A C T A U N I V E R S I T A T I S L O D Z I E N 3 1 3 F O L IA OECONOHICA 43 , 1905___________________
W ia d y s lo w M i lo * , Z b ig n ie w W o s ile w s k l
ON THE E F F IC IE N C Y OF WEIGHTED LEAST SQUARES ESTIMATORS IN THE CASE OF A GENERAL LINEAR MOOEL
1. In t r o d u c t io n
The co n ce p t o f " e f f i c i e n t e s t im a t e s ( e s t i m a t o r s ) “ woo i n t r o duced by R. F i s h e r ( c f . [ 3 ] ) f o r d e n o tin g c o n s ia t e n t a- s y m p t o t lc a lly norm al (C A N ) e s t im a t o r s w it h the a s y m p t o t i c a ll y mi n im a l v a r i a n c e .
F i s h o r ' s re a s o n in g c o n s is t e d in showing th a t
a ) i f 0 i s th e maximum li k e l i h o o d e s t im a t o r (M L E ) o f the p a ra m e te r e . then under the f o llo w in g r e g u l a r i t y c o n d it io n s .
a t ) th e d e n s it y f u n c t io n f y ( y , 6 ) o f th e d i s t r i b u t i o n f u n c t io n F y o f a random v a r i a b l e Y i s tw o - fo ld d i f f e r o n t i o b l e in 0 .
a 2) the f u n c t io n £ lo g f y C y . ^ “ u n if o r m ly c o n tin u o u s in y
the q u a n t i t y ^ ( 8 ( n ) - 6 ) has a s y m p t o t ic a l ly norma ^ d i s t r i b u t i o n w it h the p a ra m e te rs ( 0 , I 1 whore 1 ( 6 ) “ 8 [ f ^ 30 ] d o n o to ° an in fo r m a tio n q u a n t i t y g iv e n by a eample about 0 . and E d e n o te s an e x p e c t a t io n o p e r a t o r .
b ) i f { T ( n ) J d e n o te s a sequenco o f a s y m p t o t ic a l ly norm al e-a t lm e-a t e s , then
lira ( T (n ) - e » 2 > i " 1^ ) . e « R1, n-wo
where R1 i s th e s a t o f r e a l num bers.
* D r . f 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 ta -' “ -' A V i . X r -' S i i ï . S " « * -' “ 4; - . I n s t i t u t e o f E c . n o . . « r l = . and S t . - t i s t i c a , U n i v e r s i t y o f Ł ó d ź .
I f ( a ) and ( b ) b o ld , then the MLC 0/ >\n) sh o u ld bo c o n s id e ru d os tho b e st a s y m p t o t ic a lly norm al (B A N ) e s t im a t o r s in tho c la s s o f a l l a s y m p t o t ic a lly norm al e s t im a t o r s .
U n f o r t u n a t e ly , th e ro a re no e s t im a t o r s w it h th e m in im al v a r ia n c e . I t r o s u lt o from tho f o llo w in g f a c t s . L o t | T ( n ) } b® any scquenco of e s t im a t o r s and th e v a r i a b l e V n ( T ^ - 9 ) be aeym- p t o t l c o l l y norm al w it h tho p a ra m e te rs ( 0 , <S" (0 )). L e t ci2( 0 o ) * * 0 bo f o r tho f ix e d v a lu o o f 0Q. F o r
(1)
f (n)
T( n ) l f l T (n> - 8 o l > " 4 • 1 s „ t f l T ( n ) - 8 « l ‘ ^ ■
one can chock th a t the sequonce o f V n i f ^ - 0 ) i s o e y m p to tic - a l l y norm al w it h the p a ra m e te rs ( 0 , 3 2 ( 0 ) ) , where
3 2 ( 0 ) - d2( 0 ) i f 0 4 0O,
o>2( 0 ) - 0 < d2 (0 ) i f 0 - 0 ,
o o
U sin g th e above way o f e s t im a t o r Im provem ent f o r th e ML e s t im a t o r s in th e r e g u l a r i t y c a s e , one can c o n s t r u c t a sequence | l ( n ) | a s y m p t o t i c a ll y norm al e s t im a t o r s such th a t
( 2 ) . lim fe0 (VnXT( n ) - 0 ) ) 2 < I ~ * ( 0 ) , n-K»
and, f o r 8ono 9 t oven
( 2 a ) lim t e (V n (T ( n ) - 0 ) ) 2 < r a ( 0 ) . r.+oo
The e s t im a t o r s j T ^ j t h a t f u l f i l ( 2 ) an d /o r ( 2 a ) f o r 0 e 0 a re c a l l e d s u p e r e f f i c i e n t f o r convex lo s s f u n c t io n s and th e s e 0 f o r w h ich (2 o ) h o ld s a re c o l l e d s u p e r e f f l c i e n c y p o i n t s .
The f i r s t known to us improvomont o f the typ e ( l ) was p re s e n ted by 3. H o d g e s ( c f . [ 7 ] ) and i t was con cern ed w it h the ca&a when Y^ , . . . , Yn , . . . , w ere i . i . d . norm al v a r i a b l e s w it h v a r (y ^ ) = 1 , Q s R , and
^ ( n )
e ,7 ( n ) l f ,V\ n ) l < r," 1 / 4 ' | 0 ,1 < 1 '
Y (n> t f l 7 ( n ) l > n‘ 1 / 4 * 9 ( n ) 10 th0 ML" 0f e'
«h ere ^ ( n ) i s a u p c r e f f l c l e n t a t
6
*0
.Theso now f a c t o have s tlm u lc tn d nonds f o r m o d if ic a t io n s o f F i s h e r ' s d e f i n i t i o n o f e f f i c i e n c y . We r e c a l l the f o llo w in g m o d if ic a t i o n s : m l) an o o tim a to r T (n ) i s s o ld to bo a s y m p t o t ic a lly e f f i c i e n t f o r a p a r a m e tr ic f u n c t io n g (0 ) i f f : Urn i ( T (n )) -
9
(8
) . lim | v a r ( T ( n ) ) - ° ' n-»oo n+oo u n le s s * < T ( n )* v a r ( T ( n p do n ot ®x i 9 t l m2
) a f a m ily of e s t im a t o r s0
£ lo s o ld to be w£ - a s y m p to ti c a l l y e f f i c i e n t w i t h in the sub3
p ace K c0
e R1 ( a s y m p t o t ic a l ly e f f i c i e n t w it h r e s p e c t to ( w ( J ) I f f ° r “ och n on-void op>in i.ot K c K tho f o llo w in g r e l a t i o no
( 3 ) lim [ i n f sup B(eC) wt (T £- 0 ) - sup i lQ£) w£ ( § £ - 8 ) ] - 0 .
£+0 "t£ 0«Ko 9«Kq
h o ld s , where T£ i s any e s t im a t o r , ond E i s e q u a l f o r exam ple m3) a c o n s is t e n t e s t im a t o r o f 0 i s s a id to bo f i r s t o r d e r e f f i c i e n t ( f . o . e . ) i f f
( 4 ) lim V«? | T , v - 9 - a ( 6 ) * n I ■ 0» П-+00
In p r o b a b i l i t y , where a does n ot depend on o b s e r v a t io n s and
•
•
* •
’ -
...
•
?
>
. . . .
L alog f y ( y j 0>
z n “ n --- W ‘
A l l p r e s e n te d d e f i n i t i o n s o f e f f i c i e n c y e re q u a l i t a t i v e in n a tu r e and d e a l w it h th e a s y m p to tic b e h a v io u r o f e s tim a to r® . They can be g e n e r a liz e d in t o the c a s e o f 0 с R , R b e in g th e
Eu-tfiatdyatitw M ilo. 2ttl(?iiow Wasilovoki
c li d s a n ppaco w ith th e d im e n sio n k , by in c lu d in g o i l the r o e t r i - c t io n o im p lie d by the n u lt i- d im o n e io n a llt y o f the c o n s id e re d p ro blem s. Though the w hole a s y m p to tic s o f e f f i c i e n c y i s ln d is p e n s -
b io (botl> in K1 nnd R*1,) «a e brckgroun d knowlodgo f o r s m a ll sam p le a n a ly s i s o f e f f i c i e n c y , in p r a c t i c e we nood some m easures o f e f f i c i e n c y f o r s m a ll »a m p le » , and tho knowledge how th e y behave in dependence on changes o f assu m p tio n s w h ich u n d e r li e tho mo-d,>J. g e n e r a tin g the o b s e r v a t io n s Y.
Tho p urpose o f t h i s p a p e r i s to p r e s e n t one o f p o s s ib le w ays 6f m easuring the e f f i c i e n c y and to o n a ly z a sqme p r o p e r t
ie s o f the p re s e n te d ¡¿» te rm in a n ta l e f f i c i e n c y m easure.
In S 2 we p ro s o n t on a n a l y s i s o f th e p r o p e r t ie s o f weight»- ed e s t im a t o r s in the c«*»e o f a g e n e r a l l i n e a r m odel.
I n § 3 wa p r e s e n t a d e t e r m in a n t a l e f f i c i e n c y measure and p ro v e t h a t i t s range b e lo n g s to < 0 ,1 > c r | ,
I n S 4 wo d e r iv e lo w e r bounds o f th e d e t e r m in a n t a l e f f i c i e n c y m easure.
2. Some P r o p e r t ie s o f W eig h ted E s t im a t o r s in the Case o f a G e n e r a l L in e a r Model
By a g e n e r a l l i n e a r model we u n d e rs ta n d the f o llo w in g m odelt
« ( * " , k , S , Y - X(J + S , k0 - k ,n 0 - n, 9 y - Jfy ( X(3, a ) ) , w h e ra t • r 1Xi- - the s e t o f nxk r e a l m a t r ic e s , S - a p r o b a b i l i t y s p a c e , & - ( t l , * F , 9 ) , U - a s e t o f e le m e n ta r y e v e n t s , '¡F - tho t f - e o r e l f i e l d o f It s u b s e ts , 9 - a com p lete p r o b a b i l i t y m easure,
X/J - 8 ( Y ) ,
0
« A ( Y ) . fc(Y - tt(Y )) ( Y - f c ( y ) ) # - *> ( s ) , X e Rn xk, ft 6 f t * ," 9., - Ji*r ( Xf3, n ) “ = “ the p r o b a b i l i t y d i s t r i b u t i o n o f Y i s n- ••dirsanoional norm al d i s t r i b u t i o n w it h ? ( Y ) • X|3, & ( Y ) ■ n • d 20*. One of the p o s s ib le e s t im a t io n q u a l i t y f u n c t i o n a ls f o r the
( 5 ) (p - | i i ‘ 1 / 2 ( Y - X ( 3 ) J 2 . w here | • || d e n o te s E u c lid e a n norm in R1’.
About n vre assume f u r t h e r
A l ) n i e n o n s in g u la r p o s i t i v e d e f i n i t e r e n l m a t r ix , i . e .
a « Rn xn. d e t n * 0.
By c o n v e x it y o f <p and ( A l ) I t i s e a s y to f in d tb o ts
( 6 ) erg mln | i l ' 1/2 (Y - x p ) | 2 - D - ( x 'f l “ 1X ) ’ 1 x ' a " XY (3
and by the d e f i n i t i o n o f Y and assu m p tio n s o f we 9e t
(7)
A ( i ) - (x 'rf’ x)'1 x'a"1j&( Y)n~1x(x'a"lx)'’ - ( x ' a ’ x)*1
D e n o tin g L - ( x ' n “ 1x ) " 1x ' n " 1 < L% - L ♦ C . C « Rkxn we have ( 8 ) B - LY , f o r th a n - w e ig h te d le a s t - s q u a r e s (a - V IL S ) e s t im a t o r , and ( B a ) ^ - L1Y , f o r a n y o t h e r w e ig h te d le a 3 t s q u a re s e s t im a to r . To be u n b ia sed th e e s t im a t o r must f u l f i l £ (8 ^ )» L^X(3 ■ ■ (3, I . e . L^X ■ l ( j ( )* The l a s t r e l a t i o n h o ld s i f f CX ■ 0 . B y th e l a s t o o n d it lo n th e e s t im a t o r B^ can be w r i t t e n as ( 8 b ) B4 - (1 ♦ L j S . B y (8 b ) end the p r o p e r t i e s o f th e d is p e r s io n o p e r a to r Si we heve £(&:l) m
U j f t ^ - L A I . ' ♦ L A C ' ♦ C f t L ' * C f l C ' . S in c e L A C * - ( X ' i T ^ r V ( X ' o " 1X ) “ 1X ' C ' # then by CX -■ 0 we o b t a inL ft C « 0, m d by the same orgum onts C f t L ' » 0. Thus
( 9 ) ¿ ( S j ) » I f l L * ♦ C i l C ' . By (7 ) and ( 9 ) we o b ta in
U o > A ( 6 ^ - J B ( 6 ) - C A C '.
By. ( A l ) and p r o p e r t ie s o f Grom m a t r ic e s , the m a trix C n C ' I s a ls o p o s i t i v e d e f i n i t e i f f rank C » ran k X ■ k .
o We have p roved t h e r e f o r e
T h e o r e m 1. Under the model JCtll l n , th e l i n e a r e s t i m ator i s o f f i c i o n t in the aenso t h a t a n y o t h e r l i n e a r e e t lm a to r Ba ■ (L ♦ C )Y has the d is p e r s io n m a tr ix S K & j) d e fin e d by ( 9 ) and such th a t A (B ^ )> A (D ), i . e . 4 (6 ^ ) — A ( B ) i s a p o s i t i v e d e f i n i t e m a t r ix . ♦
The e s t im a t o r B i s th e fl-VVLS e s t im a t o r w it h the w e ig h t ma t r i x f t . I t can be used o n ly i f t h i s m a tr ix i s e x a c t l y known in p r a c t i c e . In most p r a c t i c a l s i t u a t i o n s we do n ot know th e m a tr ix but i t 3 a p p ro x im a tio n , i . e . th e m a trix a ^ g • where £ rung the in d ic e s o f e s t im a t o r s o f a u t o c o r r e l a t io n c o e f f i c i e n t s o f g , and 5 runs th e in d ic e s o f a u t o c o r r e la t io n schem es. The c o n c r e te form o fft^ g depends on th e assumed a u t o c o r r e la t i o n sch e mes about the components o f th e v e c t o r 2 as w e ll as e s t im a t io n methods f o r ’ the a u t o c o r r e l a t io n c o e f f i c i e n t £ in a s p e c i f ie d a u t o c o r r e la t i o n schema. The most o f t e n used schemes in p r a c t i c a l e c o n o m e tric end s t a t i s t i c a l a p p li c a t i o n s a re as f o llo w s :
- f i r s t o r d e r a u t o r e g r e s s iv e schem es, - second o r d e r a u t o r e g r e s s iv e schem es,
- f o u r t h o r d e r a u t o r e g r e s s iv e schemes ( f o r a q u a r t e r l y d a ta ), - f i r s t o r d e r m ovin g-average scheme,
- com bining a u t o r e g r e s s iv e m ovin g -averag e scheme.
The above m entioned ty p e s o f schem es, as w e l l as o t h e r sche- m'js, form the f i r s t c r i t o r l o n o f the d i f f e r e n t i a t i o n o f fta . The s e co rd c r i t e r i o n i s formed by d i f f e r e n t ty p e s o f e s t im a t io n me tho d s f o r th e a u t o c o r r e la t io n c o e f f i c i e n t £ , i . e . f o r exam ple, the f o llo w in g e s t im a t o r s !
a ) th e sample f i r s t o r d e r a u t o c o r r e la t i o n c o e f f i c i e n t (o r the cam ple c o r r e l a t i o n between the s u c c e s s iv e r e s i d u a l s )
X ! E t E t - l
A
.
---1 A ?
5 2 E 2 t-1
where 3 (u n o b s e r v a b le ) w ere ro p la c e d by tho LS r e e ld u a la E^ ■ • Yt - X* p , x ; - ( X t l ... X t k ) . B - ( X ' X ) " V y , t -
1
,...,
n - l T ' n'j b ) T h e l l ' s m o d if lc o t io n o f £j,j n (n-k) E E E A * t* 2 1 P I “ n ' (n - 1 )Ë
e 2 t - l r c ) th e e s t im a t o r n n ? ! ■ 1 - Ï - d ■ Z ( E . - S J - / Z t« 2 t - l d ) T h e ll^ s - N a g a r 's m o d if ic a t io n o f i . e . - * .f
k* f l " n2 - k2e) Dent's adjustment of Thell-Negar estimator
ex
u) *
u „ .
s — ~ 2Г
k2 — ~ (m -
U1
• * "
(ł *
m - t r [ ( X ' H X X X ' X ) " 1 ] ,
H i s tho m a trix from an a p p ro x im a tio n o f th e in v a rs © of d2F «
■ A , i . e . H t F " 1 ■ (1 H.
Gxcopt f o r v o r y r e s t r i c t i v e c o se s t h e r e i s no s u f f i c i e n t knowledge about b e h a v io u r o f tho b ia a - r o b u s t n e s s , mean sq u are o r r o r (M S E l- r o b u s tn e s s , e f f i c i e n c y r o b u s tn e s s o f A a g-WLS e s t i m ato rs on the chan g es o f assu m p tio n s d of ln in g f o r each known ( c x , 5 ) . T h ie c a l l s f o r e x t e n s iv e s t u d ie s on n u m o ric a l s im u la t io n ro b u s tn e s s . I n t h i e p a p e r, h o w e v e r, we w i l l n o t q n a ly z e the
*
p rob lem s o f s tu d y in g r o b u s tn e s s . In s t e a d o f t h i s we w i l l s tu d y oome p r o p e r t ie s o f ft^ g-W LS e s t im a t o r w it h r e s p e c t to ft-WLS- - o e t im a t o r . One o f such p r o p e r t i e s c o n c e rn s tho c o n d it io n s o f e- q u a l l t y o f d is p e r s io n m a tr ic e s o f A ^ - W L S E and A - W L S E , i . e . f o r d is p e r s io n o f Y , B ■ 6 (a ) » • ( X ' n “ 1X ) “ 1X ' F o r th e b r e v i t y o f n o t a t io n v*e d en o te a * - (-yv 5 ) , A ^ - Qq, . B y ( 7 ) and th e p r o p e r t i e s o f th e o p e r a to r it we have ■6(8^) • ( X ' û ' ^ ^ X ' û ^ û û ^ X ' n ^ X ) ' JîiO q ,; • i * A 11 a u “ a * VA i 1” 1 “ ' “ 1 and U r l * ( B ) - (X * a_ 1x) We s h a l l f i x th e c o n d it io n s o f e q u a l i t y J6 ( Bq, ) » A ( B ). Such con d i t i o n s a re g iv e n in T h e o r e m 2. I f . a ) i l i l q j — ^ot ^ i b) V I s th e m a tr ix o f th e a ig e n v e c t o r s o f û a and A , I . e . V 'V * W ' - I ; c ) V « RnXK i s o f th e form V ■ ( v - ♦ V . » . , v . ♦ O , O • JL # fl • Iv
♦ v . « ) whpre v . ( i - 1 , n ) l e th e i - t h column o f V and v ; v o ■ « e u - w i • « i n ) 1 d ) th e m a tr ix X s a t i s f i e s , ‘ a l t e r n a t i v e l y , one o f th e f o l lo w in g c o n d it io n s ] d l ) X - Vo , L y L d 3 ) X - VQG, G 'G - GG' - I ( k ) , G « R * X K î ® L v|/ d 2 ) X » V G, d e t G * O, G « R
than
(1 1 ) A ( B W ) - A ( B ) . ♦
To p ro v e t h i s theorem one needs to use c o n d it io n s ( a ) - ( d ) , the p r o p e r t ie s o f sym m etry, p o s i t i v e d e f in it e n e s s o f fta , ft, th e orem o f S t o l l ond W o n g [ 6 , p . 2 2 7 ] ond the r e s u l t in g f e c t e t h a t f t V • VA, f t '^ V ■ V A V i l 1 » A , f l a v * • V F , f t a * v " V F” 1 « v # f t a * “ F ” l v * »»her# A , F a ro th e d ia g o n a l m a tr ic e s o f e ig e n v a lu e s o f U and f t « , and A , F a re th e e ig e n v a lu e s m a trlc o o f o r f t ” ' and f t ^ . A d e t a ile d p ro o f i s g iv e n in th e work [ 5 ] . A n o th e r t r i v i a l a l t u e t i o n when JB (6 a ) - JJ(6 ) h o ld s , l a d e t e r mined by th e c o n d it io n f t a * f t , i . e . then ( X ' f t 1X ) X ft ftft x ( x ' f t “ 1x ) " 1 - ( x ' f t “ 1x ) “ 1 . 3. A P e t e c y l n a n t a l E f f i c i e n c y Measure We r e c a l l t h a t th e most e f f i c i e n t e s t im a t o r In th e co3e o f <JTcUA l a the e s t im a t o r 3 • B < ft)o n d an y o t h e r e s t im a to r In c lu d in g B tt i s le s s e f f i c i e n t . In o r d e r to a r r i v e a t t h l e c o n c lu s io n we r e p r e s e n t th e m a tr ix n a as f t a - ft ♦ A, I t 1» e a s y to f i n d t h a t ( f t * A )’ 1 - f t " 1 - f t ” 1 ( f t “ 1 ♦ A " 1 ) “ 1«,” 1 - ft“ 1 - G, w here G • f t " 1 ( f t " 1 ♦ A- 1 ) " 1 f t “ 1 . By d e f i n i t i o n o f B we now have B a - ( x # ( f t -1 - G ) x ) “ 1x '( f t “ 1 - G )Y , end s in c e ( x ' i f t “ 1 - O x ) “ 1 - ( x ' ft-1x - X 'G X )“ 1 - ( x ' a “ 1x ) " 1 - - ( X # f t ” 1X ) " 1 [ ( X ' f t ^ x ) “ 1 - ( X ' G X ) " 1 ] “ 1 ( X ' f t ^ X ) “ 1 , x '( f t -1 - G )Y - X ' f t ' V - X 'G Y ,
t h e r e f o r e
(
1 2)
Da ■ ( L ♦ C ) Y( 1 3 ) L - ( X # A * ł X ) “ V f t " 1
(1 3 a ) C ( x ' n " 1X ) “ 1X ' G ( X ' f t " 1X ) " 1 [ ( X ' f t ’ 1 X ) " 1
-- ( X ' C X ) “ 1 ] “ 1 ( X ' f t ' 1X ) “ 1 ( X ' f t " 1 -- X ' G ) .
By (1 2 ) and Thoorem 1 we o b t a in t h a t * ( B a ) > & ( § ) , i . e . 6 i e le o a e f f i c i e n t than B . F o r f ix e d f i n i t e «am ple s iz e s ( e s p e c i a l l y s m a ll sam ple s i z e s ) I t I s v e r y Im p o rta n t to havo a m easure o f e. f l c l e n c y o f th e g iv e n e s t im a t o r . Such a moaeuro I s de f ir , ad in
D e f i n i t i o n 1. The d e te r m ln a n t a l e f f i c i e n c y measu re o f the e s t im a t o r 6 ^ f o r (3 in <M<Mn I s th e q u o t ie n t o f the d e te rm in a n t o f the m a tr ix A ( B ) to the d e te rm in a n t o f the m a trix £ (8 ^ ), i . e . /.here ^ Bq, " ( X , i y i a ) d e n o te s th e d e te r m ln a n t a l e f f i c i e n c y m easure o f B a ae the f u n c t io n o f X , A , A a . ♦ L e t us assume t h a t t A 2 ) f t * I s n o n s in g u la r p o s i t i v e d e f i n i t e r e a l m a tr ix (r a n k A a -- nQ s n ) , A 3 ) f t f t a - A a a . By ( a i ) , ( A 2 ) , (A 3 ) th e f a c t s t h a t th e in v e r s e o f p o s i t i v e d e f i n i t e m a tr ix I s a ls o a p o s i t i v e d e f i n i t e m a tr ix and the p r o d u c t o f p o s i t i v e d e f i n i t e m a tr ic e s i s a l s o a p o s i t i v e d e f i n i t e m a tr ix - ( f o r p r o o f s o f t h i s s ta te m e n ts see [ l ] , c h a p t e r s 4 , 6 ) wo o b t a in t h a t ( 1 5 ) A , A a a re p o s i t i v e d e f i n i t e ( p . d . ) r e a l m a t r ic e s . (1 4 ) Ч • • ” в а ( Х ’ Л , Л “ )
__________ da t ^ U ' f l ^ X )
___
__
dot ( Х ' л ~ 1Х ) det ( X ' ^ f l f t j x ) '(1 6 ) A f l " 1 i s p . d . m a tr ix , and by th e p r o p e r t ie s o f Gram m o trlc e s (1 7 ) X 'a “ * X . X* A ^ X , X ' A ” 1 f l A ^ X o re p . d . m a tric e s . B y th e p r o p e r t ie s o f Gram d e te rm in a n ts wo g e t * (1 8 ) d e t ( x 'A " a x ) > 0 , d o t ( x ' n ^ x ) > o , Ix'a^ a a ^ x |> 0 .
From (1 8 ) ond (1 4 ) we have t h e r e f o r e
( 1 9 ) > 0
B ot and
( 2 0 ) V- . 0 i f d e t ( X 'A -1 X ) A 0 , d o t ( X 'A " ^ A A * ^ X ) 4 0,
d e t ( X ' A ^ X ) - 0.
F o r k < k , V i ■ — eo the meoeuro makes no sense and
o
0
th e r e a r i3 o s a need f o r m o d if ic a t io n s - one o f eoch m o d if ic a t io n s can bo based on ta k in g
V 6 a ■ V r i l n ^ • Now we e s t a b l i s h t h a t Vg < 1 .
Becau se f o r p o s i t i v e d e f i n i t e m a tr ix th e d e te rm in a n t i s o q u a l to th e p ro d u c t o f i t e p o s i t i v e e ig e n v a lu e s , t h e r e f o r e by the f a c t t h a t JBXBqi)
5
& ( B ) we coma to th e c o n c lu s io n(2 1 ) detteK & Q j)) > d o t(X > (B )).
Hence e f f * “ V* < l .
ot ot
We h ave p ro ve d the f o llo w in g theorem .
T h e o r e m 3. Under the assu m p tio n s o f dfcttrt, ( A l ) , (A2) th e range o f th e d e te r m ln a n t a l e f f i c i e n c y m easure V g i s the i n t e r v a l < 0 ,1>.
N o t e : I t i s e a s y to ch eck t h a t f o r A a - A , V g * 1.
I n p r a c t i c e i t i s w o rth knowing th e v a lu e s o f lo w e r bounds o f Va d e te rm in e d f o r some t y p i c a l a » ( y , 6 ) c o m b in a tio n s o f
c o r r e l a t i o n sch e n cc and o s t im o t o r s o f /p. We d e r iv e th e s e bounds In this nu;<t s e c t io n .
4 . Lower Bounda o f V *
— --- --- ---£ a
Cno o f the p o s o lb le ways o f d e r i v i n g lo w e r bounds o f V*
0 ®0f
in the C 8 6 o o f X ■ V , i . e . X X » cen k ° doc* on *he
method o f Lagrange m u l t i p l i e r s . L e t % )ml m 2 ‘• " V o ' ~ ( k ) ( 2 2 ) g - In ( V * ) -1 - 2 t r L ( v ' v rt - 2 1 ,- 0 , w h o re i I n ( Ya ) _1 In d e t ( V ' a 1V ) ♦ In d o t ( V ' f t * 1 A f l * 1V ) -O O o or
01
o - 2 In d e t ( V ' A “ XV ) o a ound L i s the upper t r i a n g u l a r m a tr ix o f k ( k + l) / 2 o f L a g ra n g e 'e m u l t i p l i e r » . From ( A l ) - ( A 3 ) I t f o llo w s (2 2 a ) n a l n “ 1 " a_1 . ^ A - ft a ” 1 - H*1 , n ” l a a " * H» H " H#* ( 22b ) ( n “ 1 A ( T * ) A " 1 - A^1 ( A “ 1 A A ; 1 ) , ( 22C ) ' ( A “ 1 A A ' J ) A " 1 - Sl~ J ( A ^ 1 A A “ 1 ), ( ¿2d ) ( A ” a A A " a ) H - H (A "a* A A " 1 ), ( A “ 1 A A fl1) « 2 - H2 ( A * 1 A a 1^1 ) , ( 22 e) ( A ^ 1 A A ^ J H ' 1 - H“ 1 ( A ^ 1 a A ^ ) .
VSte can fo rm u la te now ■
T h e o r e m 4 . I f V^VQ ■* 2 1 ^ and ( 2 2 a )- (2 2 e ) h o ld and the f u n c t io n g i s c o n tin u o u s tw o fo ld d i f f a r e n t l a b l e , then
k
(2 3 ) i n f « „ » > ¡ 1 » X . i . 1 (l> J • i - l
(2 4 ) . Vo , v ; v 0 - 2 I ( k )
w here 2H# ■ V 'H V te th e m a trix o f the form H - d ia g I h j ,
*
0 0
h j ) . ♦
P r o o f . D i f f e r e n t i a t i n g
g
w it h r e s p e c t to X ond L ( c f . [ 4 ] . p . 6 1 6 ) and p o t t in g 9g/dX - 0 and g / L - 0. where 0 deno t e s z e ro m a t r ix , wo have(2 4 a ) fta l f t A e v e ( v ; * ; 1 A r f * v e r 1
-- a"'«4(vi ûe 4 r i -- 2Vo(L + L#)‘ °*
(2 4 b ) v ; V o - 21 ( k ) - ° . P r e m u l t ip ly in g (2 4 a ) by v ' and u s in g (2 4 b ) we have ( 2 4 c ) L ♦ L ' « 0 and hence
<2*d)
0
' \
( v ;
. л^‘ л п '* » 0 ( v ;
-•
2na4(v;na4r‘-P r a m u lt ip l y i n g (2 4 d ) by Л а and u s in g the d e f i n i t i o n o f H wo ob t a i n
( 2 « ) » » д Л , ) - 1 ♦ H-‘ ye ( v ; n ; 1 nn-M
1v0 r ‘ • » .< » ; < » e r ‘ .
P r o m u lt ip ly in g (2 4 e ) by H and u s in g the r u l e s o f the a d d it io n of th e m a t r ic e s wo o b t a in( 2 4 0 H ^ C V ^ A - ^ ) * 1 - 2HV0 ( V ; Л * \ ) _1 ♦ Vfl( v ; Л ^ П ^ Г ^ О .
From Theorem 2 and c o r o l l a r y 1 (From [ 7 ] , § 6 . 6 , p . 22 8) i t f o llo w s t h a t m a tr ic e s H2 , H, a re s im u lt a n e o u s ly d ia g o n a liz o d by th e m a tr ix VQ o f th e s e m i- o rth o g o n a l tra n s fo r m a t io n V * . H ence,
о
+ v!i(v^
n ;lAft-« v!ir l - o. i - 1...k
and by v i r t u e o f the theorem , t h a t tho d e te rm in a n t o f the d ia g o n a l m a tr ix i s e q u a l to the p ro d u c t o f i t s main d ia g o n a l e le m e n ts wo g e t k < vft % ‘ v : , ) 2 ^ ; n - 1 v » t ) - ‘ ( v f a i n o i S j r » . X - l P re m u H rlp ly in g (2 4 g ) by v ^ C v ^ n “ I v ° 1 ) wo o b t a in * • (2 5 a ) >
2
h " ( v ^ n - V ^ o ' ^ y ^ - 1 f * ( v ° ; o - y ^ v j ; - o i . i ... k.V/hen A o f t h is s q u a ro d - e q u a tio n in r e s p e c t to h * l a g r e a t e r than 0 , i . e .
4 ( v ° ; n v p 2 ^ ; n - ^ y . r 2 - e ^ ; n - y ^ ; * ? a < c y t r * x >
e q u a tio n (2 5 a ) has two r o o t s , d i f f e r e n t from z e r o , tho sum o f then bning o f the form
( 23 b ) „ * . h j 2 . < v»; o - y t ) t v » ; .
and t h e i r p ro d u c t has. tho form
< 2 5 0 . J ( v » ; n ’ V , ) < v °; n ^ n r i - ^ v - ) “ .
F ro n (2 5 b ) and (2 5 c ) i t f o llo w s d i r e c t l y
< 2 M ) c v ° ; . ^ v ^ ) 2 , ( v ° ; a - y p 2 ^ ♦ h * r 2
(2 5 e ) ( v f OTj- - 2 ( y ° ' h ^ h * .
U sin g ( 25d) and (2 5 e ) we can r e w r it e (2 5 ) in th e form k
I n o rd e r to f in d I n f o f A (V wo must choose two d la jo im -OL
subgroups { h ^ ... h j j . { h j 2 ... h *2 } from th e a c t { h * . . . . . h£ } ln 8uch 8 w®y th n t tho r e l a t i o n d e f in in g e i
3
& a ’a o) in (2 6 ) re a c h e s i t s minimum. I n a cco rd a n ce w it h the c o m b ln a to ri- c e l d i s c u s s i o n ( s i m i l a r to th o t o f B l o o m ? i e l d-v< a t-a o n [2 ]) we have t h a t the e x p r e s s io n e fg re a c h e s i t s minimum f o r h * x - h * . h *2 - h * _ 1+1. i . e .
k o
i n f • f 6a( Vo ln 1n a ) * H
2
h > * _ u l (h * ♦ h * _ t+ 1 )- ‘ . i- 1w hich co m p le tes the p r o o f.
R e l a t i o n (2 3 ) may be used f o r exam ining the runs o f rnngo of tho lo w e r bound o f the d o le r m in o n tn l e f f i c i e n c y measure o f tho e s t im a t o r B a .
I t i a i n t e r e s t i n g to d e te rm in e o th o r lo w e r bounds r a t h e r than th e se from Theorem 4 f o r X » VQ. To f i x thorn on the b a s is
s t a t l o n a r l t y c o n d it io n s f o r Vg one has to f in d s o lu t io n s ot the f o llo w in g e q u a t io n s : dVA (2 7 ) - ^ - " ( 2 d e t ( x #a “ 1x ) d o t 2 ( x ' ^ 1x ) ) A ; 1x ( x 'n ’ 1x r 1 - ( d c t ( x ' A ' 1x3i), ft“ 1X ( X #0 - 1X )- 1 -< d a t< X 'a - 1X ) ) o ; 1 ^ X ) “ 1 - 0, 3Vn (2 8 ) r r r - 2 . ( d e t 2 ( x ' n ; 1x )) i l ^ X U ' n , « 1X ) ’ 1X 'ft” 1 - n t t f x , ( X ' n ; l A n ; 1X ) " 1X ' - 0, av*
(29)
_ !S !
. n - i x (x * n ^ x r Vn " 1 ♦
^ x i x ' ^ A ^ x r ^ ' f t ; 1- o.
oi* W h eth er th e re a re g e n e r a l s o lu t io n s o f ( 2 7 ) - ( 2 9 ) ? I> th e y e- x l s t , t h e y in v o lv e a d d i t i o n a l a ssu m p tio n s about tho e x te n s io n s o f d ot f u n c t io n s w it h r e s p e c t to a o r f i ^ o r X o r tho r e s t r i c t i v e form o f X. Duo to space l i m i t a t i o n s we w i l l n ot c o n tin u e the d i s c u s s io n o f t h i s problem and le B v e the problem open.5. F i n a l Remarks ■ i r f » w ; m - i i i i w i i i tw — m Tho r e s u l t s o b ta in e d in t h i s p a p e r w i l l be uaod in ou r s t u d io s on ro b u s tn e s s o f v/LS e s t im a t o r s os w o ll os f o r the c o n s tr u c t i o n o f o f f i c i o n c y t o b ie s f o r d i f f o r e n t c o m b in a tio n s o f p a i r s ( ? » 6 ) •
Tho a n a l y s i s p r e s e n te d in the p a p e r does n ot in c lu d e the co se when k < k end n < n in the model tV’tW.rt.
o o A DIBLIOGRAPHY T l ] B e l l m a n R . ( i 9 6 0 ) . In t r o d u c t io n to M a t r ix A n a l y s i s , N . Y . , M c G ra w - H ill. [ 2 ] B l o o m f i o l d P . , W a t s o n G. (1 9 7 5 ), The I n e f f i c i e n c y o f L e a s t S q u a r e s , “ B io m e t r ik o " , V o l. 62, p . 121- -128. [ 3 ] C r o m e r H. (1 9 4 6 ), M a th e m a tic a l Methods o f S t a t i s t i c s , P r in c e t o n N. 0 . , P r in c e t o n U n i v e r s i t y P r e s s . [
4
] D w y e r P ; (1 9 6 7 ), Some A p p li c a t io n s o f M a t r ix D e r iv a t i v e s in M u l t i v a r i a t e A n a l y s i s , 0. Am er, S t a t i s t . A s s o c . , V o l. •62, p . 607-625. [ 5 ] M i 1 o W ., W a s i l e w s k i Z . (1 9 7 9 ), E fe k ty w n o ś ć e stym ato ró w param etrów o g ó ln y c h m o d e li lin io w y c h . Cz. i . Work under c o n t r a c t R . I I I . 9 . 5 . 7 . [ 6 ] S t o l l R , , W o n g E . (1 9 6 9 ), L in e a r A lg e b r a , N. Y , , A cadem ic P r e s s . 17 ] Z a c k 6 S , (1 9 7 1 ), The T h e o ry o f S t a t i s t i c a l I n f s r a n c e , N. Y . , W i l e y . / /W ła d y s ła w M i lo , Z b ig n ie w W s s ile w s k l
O EFEKTYWNOŚCI WAŻONYCH ESTYMATORÓW
N AJM NIEJSZYCH KWADRATÓW
W PRZYPADKU OGÓLNEGO MODELU LINIOWEGO
Głównym calem p ro c y J e s t z a p re z e n to w a n ie jed n eg o z m ożliw ych sposobów m ie rz e n ia e fe k t y w n o ś c i w m ołych p rób ach i z a n a liz o w a n ie n ie k t ó r y c h w ła s n o ś c i ważonych e stym ato ró w n o Jw n le jo z y c h kwadratów 1 p r z e d s ta w io n e j w yzn aczn iko w ej m ia ry e f e k t y w n o ś c i. W sz cz e g ó ln o - ś c i p rz e d s ta w io n o :
a) a n a l i z ? w ła s n o ś c i e stym ato ró w ważonych w p rzyp ad ku o g ó lno -
go modelu lin io w e g o , , .
b ) dowód, że m iaro e fe k t y w n o ś c i z n a jd u je s i ę w p r z e d z ia le
< 0 . * > . X . i
