A Sequence of Improvements over the Lindley Type Estimator

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2 002 , V ol. 13, N o.2 p p . 11～19

A S e qu e n c e o f Im pro v e m e n t s o v e r th e Lin dle y T y pe E s t im at or^{1 )}

H oh Y o o B ae k^{2 )}

A B S T R A CT

In t his paper , t h e problem of estim atin g a p - v ariat e (p 4 ) n orm al m ean v ect or in a decision - th eor et ic set u p is con sider ed . U sin g a t ech niqu e of Gu o an d P al (1992), a s equ en ce of est im at or s dom in at in g t h e Lin dley t y pe est im at or is deriv ed an d each im pr ov ed est im at or is b et t er th an th e pr ev iou s on e.

K ey W ords an d P hras es : Im pr ov ed e st im at or , Lin dley t y pe estim at or , N orm al m ean v ect or .

1 . IN T R O D U CT ION

Let X = ( X ₁, , X _p) ' b e a p - v ariat e r an dom v ect or an d X N_p( , I_p) , R^p. F or any estim at or ( X ) of , t h e los s in estim at in g by ( X ) is

L ( , ) = || - ||²= ( - ) ' ( - ) . (1.1)

T h e st an dard est im at or (M LE a s w ell a s th e b est locat ion estim at or ) of is

0= X (1.2)

w h ich is adm is sible for p 2 . S t ein (1956 ) an d J am es an d S t ein (1961) sh ow ed th at

o is in adm is sible for p 3 an d it is dom in at ed by

1) T his w ork w as support ed by W onkw ang Univ er sity Res earch Fund, 2000.

2) A s s ociat e Pr ofes s or , Division of M at hem atics an d Inform ational St atis tics , W onkw ang Univ er sit y , Iks an , 570- 549, Kor ea.

E - m ail : hybaek @w onkw ang .ac.kr

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JS = (^{1 -} ^{( p - 2)}|| X ||² )^{X ,} ^p ^{3 ,} ^(1.3)

w h ich sh rink s X t ow ar d t h e origin . S u b sequ en tly a n um b er of aut h or s prov ided cla s ses of S t ein - ty p e est im at or s dom in at in g X (s ee, for ex am ple, Efr on an d M orris (1976 ), Gh osh , H w an g , an d T su i (1984 ) w h er e ot h er r efer en ces ar e cit ed ). On e com m on feat ur e of t h e ab ov e cla s s es of est im at or s dom in at in g X is th at th ey ar e all sph erically sy m m et ric shrinkin g X t ow ar d som e part icular poin t , n ot n eces sarily t h e origin .

Gu o an d P al (1992) con sidered a s equ en ce of im pr ov ed e st im at or s pr ov idin g su cces siv e im pr ov em en t ov er ^JS. T h e Lin dley (1962) t y pe e st im at or is

1= X 1 + (^{1 -} || X - X 1 ||^{p - 3} ² )( X - X 1 ) , p 4 (1.4) w h er e X = ( X ₁+ + X _p) / p an d 1 = ( 1, , 1) ' . H is est im at or pos s es se s b ett er risk pr opert ies t h an th e or din ary J am es - S t ein est im at or ov er a lar g e r eg ion of th e p ar am et er sp ace, su g g e st in g th at from a sam plin g t h eor etic v iew p oint t h e shrin k ag e sh ould b e t ak en t ow ar d X 1 a s opposed t o th e origin .

In t his paper , a sequ en ce of im pr ov ed estim at or s pr ov idin g su cce s siv e im pr ov em ent s ov er ¹ is con st ru ct ed . In S ect ion 2, su ch im prov ed est im at or s ar e deriv ed an d in S ect ion 3, th e ab ov e r esu lt s is g en er alized w h en X ～N_p( , ²I_p) , w h er e cov arian ce m at rix is ²I_p for s om e un kn ow n s calar ²> 0 .

2 . IM P R OV E D E S T IM A T OR S D OM IN A T IN G ¹ Con sider a s equ en ce of est im at or s of th e form

n= X 1 + K _n( X - X 1 ) , n = 1 , 2 , 3 , , (2.1) w h er e Kn= Kn( X ) is a suit ab le fu n ction of X . W e ch oos e K1=

(^{1 -} || X - X 1 ||^{p - 3} ² )t o m ak e t h e fir st elem ent ¹ of t h e sequ en ce { ⁿ}. Our g oal is t o con st ru ct ⁿ, n 2 , su ch th at for an y in t eg er n 1 an d p 4 ,

R ( ^{n + 1}, ) R ( ⁿ, ) , R^p. (2.2)

T o dom in at e t h e est im at or ⁿ for any n 1, defin e ^{n + 1} a s

n + 1

= ⁿ+ r^*_n( X - X 1 ) , i . e . , K _{n + 1}= K _n+ r^*_n , (2.3) w h er e r^*_n = r^*_n( X ) is a su it able r eal v alu ed fu n ct ion . Let r _n = r^*_n

( X - X 1 ) . D efin e t h e risk differ en ce ( R D ) b et w een R ( ^{n + 1}, ) an d R ( ⁿ, ) a s

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R D ( n + 1, n ) = R ( ^{n + 1}, ) - R ( ⁿ, )

= E {

p

i = 1r²_{n i} + 2

p

i = 1r_{n i}( ⁿ_i - _i) }, (2.4)

w h er e ⁿ_i, _i, an d r_{n i} den ot e th e i^th elem ent s of ⁿ, , an d r_n, r espect iv ely . T h e secon d t erm of (2.4 ) can b e sim plified a s

E{^{i = 1}^p ^r ^{n i}⁽ ⁿⁱ ^- ⁱ⁾}⁼ ^{i = 1}^p ^{E [ r}^{n i}^{{ X + K}ⁿ^{( X} ⁱ^{- X ) -} ⁱ^}]

=

p

i = 1E [ ( Kn - 1) r n i( X i- X ) + rn i( X i- i) ]

=

p

i = 1E [ ( K_n - 1) r _{n i}( X _i- X ) +

X _i r_{n i}] . (2.5) T h e ex pr es sion (2.5 ) is obt ain ed b y u sin g S t ein ' s n orm al iden tit y a s sum in g t h at r _{n i}' s ( i = 1 , 2 , , p) satisfy all th e regu larity con dit ion s of th e iden tity . Com bin in g (2.4 ) an d (2.5 ) w e g et

R D ( n + 1 , n ) = E [^{i = 1}^p {^r²^{n i}^{+ 2 ( K} ⁿ^{- 1) r} ^{n i}^{( X} ⁱ^{- X ) + 2} ² ^X ⁱ ^r ^{n i}}]^{. (2.6)}

W e n ow look for suit able r n= r^*n ( X - X 1 ) su ch t h at R D ( n + 1 , n ) 0 , n 1 .

Befor e w e deriv e t h e g en er al re sult , let u s look at s om e sp ecial ca s es . Sp ecial cas es

(1) W h en n = 1 , i.e., w e are try in g t o dom in at e ¹(Lin dley ), t ak e r^*₁= c₁|| X - X 1 || ^{- ( 2 +} ¹⁾ w h er e ₁>0 an d c₁ is a su it able con st an t . T h en

2

p

i = 1 X i

r1i = 2 c₁{p - ( 3 + ₁) }

|| X - X 1 ||^{2 +} ¹ ,

p

i = 1r²1i= c₁²

|| X - X 1 ||^{2 + 2} ¹ an d 2 ( K₁- 1)

n

i = 1r_1i( X _i- X ) = - 2 c₁( p - 3)

|| X - X 1 ||^{2 +} ¹ . T h er efor e, fr om (2.6 ) on e can g et

R D ( 2 , 1) = E c₁²

T ^{1 +} ¹ - 2 c₁ ₁ T

1 + ¹ 2

, (2.7 )

w h er e T = || X - X 1 ||² n on ce n tr a l ²_{p - 1}( ) w it h = || - 1 ||² an d

= ( ₁+ + _p) / p . It is w ell k n ow n th at T can b e t r eat ed a s a m ix t ur e of cent r al ²p - 1 + 2 U an d U P oisson (

2 ) . Let U= U + p - 1 2 , t h en

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R D ( 2 , 1) = E_U c₁² 2 ^{- ( 1 +} ¹⁾

( U- ( 1 + 1) ) ( _U)

- 2 c1 12

- ( 1 + ¹

2 ) ( U- ( 1 + ¹ 2 ) )

( _U) . (2.8)

T o m ak e R D ( 2 , 1) 0 , it is sufficient t o c₁² 2 ^{- ( 1 +} ¹⁾

( _U- ( 1 + ₁) ) ( _U)

2 c₁ ₁2 ^{- ( 1 +}

1

2 ) ( _U- ( 1 + ¹ 2 ) )

( _U) for all U, U = 0 , 1 , 2 , . H en ce , th e con dition c1 is

0 <c1< 12

( 1 + ¹

2 ) ( U- ( 1 + ¹ 2 ) )

( _U- ( 1 + ₁) ) for U = 0 , 1 , 2 , … . Let

1( p , 1) = m in

U

( _U- ( 1 + ¹ 2 ) )

( U- ( 1 + 1) ) . (2.9)

T h en a sufficien t con dit ion on c₁ is 0 <c₁< ₁2 ^{1 +}

1

2

1(p , _{1) ,} (2.10)

pr ov ided p - 1 >2 ( 1 + ₁) . In fact , th e opt im al v alu e of c₁ w h ich m in im izes

c₁² 2 ^{- ( 1 +} ¹⁾ ( _U- ( 1 + ₁) )

( U) - 2 c₁ ₁2

- ( 1 + ¹

2 ) ( U- ( 1 + ¹ 2 ) )

( U) for all U

is

c₁⁰= ₁2

1

2

1( p , ₁) . (2.11)

It can b e pr ov ed a s a part of a m or e g en er al r esu lt t h at t h e m inim um in (2.9 ) is att ain ed at U = 0 , i . e . , U= ( p - 1)

2 ( s ee Gu o an d P al (1992) ). T h e con dition t h at p - 1 >2 ( 1 + ₁) is n eces sary t o en sur e th at all th e ex pect ation ex ist . T h e follow in g r esult is im m ediat e fr om t h e ab ov e deriv at ion .

P ro p o s it io n 2 .1 . T h e estim at or ²= ¹+ ( c₁⁰

|| X - X 1 ||^{2 +} ¹ ) ( X - X 1 )

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w it h ₁>0 dom in at e s ¹(Lin dley ) un iform ly un der t h e qu adr at ic los s (1.1) pr ov ided p - 1 >2 ( 1 + 1) .

R e m a rk 2 .1 . It is int er est in g t o look at v ariou s ch oice of 1>0 in t h e ab ov e pr oposition .

(a ) If 0 < 1<0 .5 , t h en ² dom in at es ¹ for p 4 .

(b ) If ₁= 1, t h en ₁( p , ₁) = (^{p - 4}2 )

(^{p - 5}2 )^{. H en ce,}

2= ¹+ 2

|| X - X 1 ||³

1( p , 1) ( X - X 1 ) dom in at e s ¹ w h en ev er p >5 . (c ) If ₁= 2 t h en ₁( p , ₁) = p - 7

2 . In t his ca s e, ¹ is un iform ly dom in at ed by ²= X 1 + (^{1 -} || X - X 1 ||^{p - 3} ² + 2 ( p - 7)

|| X - X 1 ||⁴ )( X - X 1 ) for p >7 .

(2) W h en n = 2 , an d w e w an t t o dom in at e ²= X 1 + K2( X - X 1 ) , w h er e K₂= (^{1 -} || X - X 1 ||^{( p - 3)} ² + c₁

|| X - X 1 ||^{2 +} ¹ )^,

ch oose r^*₂= c2

|| X - X 1 ||^{2 +} ² , w h er e ₂> ₁>0 an d c₂ is suit able con st ant . S im ilar t o th e ca se n = 1, R D ( 3 , 2 ) can b e deriv ed form (2.6 ) a s

R D ( 3 , 2) = E c²₂

T ^{1 +} ² + 2 c₁c₂

T ^{1 + (} ¹⁺ ²⁾ - 2 c ₂ T ^{1 +}

2

.

F ollow in g t h e earlier appr oach , a sufficien t con dit ion for R D ( 3 , 2 ) 0 is 0 < c2< 22

( 1 + ² 2 )

2( p , 1, 2) , (2.12)

w h er e

2( p , ₁, ₂) = m in

U {⁽⁽ ^U^U^{- ( 1 +}^{- ( 1 +} ²²²^{) )}^{) )}

1 - c1 - 1

2 2

- ₁

2 ( U- ( 1 + ¹+ ₂ 2 ) ) ( _U- ( 1 + ¹+ 2

2 ) )

}

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pr ov ided t h at p - 1 >2 ( 1 + ₂) . A g ain , th e opt im al v alu e of c₂ is c₂⁰= ₂2

2

2 2

( p , ₁, ₂) .

In g en eral, con sider t h e estim at or ⁿ(in (2.1)) w it h K_n= 1 - p - 3

|| X - X 1 ||² +

n - 1 j = 1

c_j

|| X - X 1 ||^{2 +} ^j , (2.13) w h er e _{n - 1}> _{n - 2}> > ₁>0 an d 0 <c_j< _j2 ^{1 +}

j

2

j( p , _{1 ,} , _j ) , j = 1 , 2 , , n - 1 . T ak e r^*_n= cn

|| X - X 1 ||^{2 +} ⁿ , w h er e _n> _{n - 1} an d c_n is a su it able con st ant . S im ilar t o t h e sp ecial ca s es n = 1 , 2 , on e can g et

R D (n + 1 , n ) = E _U{ ⁽¹ u ) [^c²ⁿ²^{- ( 1 +} ⁾ ⁽ ^u ^{- ( 1 +} ⁿ^{) )}

+ 2 c_n

n - 1

j = 1c_j2 ^{- ( 1 +}

( j + n)

2 )

( _u - ( 1 + ^j + _n

2 ) )

- 2 c_n _n2

- ( 1 + ⁿ 2 )

( _u - ( 1 + ⁿ

2 ) )]}^. ^(2.14)

A ll t h e ex p ect ation s in (2.14 ) ex ist pr ov ided p - 1 >2 ( 1 + _n) . Defin e

n( p , _{1 ,} , _n) a s

n( p , _{1 ,} , _n) = m in

U

( U- 1 + ⁿ 2 ) ( U- 1 + n) 1 -

n - 1

j = 1c_j _n^{- 1}2

- _j 2

( _U- ( 1 + ^j + _n

2 ) )

( _U- ( 1 + ⁿ 2 ) )

. (2.15)

T h en a sufficien t con dit ion for ^{n + 1} dom in atin g ov er ⁿ is 0 < c_n< _n2 ^{1 +}

n

2

n( p , ₁, , _n) (2.16)

an d t h e opt im al v alu e of cn is cn⁰= n2

n

2

n( p , 1, , n) . T h e m in im u m in (2.15 ) is at t ain ed U = 0 (s ee Gu o an d P al (1992)). W e n ow st at e th e m ain t h eorem of t his section .

T h e ore m 2 .1 . A n estim at or ⁿ w ith K_n g iv en by (2.13 ) is un iform ly dom in at ed by

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n + 1

= ⁿ+ cn

|| X - X 1 ||^{2 +} ⁿ ( X - X 1 ) pr ov ided p - 1 >2 ( 1 + _n) an d c_n sat isfies t h e con dition (2.16 ).

R e m ark 2 .2 . N ot e t h at th e fu n ct ion s r_{n i}, i = 1, 2 , , p sat isfy t h e r egu larit y con dition of S t ein ' s n orm al iden t it y w hich en ables u s t o deriv e (2.14 ).

If w e ch oos e 0 < ₁< ₂< < _n< _{n + 1}< <0 . 5 , t h en w e g et { ⁿ}, a sequ en ce of im pr ov ed estim at or s g iv in g su cces siv e im pr ov em ent s for p 4 (sin ce,

2 ( 1 + _n) <3 n 1 ).

R e m a rk 2 .3 . T h e lim it in g v alu e of {c_n⁰} is h ar d t o fin d an aly tically du e t o th e com plicat ed st ru ctu r e of c_n⁰(see (2.15 )). H en ce, th e pr oblem of fin din g t h e close form of t h e lim it in g e st im at or of t h e sequ en ce { ⁿ} still r em ain s op en .

3 . T H E CA S E OF COV A RIA N CE M A T RIX ² Ip ( ² > 0 unkn ow n )

In th is s ect ion , w e ex t en d t h e r esult s deriv ed in sect ion 2 t o th e ca se w h er e cov arian ce m atrix is ²Ip , ²> 0 u nkn ow n . Let X an d S b e in depen dent ob serv ation s w it h X ～N_p( , ²I_p) an d S～ ² _k². H ere w e w ant t o est im at e u n der t h e los s fu n ct ion

L ( , ) = || - ||²

2 . (3.1)

A g ain th e u su al est im at or is ⁰= X an d t h e Lin dley t y pe estim at or dom in atin g ⁰ is

1= X 1 + (^{1 -} ( k + 2) || X - X 1 ||^{( p - 3) S} ² )( X - X 1 ) , p 4 . (3.2) W e con st ru ct t h e sequ en ce { ⁿ} of im prov ed estim at or s a s follow s . Let

n= X 1 + K_n( X - X 1 ), w h ere

Kn= 1 - ( p - 3) S

( k + 2 ) || X - X 1 ||² +

n - 1 j = 1

cjS

1 + ^j 2

|| X - X 1 ||^{2 +} ^j ,

n - 1 > _{n - 2} > > ₁ > 0 (3.3)

an d

n + 1

= ⁿ+ r _n= ⁿ+ r_n^*( X - X 1 ) .

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T h en ,

R D ( n + 1 , n ) = R ( ^{n + 1}, ) - R ( ⁿ, )

= 1

2 E [i = 1^p r²_{n i}+ 2

p

i = 1r_{n i}( _iⁿ- _i)]

= 1

2 E [^{i = 1}^p ^r²^{n i}^{+ 2 ( K}ⁿ ^{- 1) r} ^{n i}^{( X} ⁱ^{- X ) + 2} ² ^X ⁱ ^r^{n i}]^. ^(3.4)

T h e la st ex pr es sion follow s from S t ein ' s n orm al iden tit y a s sum in g th at all t h e

ex p ect ation s ex ist . By t akin g rn^*= c_nS ^{1 +}

n

2

|| X - X 1 ||^{2 +} ⁿ , w h er e cn is a suit ab le con st ant an d n - 1 > n - 2 > > 1 > 0 , on e can g et

R D ( n + 1 , n ) = 1

2 E c_n² S^{2 +} ⁿ

|| X - X 1 ||^{2 + 2} ⁿ + 2 ²c_n( p - ( 3 + _n) ) S ^{1 +}

n

2

|| X - X 1 ||^{2 +} ⁿ - 2 cn

( p - 3) S^{2 +}

n

2

( k + 2) || X - X 1 ||^{2 +} ⁿ + 2 cnS

1 + ⁿ 2 n - 1

j = 1cj S

1 + ^j 2

|| X - X 1 ||^{2 +} ^j⁺ ⁿ

= E _{T , S} c_n² 1

T ^{1 +} ⁿ ( S

2 )^{2 +} ⁿ + 2 c_n( p - ( 3 + _n) ) 1

T

1 + 2

( S

2 ) ^{1 +}

n

2

- 2 c_n p - 3 ( k + 2) T

1 + ⁿ 2

( S

2 )^{2 +}

n

2

+ 2 c_n

n - 1

j = 1c_j 1

T ^{1 +}

( j+ 2) 2

( S

2 )

2 + ( j+ n)

2 ,

w h er e T = || X - X 1 ||

2 , ( S

2 )～ k²( ) w it h = || - 1 ||

2 , ( S

2 )～ k², an d t h ey ar e in depen den t . S im ilar t o th e pr ev iou s s ect ion a sufficient con dition for

R D ( n + 1 , n ) 0 is

0 < c_n < _n k + p - 1

k + 2 ⁿ( p , ₁, , _n) , (3.5)

w h er e

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n( p , ₁, , _n) =

( ( p - 3 - _n)

2 ) ( k

2 + 1 + ⁿ 2 ) ( ( p - 3 )

2 - _n) ( k

2 + 1 + _n)

{

^{1 -} ⁿ^{2 ( k + 1)}( k + p - 1)

n - 1 j = 1cj

( k

2 + 2 + ( _j+ _n)

2 ) ( ( p - 3)

2 - ( _j+ _n)

2 )

( k

2 + 1 + ⁿ

2 ) ( p - 3 - n

2 )

}

^. ^(3.6)

T h e opt im al v alu e c_n is c⁰_n = _n k + p - 1

2 ( k + 2 ) ⁿ( p , ₁ , , _n) .

T h e o re m 3 .1 . A n est im at or ⁿof t h e form (3.3) is un iform ly dom in at ed b y

n + 1

= ⁿ + ( c_nS ^{1 +}

n

2

|| X - X 1 ||^{2 +} ⁿ ) ( X - X 1 ) un der th e los s (3.1) pr ov ided p - 1 >2 ( 1 + n) an d cn s at isfies t h e con dit ion (3.5 ).

R e m a rk 3 .1 . N ot e t h at t h e fu n ct ion rn i, i = 1 , 2 , , p , sat isfies t h e r eg ularit y con dition of S t ein ' s n orm al iden t it y w hich en ables u s t o deriv e (3.4 ).

H er e also t h e qu est ion of conv er g en ce of { ⁿ }r em ain s op en .

R e f ere n c e s

1. Efr on , B . an d M orris , C. (1976 ). S t ein `s est im ation rule an d it s com pet it or s - an em pirical Bay es appr oach , J ou rn al of th e A m erican S t at ist ical A s sociat ion , 68, 117 - 130.

2. Gh osh , M ., H w an g , J . T ., an d T sui, K . W . (1984 ). Con stru ct ion of im pr ov ed estim at or s in m ult iparam et er estim at ion for cont in u ou s ex pon en t ial fam ilies , J ou rn al of th e M ultiv ariat e A n aly sis , 14. 212- 220.

3. Gu o, Y . an d P al, N . (1992). A s equ en ce of Im pr ov em en t s ov er t h e J am es - S t ein E st im at or , J ourn al of t h e M ult iv ariat e A n aly sis , 42 302- 317.

4. J am es , W . an d S t ein , C. (1961). E stim at ion w ith qu adr at ic los s , In P r oceedin g s of th e F ou rth Berk ely S y m posiu m on M ath em at ics , S t atist ic s an d P r ob ability , 1, 361- 380, Un iv . of Californ ia P re s s , Berk eley .

5. Lin dley , D . V . (1962). Dis cu s sion of paper by C. S t ein , J ourn al of R oy al S t at ist ical S ociety , B , 24, 256 - 296.

6. S t ein , C. (1956). In adm is sibilit y of t h e u su al estim at or for t h e m ean of a m ult iv ariat e n orm al dist ribu tion , In P roceedin g s of t h e T hir d Berk eley S y m posium on M at h em atics , S t at istics an d P r ob abilit y , 1, 197- 206, Univ . of Californ ia P r es s , Berk eley .

[ 2002년 3월 접수, 2002년 5월 채택 ]