e t u r n to r o b u s t n e s s ? Q7) To what d e g re e som ething I s r o b u s t ?
Q8) w h eth er th e c o n c e p ts " s t a b i l i t y o f so m e th in g " and " r o b u s tn e s s o f so m eth ing * a re d i f f e r e n t In m eaning?
B r i e f q u a l i t a t i v e an sw ers a re as f o l l o w s : Ad Q l) Robust i s : a ) a s t a t i s t i c a l (e c o n o m e tr ic ) method o f e s t im a t io n , p r e d ic t i o n , t e s t in g (a n e s t im a t o r , a t e s t , a p r e d i c t o r ) , b) a s t a t i s t i c a l m odel, c ) a n u n e r ic a l a lg o r it h m o f a g iv e n method. Ad Q2) IVe p o s t u la t e th e ro b u s tn e s s o f a g iv e n method ( o r an a lg o r it h m ) a g a in s t the changes o f : c h i ) a ssu m p tio n s of s t o c h a s t i c s t r u c t u r e o f th e g iv e n s t a t i stic:.* 1 m odel,
c b 2 ) assu m p tio n s of n o n s t o c h a e t lc s t r u c t u r e o f the g iv e n mo d e l ,
P o s s ib le changes ( c h i ) c o n c o rn : c h l . l ) p r o b a b i l i t y m easures,
c h i . 2 ) s h i f t s in the p ern m o ter v a lu e of p r o b a b i l i t y . ¡ is t r ib u - t i o n , c h l . 3 ) r e p la c in g s ta n d a rd d i s t r i b u t i o n by a s u p e r p o s it io n o f d i s t r i b u t i o n s b e lo n g in g to d i f f e r e n t c l a s s e s , c h i . 4 ) d is t a n c e s botween d i s t r i b u t i o n s . P o s s ib le changes (c h 2 ) c o n c e rn : c h 2 . l ) l e v e l s o f b o d - c o n d itio n in g of dr»to m a tr ix ,
c h 2 .2 ) shapes o f r e l a t i o n s h i p s ( e . g . from l i n o o r in t o n o n - li- n e a r ).
P o s s ib le chonges (c h 3 ) co n co rn tho e le m e n ts of c r o n s - c le a a i- f i c a t i o n o f ( c h i ) and (c h c ) .
Ad Q 3 ) Wo p o s t u la t e th a t the r o b u s tn e s s sh o u ld m a n ife s t in tha in v a r lo n c o (a g o in o t changes ( c h l ) - ( c h 3 j ) o f one o r moro o f the f o llo w in g p r o p e r t i e s : p i ) u n b ia s e d n e s s , p 2 ) e f f i c i e n c y , p 3) p r e c i s i o n , p 4) c o n s is t e n c y , p 5 ) s u f f i c i e n c y , p 6 ) s t a b i l i t y , p 7 ) shape o f d i s t r i b u t i o n .
C o n firm in g o r n o t - c o n fir w in g the in v a r ia n c e o f some anum arat- ed p r o p e r t i e s i s n o th in g more than e q u a l i t a t i v e a n a l y s i s . I t h e lp s to s o y t h a t som ething i s o r i e n o t robu9 t in a sen se and no th in g moro.
Ad Q4) We p o s t u la t e t h a t tho r o b u s tn e s s w i l l m a n ife s t in the r e l a t i v e s t a b i l i t y o f th e range f o r tho f o llo w in g f u n c t io n s o f s t a t i s t i c a l p r o c e d u r e s : f l ) t o t a l b io 3 , f 2 ) e f f i c i e n c y , f 3 ) t o t a l mean sq u a re e r r o r , f 4) s ta n d a rd e r r o r , f 5 ) a v e ra g e r e l a t i v e a b s o lu te p r e d ic t i o n e r r o r , f 6 ) w e ig h te d t o t a l sum o f p r e d i c t i o n e r r o r s , f 7) speed o f co n ve rg e n c e o f i t e r a t i v e s t a t i s t i c a l p r o c e d u r e s , f 0 ) d e v ia t i o n between the p - th q u a n t i le s o f d i s t r i b u t i o n s of s t a t i s t i c s .
f o ) d is t a n c e s botween p r o b a b i l i t y d i s t r i b u t i o n s o f s t a t i s t i c s (m easured by d is t a n c e s between d i s t r i b u t i o n f u n c t io n s o r c h a r a c t e r i s t i c f u n c t io n s o r moment q cn rjr.n in g f u n c t i o n s ) ,
f lO ) v a lu e s o f G Steaux d o n v a t i v e oi s t a t i s t i c a l p ro ce d u re o r lo g a r ith m s o f s t a t i s t i c s ' v a r i a n c e .
Ad Q5) R o b u stn e ss m easures w i l l be d e fin e d by u sin g ( f l ) - - ( f l O ) . T h ere a r e , among o t h e r s , the f o llo w in g o p t io n s :
- a ro b u s tn e s s measure i s tho d ia m e te r of one o f th e fu n c t io n s ( f l ) - ( f l O ) ran g in g f o r tho g iv e n v a r i a n t o f changes (chl)- - (c h 3 ) o r t h e i r m ix tu r e s ,
- a ro b u s tn e s s measure i s th e r a t i o o f thp d lu m o te r of one of tho ran ges of f u n c t io n s ( f l ) - ( f l O ) and changes v e r s io n s ( c h i ) - - ( c h 3 ) f o r a g iv e n s e t of s ta n d a rd model n eig h b ou rh oo d s to the d ia m e te r o f tho s e t of ran g es o f d is t a n c e s between n e ig h b o u r hoods of a s ta n d a rd model ( o r a ro b u s tn e s s measure i s th e l i m i t o f t h i s r a t i o when the d en o m in ato r ap p ro ach es to z e r o ) ,
- a ro b u s tn a s s measure i a tho d if f e r e n c e between th e sup re- Mun o f one o f tho ran g es c f f u n c t io n s ( r l ) - ( f l O ) end the c o r resp o n d in g infim um d e fin e d on e le m e n ts o f p a r t i c u l a r n e ig h b o u r hood o f s ta n d a rd m odel,
- a ro b u s tn e s s measure i s the d is t a n c e between u v a lu e of one o f the f u n c t io n s ( f l ) - ( f l O ) c a lc u la t e d f o r tho g iv o n s ta n d ard model ond the range o f the c o rre s p o n d in g f u n c t io n c a l c u l a t ed . f o r th e e le m e n ts of p a r t i c u l a r neighbourhood o f tho s ta n d a rd m odel,
- a ro b u s tn e s s measure i s the d is t a n c e between the s o t o f j l l ran ges of f u n c t io n s ( f l ) - ( f i O ) c a lc u ln t « d f o r the g iv e n s ta n d ard model and the s e t o f a l l ran g es o f f u n c t io n s ( f l ) - ( f l O ) c a l-lila t e d f o r the s e t o f neig h b ou rh ood s o f s ta n d a rd model ( s e t of a u p e rm o d e ls ).
Ad Q6) Robust msthods a r e , in g e n e r a l, more n u m e r ic a lly complex than n o n - rcb u st on u s. I t means th a t the n u m e r ic a l c o s t i f u sin g them ( a s measured by e . g . com puter tim e u sa g e ) i s g re a t e r . We a re not in a p o s it io n to g iv e re a s o n a b lo com p ariso n s ©f o e n e f lt s and lo s s e s when u s in g ro b u s t m ethods.
Ad Q7) A d egree of r o b u s tn e s s sh o u ld be f ix e d on the grounds i n d ic a t io n s g iv e n by the m easures proposed in Ad Q5.
Ad Q8) The co n ce p t o f " s t a b i l i t y " was d e fin e d p r e c i s e l y i t h i n m echanics by P o in c s r ® and n ext by L a p u n o ff. U 1 a m [1 9 ]
d is c u s s e s I t In a more g e n e r a l c o n te x t o f s t a b i l i t y o f m ath em ati c a l theorem s. He u n d e rs ta n d s by i t the p r o p e r t y th a t w h ile chang ing assu m p tio n » of a th o o re n "a l i t t l e " , th e t r u t h of the th e orem I s f u l l y p r e s e rv e d o r ''a p p r o x im a te ly " p r e s e r v e d .
I n t e r e s t i n g p r o p o s it io n s o f d e f in in g s t a b i l i t y wore g iv e n by Z o l o t o r o v [2 2 , 2 3 ] and extended by B e d n a r e k - * K o z e k, K o z e k [ 2 ] , and B a r t o s z y r t s k l , P l e s z c z y r t s k o [ l ] .
H ere we a re t r y in g to maka the co n ce p t of s t a b i l i t y more c o n c r e t e . P r o p o s it io n s o f c h a r a c t e r iz a t io n s and ro b u s tn e s s moa- ourea p re s e n te d In <j 3 and 4 were f i r s t fo rm u lo to d by M ilo In the work o f M i l o , W a s l l u w s k l 1 1 5 ]. A s p e c i f i e d k in d o f g iv e n m eth o d 's r o b u s tn e s s a g a in s t eotne changes o f s ta n dard modol would be d e fin e d by u sin g th e co n ce p t o f s t a b i l i t y o f m easures b e lo n g in g to th e m easures ty p e s (M 1 )- (M 5 ).
3. A C h a r a c t e r i z a t io n
mmmmmmmmmmftaaCaa ■ — bb— b — — —i
o f L in e a r S ta n d a rd M o d e l's Neighbourhoods — Tr •• a
1
n 'n w i ini ■'imaaaMBCTaaa—F i r s t we need to d e fin e n eigh b ou rh oo d s o f Y . They can bo de f in e d In term s o f L e v y 's , P r o c h e r o v 's , M e e h a lk ln 's , Kolmogo r o v 's o r t o t a l v a r i a t i o n m e t r ic s .
Duo to the d e f i n i t i o n o f random v e c t o r Y as
Y
1
(11, 7 , *>) --- ► (<ftn . 7 n , <Py ) 9.we w i l l use f u r t h e r the p r o b a b i l i t y s p a c* (9.n , rf n , $>y ) . We w i l l
ro« t r l z e th e space Rn. ®
A f t e r m e tr lz ln g the space R ° becomes the m e t r ic space ( R 0, £ n ) (o r s h o r t l y , i f p o s s ib le , w r i t i n g Rn ,£ > ). . ®
L e t
^2
ke two r.o n - n e g a tlv e f i n i t e m easures d e fin e d on ^ „ ( 7 i s d - f i e l d o f B o r e l s u b n e ts o f ( R n , p ) ) . Then we con*n ' -r. u
d e fin e th e above m entioned m e t r ic s os f o llo w s a ) M e s h a lk ln 'e m e t r ic s [ l 3 ] i
f}-1 f}-1 0 Wi a d y u Ui wMl i.j, Z b l c p l e w W u s i l e w - k i w b oro: A - an I n t e r s e c t i o n o f a t moat n h o lf - s p a c o e , a h a lf- e p a - ce to an a r b i t r a r y s o t o f p o in t s In R n d e te rm in e d by n { ( y j , y n ) ' « Rn ! < b | ; « i t b « R1 } 1»1 b ) L e v y 's m e t r ic s £ L C*»4 . \>2 ) - "»ox [ e * ( fJ 1 . H 2) . { J j ) } , « « w here i £ * ( * V tl2 ) * l n f
{ 5
: ^1 { ( " ° ° » V î } <^2 O “ 90* V i 6 } * S • V Y e R " ] * 5>0 **1* " in f { 8 s **2 f 1" 00» ***1 { ( “ ® . y ] 5 } + 5 , Vy e Rn }, 5>o ( -00, y f - I z « Rn : |0^n ( z , (-CJO, y ] ) <£ | c ) P r o c h o r o v 's m e t r ic s e P h2) • max { e j ( t v h 2 >» e p * ( ij 2 ‘ tJ i ) } .»hare Sp ( I V <J 2 ) * ln f * lJ l (A ) < H 2 ( a 5 ) + S » VA e 9= I , 5>0 R J B p * ( t*2* ¥%> " in f f S : Ï 2 ( A ) < 1*1(a5)+ S > VA * 7 n}> 5=0 R A5 - | y € Rn I A n ( y , A)< 5 j-d ) K o lm o g o ro v 's m e t r ic s£ k ^ 1 * ^ * 8up " P2^A ) l 1 A " f -00' y]* y 6 R ° }>
e ) t o t e l v a r i a t i o n m e t r ic s
e ™ ( P l ' t12) " 3Up - i*2< A )l}-A « y
R
In fo rm u la tin g e x p e rim e n ts schemes one can r e p la c e p j , j j2 w it h the c o rre s p o n d in g d i s t r i b u t i o n f u n c t io n s F ^ , F 0 ( o r c h a r a c
t e r i s t i c f u n c t io n s ( ^ ( t ) , < p „ (t )). The enum eratud m e t r ic s d id not ex h a u st the l i s t o f e l l p o s s ib le m c t r ic s o f p r o b a b i l i t y space ( Rn , 7 n , i>y ) . They e n a b le u s , h o w e ver, to d e f ln o r j^ ,
R
i . e . the 1-th neighbourhood o f measure ■ P y ■ ctfy. w ith r e s p e c t to th e m e t r ic s 1 (w h ere i - f o r o g iv e n m e t r ic s 1 de n o te s th e in d ox number a tt a c h e d to the g iv e r. 1-th v a lu e o f n . « ■■ m ■ i Q ■§ e . g . , f o r 1 • 1 , 3 ■ 1 , 2, 3 we have n ._ ■ 3 • 10 , V .
•
•1 „ o »11 *21 • 3 • i o , « •3
«10
z ) . Thua. D e f i n i t i o n 1, B y i j A1 - neighbourhood o f mea su re f>Y m ■ JPy (X(3, <J2I ) we u n d e rs ta n d ouch s e t o f d i s t r i b u tio n s th a t the e le m e n ts o f U a r e d i s t a n t from th e moaaureo ^ 2 bY the v a lu e *>•••U^ » {^ 2 * ^1 ^ 1 * ¿*2^ ^ ^11 } ’ 1 ■ M, L , P , K, TV.
U sin g D e f. 1 we can d e fln o - neig h b ou rh oo d o f tho s ta n d ard model c*VM.0 as f o llo w s t
D e f i n i t i o n 2. B y - neighbourhood o f model
<^cU.0 we u n d e rstan d such a s o t t *10t f ^ o l n s jw nxl\ * , Y
-X f l + 2 , k0 - k , n0 - n , ♦ 11 11
W h ile u s in g th e name o f jdfcM.0j ^ i n d e f in in g the ro b u s tn e s s me a s u r e s I s v o r y f r u i t f u l In c o n s t r u c t in g schemes o f e x p e rim e n ts , I t tu r n s o u t on the o t h e r hand, t h a t o t h e r names a re a lc o con v e n ie n t ( e . g . ,,T
2
j2
- s e t o f su p erm o d els w it h r e s p e c t to tho s ta n dard model o r - s e t o f p r o b a b i l i s t i c e x t e n s io n s o f the s ta n d a rd model JTcU.0" ) .N o te i I f i ■ 1 ,1 0 , th an under D e f. 1 ,2 t h e r e a r e 10 x 5 v a lu e s o f
12
A1 t h e r o f o r e I t I s 50 s e ta o f sup erm o d els f o r the oP<M.0.{ W ^ i l " (RnXl<’ * , Y " X (J* S ' ko - k * " o » " . ^ u )» th a t 13 - neighbourhood o f w td e l .K’ iM.j.
A. Robu s t n e ss lionoures of E c t lm a t o r s A g a in s t Chungee of__the Model Neig h b o u rhoods
L e t b in o ( B j ) • - (3 bo the b la e o f t h e 'j - t h k in d of e s t im a t o r B f o r the p o rsm o te r v e c t o r (3 from JC<M,0 o r a model be lo n g in g to the s e t {ji’cW.J .Thus e t o t a l b ia s la t o b ia s ( ) * 8 £ ( B . ) - P 1 where | * | d e n o te s e u c lid e a n v e c t o r n o n » /
J ( j )
L o t ° V i l donot6 the range o f t o b la a (B ^ ) » t o b ia e
c a l c u la t e d f o r the p a r t i c u l a r d i s t r i b u t i o n s b e lo n g in g to the U, - neighbourhood o f ^ ) and th e c o rre e p o n d ln g m odels be lo n g in g to the s e t {«v*cM.0}-^
In o r d e r to be more p r e c is e we need to in tr o d u c e one wore in d e x ” d " co n n ecte d w it h c o u n tin g the c o n s e c u t iv e numbor o f "pu r e " o r “ c o n ta m in a te d ’' d i s t r i b u t i o n (b e lo n g in g to th e U„ ) f o r example : d - i -- ► (1 - v)JCy ( X ( l, d 2I ) ♦ vJTy (X|3 ♦ V ^ . d ^ D ^ w h e r e
mcN
u CH
Vra i s an a t y p i c a l i t y q u a n t i t y o f th e ra-th component o f v e c t o r Y, j n d e n o te s the u n it v e c t o r o f th e ro-th c o o r d in a t e a x i s . i s the c e t o f a t y p i c a l r e s u l t s o f o b s e r v a t io n s ' i n d i c e s . L e t the range of d ba d - 1 , . . . . d . Then
0
j t o b i a s j , d - X , d,(
U nder the above n o t a t io n s and d e f i n i t i o n s we have
D e f i n i t i o n 3. The e s t im a t o r B (;,) o f (3 from ,ICM0 w i l l be c a l l e d b ia o - ro b u s t in the ^ -n e ig h b o u rh o o d o f th® standc-rd r.cd el c/ftU0 i f th e f o llo w in g i m p lic a t io n
(d ) T i l
<y( | ) n,</P( b ( i j . -
0
) ) .VEXD 3 _ > 0 V l Vd : ( u . e U*d) ) — *• £> (0 ' ^ d)ff(i >)< t,
^ i i 2 r 2 ^ ^
i l l h o ld s . ♦
No t o : the abovo im p lic a t io n con be ro p lo c o d , o .g . by V £ >0 > 0 Vu Vd : ( u e l / d ) ) — ► (diom
)
< 0 »H i
f o r diam A ■ oup £ (o^. , a2 ^ ' Oj . 02 e A
Aa b ia 3 ~ ro b u 9 tn a a s mensuros f o r th e e s t im a to r ono con prop oca a ) DROMOjl b) BROM 0^2 (j ,d) diem ,
11
diam --- S l L _ , diom M l dinm O'.(j.d) c ) SwOH B .3 - lim J .... .. . n d ia n U diam Un — ►O 1 M l d ) BROM B..4 « 3 u p -{Tobias (B (i l ,0 ^1 1;,‘ v)
in f d T o b ia s (o ,9 e ) BROM B .5 - in f p J d u v ^ i l ' f ) BROM 8^6 aup - iL _ (j. d ) 9UP d - in f AK
“ }
r a . d) I x L . V/a s a y t h a t t h e e s t i m a t o r B ^ i ‘- m o r e b i a a - r o b u 3 t t h a n t h o c o - t i m a t o r B g i f BROM B^ r < BROM B2 r , r • 1 ,6 . I f someone i s in t e r e s t e d in tho p r o c is io n o f e s t im p to r s he sh o u ld s tu d y the p e rfo rm a n ce o f B^ w it h r e s p e c t to th e m ean-aquare-er- r o r M S E (B . ) , th a t i s th e c h a r a c t e r iz a t io n o f M S E (O j) » t \ B^ - (311 . VJe would be in t e r e s t e d in tho b e h o v io u r o f M S E (B j) w it h in the p r o b a b i l i s t i c neighbourhood o f eAfcW.0 , i*®.* { * ’***■ o } ^ •L e t d eno te th e rango o f M S E iB ^ ^ ) c a lc u la t e d f o r the d i s t r i b u t i o n s o f Y b e lo n g in g to the n eighbourhood o f the d i s t r i b u t i o n c ^ y U f i, d 2l ) . (¿ j
F o r the g iv e n d i s t r i b u t i o n moasure H2e U l2ii,d “ l f da from t h i s neighbourhood the e s t im a t o r 0^ c o rre s p o n d in g to w i l l be denoted by 6 ^ * . Then
Ф f / ( b«J>\ . , 1 •K - \ MSEVB« d - 1 ... d f .
•11 v u >
Thus
D e f i n i t i o n 4 . The e s t im a t o r B^ o f v e c t o r (3 from jTcM. w i l l he c o l le d M SE-robust In the neighbourhood {Uf(tt0} i f the
° 1 J ‘l l f o llo w in g im p lic a t io n , (d>4 / (i,d ) O K , V O O S i j ^ O V ^ V d : (t» 2 6 )-->e ' * * ц ' * ) < l • * • h o ld s . ♦ Note i a n o th e r o p tio n l a V O O a ^ O V ^ V d « ( J i2 « ) — ► (d .o m i- K ^ ) < С ) .
On the ground o f O e f. 4 we can d e f in e M SE-rob u etn eea m easure», e . g. as f o llo w ? •
1
) MSERll B ^ l ■ d la e » d la ® Ы> H3ERH 0 j 2 - O T T D j f • d la . r i c c l ) HSERM В 3 . l l n m a i n j r 1- • 3 diem uj— *11 d l ) MSERM B 44 - eup ( ’K (1^ # d )|- i n f } d • 1 ... d , J d t ¿ i l J d *- * 11 e l ) HtfcRM B j5 • i n f g { * K ^ , d ) , i f l ) MSERM B fi - 5 Д . J * UP a m A r e l a t i v e l y s y n t h e t i c r o b u s tn e s s measure l a в ro b u e tn e e e л е з su re of B j a g a in s t the chan ges o f d is t a n c e betw een the d i s t r i b u tio n ¿ Jj - P Y end o t h e r d i s t r i b u t i o n s o f Y w it h r e s p e c t to th e changes of d is t a n c e s betw een c o rre s p o n d in g d i s t r i b u t i o n s offe , Wo need ( B j )
D e f i n i t i o n 5. By the neighbourhood U.-^ o f ¿j3 • - P r ►«« u n d e rs ta n d such a s e t o f p r o b a b i l i t y d i s t r i b u t i o n s of
Dj w hich a re not moro d is t a n t from f*pj thnn by
0
q u a n t it y i j f 1«6* uf J - \ i»4: e ( f3 »^4
< Ei }• ♦1
*■ 4 Hence D e f i n i t i o n 6. The e s t im a to r B^ w i l l be c a l l e d d i- s t r i b u t i o n a l l y - r o b u a t a g a in s t changes o f d i s t r i b u t i o n s of ( w it h i n the neighbourhood u j ? ' i f th e im p lic a t io nM l
( Y1 i \
v E ^ o 3 , ^ » ^ v d , ( t . 2 « \ { ) ~ e \ V» h o ld s . ♦
Under th e D e f. 0 we con f o r m u la te , among o t h e r s , tho f o llo w in g d i s t r i b u t i o n r o b u s tn e s s m oasuros:
(B ) 82) DIRM B . l - dium Uj- J ,
j X
dlam U ^ J 5 b 2) DIRM B .2 --- f t y - ,
J diom U „ ' H I
c 2 ) DIRM D .3 » lira DIRM Bj2 , J I « , № +
0
J ^ il d 2 ) DIRM B j4 - sup [<0
( ^3
* ^4
) } “ 1^ f { d ^ 3 » **4,d^ } * ( B ) 1*3 ■ V j * t)4 , d ’< P 3 ' ^ . d a U £ 1 • / \ o 2 ) DIRM B .5 - i n f ) , J d 1 DIRM 0 4 f 2 ) DIRM D.6 » --- * J r u n f /I n th e above d e f i n i t i o n s the sam ple s iz e n was f ix e d . H a m p e l [ 6 ] p ro p o se « to v a r y n and c o n s id e r th e r o b u s tn e s s of sequence o f sam ple e s t im a t o r s w it h the v a r y in g sam ple s i z e . Be fo r e p r e s e n t in g h is id e a l e t us In t r o d u c e th a f o l l o w ing n o t a t io n :
•f « { f , G , . . . , P . Q } - th e s e t o f a l l d i s t r i b u t i o n s mea s u re s d e fin e d on ( <U ,'5 F ){
*V., *» |i-n , Gn , . . . . P(1, Qn , n > 1 | - tho s o t of o i l d i s t r i b u tio n s -fjaoures d e fin e d on 7 n ) ;
n
{ B ( n ) , n > l j - th a soquonce o f 8 ^ t ( U , V )- * (S tn ,
F o r ouch sequence o f n-olem ent sam ple g e n e ra te d from F e *V i t c o rre s p o n d s to tho e m p ir ic a l d i s t r i b u t i o n from cVn .
The mapping G ^ In d u c e s the c o n d it io n a l d i s t r i b u t i o n
t h a t i s , the d i s t r i b u t i o n o f e s t im a t o r R ,_ v under tho (n J c o n d it io n th a t the sample was g o n e ra to d from th e d i s t r i b u t i o n F .
D o . f l n l t l o n 7. Tho sequence { ° ( n )» n > 1 } w i l l be c a lle d ro b u s t w it h r e s p e c t to changes In e m p i r ic a l d i s t r i b u t io n s i f th e f o llo w in g im p lic a t io n VCn >03T2nVOVn , ( e ( F , 0 ) < TL n ) - + ( l i < x F ( B ( n ) ) , -cG( B ( n ) ) ) < h o ld c . ♦ D e f i n i t i o n s 6 and 7 d e te rm in e th e r o b u s tn e s s w it h r o s p e c t to tho e s t i m a t o r 's d i s t r i b u t i o n os such a p r o p e r t y w hich cnusos t h a t the c o n d it io n a l d i s t r i b u t i o n o f t h i s e s t im a t o r i s ch an g in g ” u l i t t l e " I f we change th e d i s t r i b u t i o n g e n e r a tin g th e sample u n d a r ly in g i t . These f a c t 3 a ls o d e te rm in e p o s t u la t e s t h a t sh o u ld ba a d d re sse d to ro b u s t e s t im a t io n m ethods. Due to sp ace l i m i t a t io n and a h om ogeneity ro q u lre m o n t we w i l l n ot c o v e r t h i s t o p i c .
5, F i n a l Remarks
The p a p e r c o n t a in s some o r d e r in g o f th e known s t a t i s t i c a l c o n c e p ts co n n ected w it h th e p r o p o s it io n s o f d e f i n i t i o n s o f ro- w u itn e s s m easures f o r the e s t im a t o r s . We have n o t p re s e n te d r e s u l t s o f e x p e rim e n ts c a r r ie d o u t by Z . IV a s lle w s k i co n n ecte d w ith
be p er f or ma n c e o f chosen e s t im a t io n methods t h a t would be p re- ic n te d in the n ext p a p e r. We have n o t d e s c r ib e d , among o t h e r s ,
icu au res o f ;
- o f f i c ie n c y - r o b u s t n a s s o f e s t im a t o r s ,
- f o r e c a s t in g p o vje r- ro b u o tn ess o f e s t im a t o r s ,
BIBLIO GRAPH Y
[ 1 ] B a r t o s z y r i s k i R. , P l e s z c z y r i e k a E . (1 9 8 0 ), R e d u c i b i l i t y o f S t a t i s t i c a l S t r u c t u r e s ond D e c i s io n P ro b le m s, M a th e m a tic a l S t a t i s t i c s Danach C o n te r P-ubll~ c a t i o n a . V o l. 6 , W arsaw, PWN, p. 29-38, [ 2 ] B o d n a r e k-K o z e k B . , K o z o k A. (i9 6 0 ), S t a b i l i t y , S e n s i t i v i t y and S e n s i t i v i t y o f C h a r a c t e r i z a t i o n s , M a th e m a tic a l S t a t i s t i c s Banach C e n to r P u b l i c a t i o n s , V o l. 6, W arsaw , PiVN, p . 39-64. [ 3 ] B o r g e r D. (1 9 8 0 ), S t a t i s t i c a l D e c is io n T h e o ry , B e r l i n , S p r in g e r - V e r la g . [ 4 ] B i c k e l P . (1 9 7 6 ), A n o th e r Look a t R o b u s tn e s s , Scan- d ln . J . o f S t a t i s t i c s , p . 145-168. [ 5 ] B o x G . , A n d e r s o n S . (1 9 5 5 ), P e rm u ta tio n The o r y In th e D e r i v a t io n o f Robust C r i t e r i a and th e S tu d y of D e p a rtu re s from A s s u m p tio n s , D. Ro y. S t a t i s t . S o c . , S e r . B , P . 1-34. [ 6 ] H a m p e l F , (1 9 7 1 ), A G o n e ra l Q u a l i t a t i v e D e f i n i t i o n o f R o b u s tn e s s , Ann. M ath. S t a t i s t . , p . 1887-1896. [ 7 ] H a m p e l F . (1 9 7 3 ), Ro b ust E c t im n t io n , Z e i t . f ü r W a h rsch . und V erw an d te G e b ie t e , p . 87-104. T B ] H o g g R . V. (1 9 7 9 ), An I n t r o d u c t io n to Robust E s tim a t i o n , [ i n s ] Rob uotn oss in S t a t i s t i c s , ed . L a u n e r R. L . , N. Y . A cadem ic P r e s s , p . 1-18. [ 9 ] H u b e r P . (1 9 6 4 ), Robust e s t im a t io n o f a L o c a t io n Pa ra m e te r , Ann. M ath. S t a t i s t . , p . 73-101. r 103 H u b e r P . (1 9 7 2 ), Robust S t a t i s t i c s : a X o v ie w , Ann, M ath. S t a t i s t . , p . 1041-1067. [ 1 1 ] H u b e r P. (1 9 7 7 ), Robust S t a t i s t i c a l P r o c e d u r e s , P h i l a d e l p h i a : S o c i e t y f o r I n d u s t r i a l and A p p lie d M a th e m a tic s . [ 1 2 ] H u b e r-C a r o 1 C. (1 9 7 8 ), Q u elq u es p rob lèm es de s t a
t i s t i q u e ro b u s te e t n o n - p a ra m o triq u e . These l e g rad e da d o c te u r en s c ie n c e s , U n i v e r s i t é de P a r i s S u d : C e n tre d 'O r s a y . t l 3 ] M o s h a l k i n L , (1 9 6 8 ), On the R o b u stn e ss o f Some C h a r a c t e r i z a t i o n s o f th e Norm al D i s t r i b u t i o n , Ann. Moth. S t a t i s t . , p . 1747-1750.
estym ato ró w o b c ią ż o n y c h param etrów m o d e li lin io w y c h . P . l : E s ty m a to ry r e g u la r y z u j ę c e , P a p e r w it h i n th e G ra n t R . I I I . 9 . 3 . 1 . [1 5 ] M i l o W, , W a s i l e w s k i Z . (1 9 0 0 ), Metody oa-
ty m o c ji poram etrów m o d e li lin io w y c h w przyp ad ku w ystęp ow an ia o b s e r w a c ji n ie ty p o w y c h . P a p e r w it h in th e G ra n t R . I I I . 9 .5 . [1 6 ] P o l l o c k K. (1 9 7 0 ), In f o r o n c o R o b u s tn e s s v s . C r i t e
r io n R o b u s tn e s s : an Gxomple, The Amer. S t a t i s t . , November. [1 7 ] R e y '.V. (1 9 7 8 ), Robust S t a t i s t i c a l M etho d s, L o c t . N otes
in M ath. 690, B e r l i n , S p r in g e r - V e r la g . [1 8 ] R o u e s e u w P . (1 9 8 1 ), New I n i f l t e s i m a l M ethods in R o b u st S t a t i s t i c s , P h . 0 . T h e s is , B r u s s e l s , F r e e U n i v e r s i t y o f B r u s s e ls . [1 9 j U 1 a m S . ( i 9 6 0 ) , A C o l l e c t i o n o f M a th e m a tic a l Problem s, V o l. 8 , N .Y . I n t e r s c i e n c e P u b l is h e r s , [2 0 ] Z i e l i n s k i R . (1 9 7 7 ), R o b u s tn e s s : a Q u a n t it a t i v e A p p ro a ch , B u l l . A cad . P o lo n a is e de S c ie n c e s , S e r i e des S c i. M ath. A s t r . e t P h y s . , V o l. XXV, No. 12, p . 1281-1286. [2 1 ] Z i e l i ń s k i R. (1 9 8 0 ), R o b u s tn e s s : a Q u a n t i t a t iv e A p p ro ach , Bonach C e n te r P u b l i c a t i o n s , V o l. 6 , M a th e m a tic a l S t a t i s t i c s , Warsaw , PWN, p . 353-354.
[2 2 ] Z o l o t a r e v V . (1 9 7 5 ), 0 n ie p r e r y v n o s t i s t o k h e s t i- c h e s k ik h p o s le d o v a t e ln o s t e y porozhdayem ykh re k u re n tn y m i p ro c e d u ra m i, T e o r ia V e ro y . i y e yo P r im . , V o l. 4 , p . 834-847. [ 2 3 ] Z o l o t a r e v V . (1 9 7 6 ), E f e k t u e t o y c h lv o e t i ch ra k -
t e r l z a y l r a s p r e d e le n lt y t Zap. Nauch. Sea. LOMI AN SSSR, p. 38-55. • a •
.
i • ■ • * . . . . x .. «
W ła d y s ła w M i lo , Z b ig n ie w W a s ile w s k i
WYBRANE PROBLEMY 00P0RNYCH METOO ESTYM ACJI PARAMETRÓW MOOELI LINIOWYCH.
CZ. I l CHARAKTERYSTYKA ODPORNOŚCI
P r a c a z a w ie r a :
1) o p is in t u ic y jn e g o i h e u ry s ty c z n e g o z n a c z e n ia o d p o rn o ś c i, 2 ) o p is Ja k o ś c io w y m ia r o d p o rn o ś c i,
3) forroaln ę c h a r a k t e r y s t y k ę o t o c z e n ia stan d ard ow eg o a o d e lu lin io w e g o ,
4 ) d e f i n i c j e m ia r o d p o rn o ś c i e sty m a to ró w na zm iany o to c z e n ia danego modelu stand ard ow ego ( t j . m ia r o d p o rn o ś c i o b c ią -ż e n ią i m ia r o d p o rn o ś c i b łę d u ś r e d n ie g o ) ,
5) d e f i n i c j e o d p o rn o ś c i w o d n ie s i e n iu do em p iryczn eg o r o z k ła du e s ty m a to r a .
