(i) fe(x) =Lpe(xla) dG( a).For any second differentiable function r of 0 let fa and fr' be its first and second deriva-

Bull. Kyushu Inst. Tech.

(M. & N. S.) No. 28, 1981, pp. 7-10

NOTE ON OPTIMAL ESTIMATING FUNCTIONS FOR A MODEL WITH A NUISANCE PARAMETER

By

Akimichi OKuMA

(Received Oct. 30, 1980)

The problem of estimating equations has been studied in many papers ([1], [2], [4], [5] etc.). The purpose of the present paper is to give a generalization of the results in [3], where the relations between the class of all optimal estimating functions for any fixed nuisance parameter value and the class of all optimal ones for the probability model with a prior distribution on the nuisance parameter space were discussed.

In the most of the papers on estimating equations, the class of regular estimating functions is restricted by the condition which requires the expectation of the estimat- ing function to be zero. This condition is rather strong. For example, for any unbiased

estimator 0A (X) of 0, we define h(x, 0) = 0- 0- (x). The equation h=O gives the estimator

0-

(X), and satisfies the condition E(h(X, 0))=O. It sujests us that the class of regular estimating functions includes the estimating function corresponding to the unbiased esti- mator. On the other hand, for any biased estimator it is hard to define the estimating function in the same manner. Therefore in this note we do not require the condition as the regularity condition.

Let X be a random variable with density function pe(x1 ev) with respect to (abbrevi- ated by wrt in the sequel) some o-finite measure u on the sample space (Sif, jB) where 0ee, aeY and, 0 and .s4 be open intervals of the real line. And let 0 be the parameter to be estimated and a be the nuisance parameter. We assume a prior distribution G on J4. Let

(i) fe(x) =Lpe(xla) dG( a).

For any second differentiable function r of 0 let fa and fr' be its first and second deriva- tives wrt 0 respectively. For any set A, let IA be the indicator function of A.

AssuMpTioN 1. For each a6J4 Pe(x1a) exists and ds continuous foralmost everywhere

Jct .

AssuMpTIoN2. ForeachaeJa and each 0ee (2) OÅq Ee({ l'e (XI a) }21 a]Åq oo

where le (x 1 a) == log pe(x 1 a) • Ic(x), and C = {x ; pe(x 1 a) År O} does not depend on the pa ra- meters.

DEFiNITIoN 1. A real valzaed function g(x, 0, a) dejined on AfÅ~ (E)xJ4 is said to be

the regular estimating function for the conditional Problem zffor each /ixe4 a 6Y ,

8 Akimichi OKuMA

(gl) OÅqEe({g(X, 0, ev)-M(gla)}21a)Åqoo

where M(g l a) = Ee (g(X, 0, ev) 1 a) ,

(g2) g and M(g1 a) exist and are continuous functions of 0 for almost everywhere u,

(g3) 0ÅqIEe(g(X, 0, a)-M(gla)1a)1Åqoo

(g4) fxg(x, 0, a)pe(xla)dpt is dzlfferentiable wrt 0 under the integralsign. and

For each fixed aEs4 let EY. be the class of all regular estimating functions for the conditional problem.

DEFINITIoN. 2. A real valuedfzanction h(x, 0) defined on WÅ~ (E) is said to be the regu- lar estimating function i

(hl) OÅqEe[{h(X, 0)-M(h)}2]Åqoo

where M(h) =Ee(h(X, 0)] ,

(h2) h andM(h) existandare continzaozasfunctions ofOforalmosteverywhere pt,

(h3) OÅqIEe(h(X, 0)-M(h)]lÅq oo

(h4) fxh(x, 0)fe(x)du is dzlfferentiable wrt0under the integralsign. and

Let or be the class of all regular estimating functions. For any function r, let K(r, 01 ev) and K(r, 0) be the following,

(3) K( r, 01 ev) = Ee[{ r- M(rl a)}21ev){Ee(di-M(rl a)la)}h2 and

(4) K(r, 0)=Ee({r-M(r)}2){Ee[of-M(r))}-2

provided that the right hand sides of (3) and (4) are well defined.

For any estimator 0(Xl a), which has the finite conditional second moment and may depend on the fixed nuisance parameter value a, we are able to construct a regular es- timating function. Let

(5) ge (x, 0, a)=0- 0(xla)

then go(x, 0, ev) = 1 for all xeAr, 0e0 and ae", and then the conditions (gl)-(g4) are all satisfied, i. e., gee g. for each aes4. Furthermore

(6) K(ge, 01 ev) = Ee ({ ge- (X, 0, ev)- M(go- l a)}2 l a] Å~ {Ee (ge(X, 0, ev) - M(g,- l a) l a)}-2 = Ee({ 0A(X1 ev)-M( 0"1 ev)}2l a]/{M( 0'Sl a)}2 .

For any estimator 0(X) with finite second moment, let

(7) hb-(x, 0)=0- 0A(x).

The conditions (hl)-(h4) for he are satisfied. Therefore heA'E evand hence

(8) K(hO', 0)=Ee[{i(X)-M(0A-)}2)/{M(0A')}2.

If 0(Xla) and 0(X) are unbiased, thenK(ge-, 01ev)and K(he=, 0) are conditional variance of 0(X1 ev) and variance of 0(X) respectively.

Let '

Note on Optimal Estimating Functions for a Model with a Nuisance Parameter 9

(g) g*(x, 0, cr)={Pe(xla)/Pe(xla)}'Ic(x)

and

(10) h*(x, 0)={fe(x)/fe(x)}'Ic(x).

THEoREM 1. For any ge 9.

(11) K(g, 01ev)={V(g.la)R2(g, g.la)}-i

2 K(g*, 01 a) = {Ee(gi l a)}Ti for each aeJ4. And for any hE ,sg

(12) K(h, 0) =={V(h*)R2(h, h*)}r'IK(h*, 0)={Ee(h2*)}-i

where V(•1a) and V(•)are the conditional van'ance given ev and the variance respectively, and R(•,•1a) and R(•,•) are the conditional correlation coefiicient given a and the

correlation coefiicient resPectively.

PRooF. From the definition of .lt4(•1a)

Ee(g(X, 0, a)-M(gla)) =O.

By taking the derivatives of the both sides of the above equation, we have O = Ee [g(X, 0, ev) - M(g 1 a) i a] + Ee [{g(X, 0, ev)

-M(g1ev)}pe(X1a)/pe(Xla)1a] . Hence

{Ee(g(X, 0, ev) - M(g I a) 1 a) }2

== {Ee[{g(X, 0, a) - M(g l ev)}g. (X, 0, ev) l a) }2

={Cov(g, g,1a)}2 ,

where Cov(g, g. 1 a) is the conditional covariance of g and g. given ev. And from V(g 1 a) = Ee ({g(X, 0, a) - M(g 1 a)}2 i ev)

we have the equation of (11). Furthermore, since M(g*l ev) = Ee[g. 1a) = O and g*(x, 0, a)

= pe(x I a)/pe(x 1 a) - gZ (x, 0, ev), then

K(g*, 01 ev) == Ee({g* - M(g* 1 a)}2 la)/{Ee(g. -M(g. 1 ev)l a)}2 = Ee(gi l a)/{fffpe(x 1 a)dp"Ee(gZ 1 ev]}2

= {Ee(g*2(X, 0, a)l ev }"' = { V(g. 1 a)}-i

From O S. R2(g, g.Ia) S. 1, we have (11). As the same argument, we have (12).

DEFINITioN 3. For each .tixed a, goEfl7. is said to be oPtimal in El. z;ICforanyge fg.7 K(go, 01 ev) S- K(g, 01a)

for all 0e0.

DEFINITIoN 4. ho E M is said to be oPtimal in ev z;ICforany he dS2?,9 K( ho , 0) $ K(h, 0)

for all 06e.

From Theorem 1, g. and h. are optimal in Sf'. and c?0 respectively.

AssuMpTioN 3. There exists a Partly szafiicient statistic T of0for .9i[= {Pe(•l a); 0e (E),

aEY} (see [3]), and its density function, peT(t1 a), is continuozesly dzlfferentiable wrt 0 almost

everywhere y for each fixed ev e.s4 where y is the induced measure from pt on the samPle space

(sr,s) of T.

10 Akimichi OKuMA

AssuMpTioN 4. p;(tla) does not dePend on a.

THEoREM 2. Under the assumPtions 1-3, forany ge 9., let

(13) gi(t, 0, ev)=Ee(g(X, 0, a)lt, a) .

Then gie gl. and for all 0e0

(14) K(gi,01a) S- K(g, 01a).

PRooF. Let g(x, 0, a) = g(x, 0, a)- M(g l a). Then Ee (gl ev) = O. Therefore g satisfies the conditions (i)-(v) in [3]. From (13),

gi(t, 0, a) = Ee(g(X, 0, a)1a]+M(gla) =Ee[g(X, 0, a)1a]+M(gila) .

Set gi(t, 0, a) = gi(t, 0, a)-M(gila) and apply Theorem 1 in [3] for g. We have K(gi , 0I a) = Ee({gi - M(gi 1 a)}2 1 a)•Ee[gi -M(gi l a)1 a)}-2

= Ee(g?la)'{Ee(gila]}-2 S- Ee[g2lev]'{Ee[gla]}m2

= Ee({g- M(g I a)}2 l a] '{Ee(g- M(g l a) I a]}-2 = K(g, 01 a) .

THEoREM 3. For each aEJ4, let g' be an oPtimal estimating function in [Y. szach that K(g*, 01 ev)Åq co, and let

(15) h'(x, 0) == Ee(g'(x, 0, a)lx].

Under the assumPtions 1-4, h* is oPtimal in ,;Y.

Proof is similar to the proof of the previous theorem, i. e., let g'(x, 0, a) = g"(x, 0, a) - M(g' 1 a) and h'" (x, 0) = h' (x, 0) - M( h" ), and apply Theorem 2 in [3].

References

[1] V. P. BHApKER, On a measure of efiiciency of an estimating equation, SANKHYA, Ser. A, 34,(1972), 467-472.

[2] V. P. GoDAMBE, An optimal property of regular maximum likelihood estimation, Ann. Math. Sta- tist., 31, (1960), 1208-1211.

[3] A. OKuMA, Optimal estimating equations for a model with a nuisance parameter, Tamkang Jour.

Math., 6, (1975), 239-249.

[4] A. OKuMA, On invariance of estimating equations, Bull. Kyushu Inst. Tech. (M. & N. 9. .), No. 23, (1976). 11-16.

[5] A. OKuMA, Some applications of a partly sufficient statistic to estimating equations in the presence

of a nuisance parameter, Bull. Kyushu Inst, Tech. (M. & N. S.), No. 24, (1976), 29-36.