Metrika (1992) 39:185-197

Minimax Estimators and F-Minimax Estimators for a Bounded Normal Mean Under the Loss/p (0, d) = 10-d[

p

W. B i s c h o f f a n d W. Fieger I

Summary: Let the random variable X be normal distributed with known variance 02>0. It is supposed that the unknown mean 0 is an element of a bounded interval O. The problem of estimating 0 under the loss function/p(0, d) = [0-dl p where p _ 2 is considered. In case the length of the interval O is sufficiently small the minimax estimator and the F(fl, r)-minimax estimator, where F(fl, r) represents special vague prior information, are given. Key words and phrases: Minimax estimator, F-minimax estimator, bounded normal mean, Lfloss, least favourable two point priors.

1 Introduction and Notations

T h e p r o b l e m considered is t h a t o f e s t i m a t i n g the m e a n 0 o f a n o r m a l d i s t r i b u t i o n N(O, o 2) u n d e r the a d d i t i o n a l a s s u m p t i o n t h a t the m e a n / 9 lies in s o m e b o u n d e d interval O = [0,c] where c > 0 is fixed. T h e v a r i a n c e t r 2 > 0 is s u p p o s e d to be known. T h e loss f u n c t i o n is a s s u m e d to be ~ ( O , d ) = [O-dl p for s o m e fixed p >-2 (we m a y designate this loss as L f l o s s ) ; the case o f squared e r r o r loss is i n c l u d e d (17 = 2). In this p a p e r for sufficiently small c the m i n i m a x e s t i m a t o r o f O is d e t e r m i n e d . F u r t h e r l e t / 7 b e the set o f all priors, t h a t is, t h e set o f all Borel p r o b a b i l i t y measures o n t h e p a r a m e t e r space O. I f c is a g a i n sufficiently small, then we also d e t e r m i n e t h e F - m i n i m a x e s t i m a t o r o f O as well as least favourable p r i o r s in F in case 1 Dr. W. Bischoff and Prof. Dr. W. Fieger, Institut fllr Mathematische Stochastik, Universitat Karlsruhe, Englerstrafle 2, D-7500 Karlsruhe 1.

0026-1335/92/3-4/185-197 $2.50 © 1992 Physica-Verlag, Heidelberg

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W. Bischoff and W. Fieger

r = r0~,r) ={ne/-/ln=/~lq+(1-fl)~z2;

~zl,n2•H;

rq([0,r])=lt2([r,c])=l}

where B • (0, 1) and r • (O,c) are some fixed constants; F(fl, r) consists of all priors that give probabilities fl and 1 - f l to the intervals [0,r] and It, c], respectively. This class of priors seems to be realistic in applications (cf. Berger (1985), p. 217). Bischoff (1991) has determined the minimax estimator and the F-minimax estimator for functions of the bounded parameter of a scale parameter family with respect to the above loss. For more details we need the following definitions and notations: Let d denote the set of all (non-randomized) estimators for 0, i.e. the set of all Borel measurable functions ~: ~ [ 0 , c ] . Under the loss function/u(O,d) = IO-dl p the risk function of an estimator ~ • A is given by

R p ( O , a ) = ] / ~1n ~l l 0 - ~ ( x ) l U ' e x p l

(without loss of generality let a prior n • H is defined by

0 -2 =

1

- ~1( x - O ) 2 d x ,

O•O,

1) and the Bayes risk of J • d with respect to

rp(n,a) = J Rp(o,a)n(ao). o

If a precise prior information n e H about the mean 0 ~ 0 is given, then we have a Bayesian decision problem. An estimator ~* • d with rp(lt,~*) = rain rn(n,J )

is called Bayes estimator of 0 with respect to the prior ft. If on the other hand no prior information is available, then the minimax principle can be used. J* E A is called minimax estimator of 8 if sup rp(n,~*) = min sup rp(n,t~) . neH

6cA h e / 7

The/"-minimax principle is an intermediate approach between the Bayes and the minimax principle. It is used if vague prior information is available, which can be described by a subset F of H. An estimator J* e A with

Minimax Estimators and F-Minimax Estimators for a Bounded Normal Mean

187

sup rp (n, ~ *) = min sup rp (n, ~ ) ~reF

t~A n ~ r

is called F-minimax estimator o f 0. Obviously, a F-minimax estimator coincides with a Bayes estimator and a minimax estimator if F = In} and F = 17, respectively. Eichenauer et al. (1988) proved: Let P---T, and let squared error loss be given. Then there exists a constant c0(fl)>0 and for c e (0,c0(fl)] there exists a constant r0(fl, c ) e [0,c) such that the following statement holds: For each r e [Zo(fl, c ) , c ] the two point prior n# = #e0 + ( 1 - # ) e c (e t denotes the one point measure on t) is least favourable in F(fl, r) and the Bayes estimator with respect to n/~

d~a(x)=c"

1+

exp

-

2cx-c

2

,

x e ~, ,

1

is F(fl, z)-minimax. Since Lemma 4.1 one may assume fl_<~- without loss of generality. The result o f Zinzius (1981) on minimax estimators is contained in the above statement if we put # = 7. In this paper the above result is generalized to the loss function/p(0, d), p >__2 fixed. In Section 2 the Bayes estimator of 0 with respect to a two point prior is determined; in Section 3 the minimax estimator and the F-minimax estimator of 0 as well as least favourable priors are given. These results are obtained by extending the main idea of Zinzius (1981). By some more tricky evaluations we get a result for the loss function ~ (0, d), p_> 2. For p = 2 our result even improves that o f Zinzius (1981) for the minimax estimator as well as that of Eichenauer et al. (1988) for the F(fl, r)-minimax estimator. (In case o f squared error loss t2(O,d) = [ O - d [ 2 the best known result for the minimax estimator is stated by Casella and Strawderman (1981).)

2 Bayes Estimator of 0 with Respect to a Two Point Prior

In the following proposition we give the unique Bayes estimator o f the normal mean Oe [O,c] with respect to the prior ~ t B = B . e o + ( 1 - f l ) e c, O < f l < l , in case the loss is lp(O,d) = [ O - d [ p, 1 < p < oo. Let A be the Lebesgue measure on ~, and

188

W. Bischoff and W. Fieger

let ~ be the Borel field induced by the Euclidian topology of f2 = Ox IR. We consider the measure space (g2,~, m) where m is the measure on (f2, ~R)that has a density fo(x) with respect to the product measure rtB® A. Lp denotes the Lebesgue space of all realvalued measurable functions g(co) on f2 with finite norm IIg ll = [~IglPdm} I/p. Further let ~ be the product field of {~,(9] and the Borel field of ~, and let Lp(2]) denote the subspace of all Y-measurable functions. Since Lp(~) is a closed linear subspace of Lp the projection of each g ~ Lp onto Lp(D) with respect to I1"11 exists and is unique 2 - a . s . Obviously, the Bayes estimator of 0 is the projection of g(O,x)=--0 onto Lp(D).

Proposition 2.1: The A - a.s. unique Bayes estimator J~.p of 0 e ~9 with respect to the prior n# = fl'e0+(l -fl)ec and the loss function $(0,d), 1 < p < oo, is given by J~;p(x) = c" [1 +(]3(1 _ f l ) - I exp {{-(c2 -2cx)}) 1/(p-1)] -1 .

Proof." Following the proof of Theorem 1 of Ando and Amemiya (1965) the projection of g(0,x)------0 onto Lp(~) - i.e. the Bayes estimator J = J#,p of 0 fulfills

hA(O~):= I I e - - a ( X ) + a I A ( X ) I I P > _

IIO--,~(X)II p ,

a ~

,

for every O x A ~ ~ where 1A (x) is the indicator function of A. Hence for any O x A e ~ the derivative of hA(a) must vanish at a = 0, that is I [ sgn (0- a(x))" I0- J (x)IP-"exp {--~(x- O)2}dxx#(dO) = 0 OA

for every Borel set A of R. So we get .[ fl.sgn ( - J(x)) • IJ(x)lP-t.exp {I T X }2 A

+(1 - f l ) ' s g n ( c - J(x))- Ic - J(x)[p-i .exp [-½(x-c)2]dx = 0 for every Borel set A of ~ and consequently ~21 p . s g n ( - J ( x ) ) ' l a ( x ) l p-l" exrl ,. [-- T1 ~ ,

+(1 - p)" sgn ( c - J(x)) Ic - J(x)I p-1 .exp {- {-(x- c) 2} = 0

A-a.s.

Minimax Estimators and F-Minimax Estimators for a Bounded Normal Mean

189

Hence J ( x ) ~ (0,c)

,l-a.s. ,

and ~.~(x)p -1 .exp {-~-x21 = ( 1 - p ) ( c - ~ ( x ) ) p -1 • exp { - ~ ( x - c)21

,~-- a.s. ,

[]

thus the assertion follows.

3 Minimax Estimator and FOg,'c)-Minimax Estimator of 0

With the abbreviation

r(x) := [ft.(1 _ f l ) - i .exp {~-(c2 -2cx)}] 1/(p-1) we may write JB;p(x) = c" [1 + r ( x ) ] - I . Further we define for p >_2: r 12(p- 1 ) ~

1/2

Let p~(0,½] and p _ 2 be fixed. For c e (0,c(p)] it will be shown in Section 4:

Rp(',J~,p) is convex on [0,c] ,

(3.1)

alz0 = T0(c,P,B) ~ [O,c):Rp(zO, J~,p) = Rp(c,J~,p) ;

(3.2)

whence for ~

[%,c]:

Rp(O,J~p) = max Rp(0,J~,p) 0~[0,~]

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W. Bischoff and W. Fiegcr

as well as Rp(¢,t~f;,p)

=

max Rp(O,~f;,p)

.

0e[~,c]

This and Lemma 1 in Chen and Eichenauer (1988) yield

Theorem 3.1: Let Be(0,½] and p > 2 be fixed. If 0
np =/~'e0 +(1-/~)'Cc is least favourable in F(fl, z) and the Bayes estimator ¢

~p;.(x)---, 1 +r(x)

xeR,

with respect to ~zp is F(fl, O-minimax. It is not certain so long that the statement of Theorem 3.1 is also true for p ~ (1, 2); the proof given for Lemma 4.3 does not work for p ~ (1,2). With Lemma 4.1 we get a similar result for f i e (~-, 1). Observe that sup

neFq~,~)

sup | Rp(O,~) rp()z,~)= fl. 0~[0,~] sup Rp(O,t~)+(1--fl)'Oe[T,c

and R. (0, ~I/2;.) = R. (c- O, ~1/2;.) • Hence for c _ c(p) and arbitrary ~"~ [0,c] the F(½, Q-minimax estimator equals the ordinary minimax estimator for a bounded mean with respect to the loss ~(0,d). So we get

Corollary 3.2: Let p_>2 be fixed, and let c<_c(p). Then the two point prior 1. l__.~ 7[1/2 = T ~0 "[- 2 c

is least favourable i n / / a n d the Bayes estimator

Minimax Estimators and F-Minimax Estimators for a Bounded Normal Mean

¢~1/2;p (X) = C" [ 1 +

191

exp {~-(C 2 -- 2 cx)] 1/(p- 1)] - 1

with respect to 7tl/2 is minimax. R e m a r k 3.3: J. Eichenauer-Herrmann pointed out to the authors that the asser-

tion of Theorem 3.1 and Corollary 3.2 remains true if the constant c(p) is replaced by the greater constant c* (p) = min ( p - 1)3/4 (1 + r)- [(r. I r - 1 l" (I + r)) 1/2 + r ( p - 1)1/21- 1/2 . re(O, ~)

The constants c(2) and c* (2) are greater than the constant determined by Eichenauer et al. (1988) and Zinzius (1981), respectively. Note that ~ / - 3 < c ( 2 ) < ] / ~ and c* (2) = ]/~-. By computer calculation one gets that the constant of Theorem 3.1 can be enlarged to c** (p) given in the following table; for c > c** (p) the statement of Theorem 3.1 does not hold.

p

c(p)

c*(p)

c**(p)

p

c(p)

c*(p)

c**(p)

2.0 2.2 2.4 2.6 2.8 3.0

1.33 1.50 1.65 1.79 1.93 2.06

1.41 1.59 1.75 1.90 2.05 2.19

2.11 2.33 2.53 2.72 2.90 3.07

4 5 10 15 30 50

2.64 3.14 5.04 6.48 9.70 12.90

2.81 3.34 5.34 6.84 10.16 13.42

3.73 4.23 6.16 7.61 10.86 14.07

R e m a r k 3.4: Let the mean 0 ~ O = [0, c] to be estimated in case we observe n independent random variables XI_/---. . . . . Xn distributed according to N(O, a2), where O-2>0

is

known. Then X = aI/-n- ' X- = aVn~-~i=l 1 X i is a sufficient statistic of 0.

Hence the problem of estimating the mean 0 ~ O =

[0,c] of the variables _/--

X1 . . . . ,An is equivalent to the problem of estimating the mean ~ = V n . 0 ¢ 0, ¢-~. (0, 1/2],

of X that is distributed according to N(~, 1). Hence in case B ~

c_~n •c(p),

and

~nn"'

.c,p,

<__z<_cthe F(fl, r)-minimax estima-

tor o f 0 is given by

I~ '(Xl .....

Xn)=C"

1+

I~G2 (1-')-Iexp

-~\ I/(p-I)"I-I "(C2-2CX)I) l

192

W. Bischoff and W. Fieger

x i.

where ~ = - . /'/ i = 1

Remark 3.5: The case that the loss is measured by

(9,~.lO-dl p

:

Y2"]O-d] p

if if

d>_O d
where Yt, ~'2> 0 are fixed, can easily be reduced to the case that the loss is given by ~(0,d): Let be ]~:= Y2+ B" fl~'t 0'1 - 72); then a F(~, z)-minimax estimator of 0 with respect to the loss tp(0, d) is a F(fl, O-minimax estimator of 0 with respect to the loss 4(0, d).

4 Proofs

Since the following lemma we may restrict us to fl c (0, 1/2].

Lemma 4.1: For every J c A

put 3 ( x ) : = c - J ( c - x ) c A ;

v OcO:R(O,t])=R(c-O,J)

(a) v J c A (b) V J c A :

sup 7t¢F(fl, r)

r(n,J)=

sup

then we have

, r(lr,~) ,

rt ~F(1-fl,c-'/')

(c) J c A is Ftfl, O-minimax ** 3 is F ( 1 - # , c - r ) - m i n i m a x . The easy proof is omitted. In the following lemmas we investigate the risk

Lemma 4.2: For all 0 e (O,c) and p > 1 we have O

-

P

I sgn(O-Jp, p(X))" lO-Jl~p(X)lP-l"exp

Ill

-~(x-O) 2 "(1-6~p(x))dx

Minimax Estimators and F-Minimax Estimators for a Bounded Normal Mean

193

d where ~;p(X) = - ~p.p(X). dx '

Proof." By Lebesgue's dominated convergence theorem and integration by parts we obtain:

' > ' l "n<' ' x . { '

'>=P{

'>'l'x =p {

<,

[]

L e m m a 4.3: For all 0 ~ (0, c) and p >_2 we have 02

-~R(O,~#,p)

= p" [ ! ( p - 1)" IO-~,p(x)[p-2"fo(x)'(1-2<~'~;p(x))dx

- ~I sgn (0- c~,p(x)). Io- ~,p(x)I p- '.
Proof." For p_> 2 we get by Lebesgue's dominated convergence theorem and integration by parts:

194

W. Bischoff and W. Fieger

2V~ 0 2 R(O,$~,p) p 002

+ ~ sgn ( 0 - 6~p(x))" I O- d~p(X)I p-1

opI - ~ sgn (O-t~,p(X))lO-t~,,p(x)lP-l (x-O) cxP I-~ (x-O)21~"°;P(X)dx "[-] Using I.emma 4.1 we get

Lemma 4.4: R(O,$B;p)>R(c,,~. p) for all fl~ (0, 3] and p > 1.

Proof.." Since v x e ~,:al_~,p(X) = c-J~,p(C-X) we get by Lemma 4.1 (a) v 0~ e : R ( 0 , $ 1 _ ~ ; p ) = R ( c - O , $ , ; p )

.

Hence we obtain

R(O,~,p)-R(c,(~,p) = R(O,~,,p)-R(O,¢~l_p; p)

Minimax Estimators and F-Minimax Estimators for a Bounded Normal Mean

Because c~p;p(x) is strictly decreasing in B the assertion follows.

195

[]

For the following two lemmata we need

t~;p(x)

= dc~p.p(X) = -c'r'(x)" [1 +r(x)] -2 = c2(p - 1) -lr(x). [1 +r(x)] -2 dx '

and d2

g/~;p(x) = ~ x 2 g/p;p(x) = c2(p - 1)-l[r'(x) • [1 +r(x)l-z -2r(x)-[1 + r(x)]-3r'(x)]

= c 2 ( p - 1) -1 r t ( x ) [ 1

= C3(p - 1)-2r(x)(r(x)

+ r(x)1-3 -{1 + r ( x ) - 2 r ( x ) }

- 1)(l+r(x))

-3

.

Lemma 4.5: -~-~R(O,c~p)lo= c > 0 for c2_<4(p - 1) and p > 1.

Proof." Since t~;p(X)_< C2(p - 1)- 1-max r-(1 + r ) -2 = ¼c2(p - 1) -~ r>0

[]

the assertion follows by Lemma 4.2.

Next we show that R(.,c~,p) is convex on [0,c], if c is small enough. We 02 choose c = c(p) so that ooER(O,~B..p)>O., for all 0 e ( 0 , c ) . (12(p- 1 ) ~ 1/2 --~+---~/] . Then

Lemma 4.6: Let p___2 and let 0 < c _ < c ( p ) = \ . ~ 2 2 ~2

802R(O,t~)>O for Oe(O,c). Proof." Integration by parts of the second term of the expression in Lemma 4.3 yields

196

W. Bischoff and W. Fieger

= J ( p - l ) . [O-O~,p(x)lP-Z.exp

-

Ilt

- ~ ( x - O ) 2 .(1-2tJj;p(x))dx

J sgn (O-O~,p(X))" [O-tJ~,p(x)lP-"tY~;p(X)'exp R

Ill

--~(x-O) 2

dx

where

h(x) = h(x;fl, p) = (p-1).(l-O~p(x))2-c • IO~;p(X)] . Using the above equations for 0~;p(X) and 0~;p(X) we get h(x) = ( p - 1)(1 _ c 2 ( p _ l ) - i r ( x ) . ( | +/(x))-2)2

- c 4(p- 1)-2r(x)lr(x)- 1 [(1 +r(x))-3 Hence h (x)> 0 for all x ~ [R if and only if W r e (0, oo): ( c - 2 - ( p - 1)-lr.(1 +r)-2)2-(p-1)-3rlr-1 ](1 + r ) - 3 > 0 . (4.1) Since

V rE(O, oo):r.(l+r)-2<~ , r. I r - l l ' ( l + r ) -3
the assertion follows easily.

[]

Proof of (3.1) and (3.2): It is easy to check that C2(p)<4(p-- 1) ;

hence the Lemmas 4.4, 4.5 and 4.6 imply (3.1) and (3.2) of Section 3.

[]

Minimax Estimators and F-Minimax Estimators for a Bounded Normal Mean

197

P r o o f o f Remark 3.3: Let b(r,p):= ( p -

1)3/4(1 +r)"

[(r. I r - 1 I "(1 + r ) ) l / 2 + r ( p

-

1)1/21-1/2,

r e (0, oo), p > 2; then

c*(p) = min b(r,p)<_b(1,p) = 2 " V p - 1 . re(O,oo) Since (4.1) we get that h(x)>~O for all x ~ [R iff V re(O, oo):c<_b(r,p) .

Consequently Remark 3.3 follows.

[]

Acknowledgement: The authors would like to express their gratitude to J. Eichenauer-Herrmann for valuable hints and suggestions.

References

Ando T, Amemiya I (1965) Almost Everywhere Convergence of Prediction Sequence in Lp (1 < p < oo). Z Wahrscheinlichkeitstheorie verw Gehiete 4:113-120 Berger JO (1985) Statistical Decision Theory and Bayesian Analysis, 2nd ed. Springer, Berlin Heidelberg New York Bischoff W (1991) Minimax Estimation and F-Minimax Estimation for Functions of the Bounded Parameter of a Scale Parameter Family under "Lp-loss". Statistics and Decisions (to appear) Casella (3, Strawderman WE (1981) Estimating a Bounded Normal Mean. Ann Statist 9:868-876 Chen L, Eichenauer J (1988) Two Point Priors and F-Minimax Estimating in Families of Uniform Distributions. Statistical Papers 29:45- 57 Eichenauer J, Kirschgarth P, Lehn J (1988) Gamma-Minimax Estimators for a Bounded Normal Mean. Statistics and Decisions 6:343- 348 Zinzius E (1981) Minimaxsch-~ttzer ftlr den Mittelwcrt 0 einer normalvertcilten ZufallsgrOBe mit bekannter Varianz bei vorgegebener oberer und untcrer Schranke ftlr 0. Math Operationsforsch Statist, Ser Statist 12:551-557 Received April 2, 1991 Revised Version July 2, 1991