CONVERGENCE RATES FOR EMPICAL BAYES TWO-ACTION PROBLEMS- THE UNIFORM br(0,#) DISTRIBUTION

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1 Jurnal fapplied Mathematics and Stchastic Analysis 5, Numer 2, Summer 1992, CONVERGENCE RATES FOR EMPICAL BAYES TWO-ACTION PROBLEMS- THE UNIFORM r(0,#) DISTRIBUTION MOHAMED TAHIR Temple University Department f Statistics Philadelphia, PA 1912e ABSTRACT The purpse f this paper is t study the cnvergence rates f sequence f empirical Bayes decisin rules fr the tw-actin prlems in which the servatins are unifrmly distriuted ver the interval where is a value f a randm variale having an unknwn prir distriutin. It is shwn that the prpsed empirical Bayes decisin rules are asympttically ptimal and that the rder f assciated cnvergence rates is O(n- a), fr sme cnstant a, 0 < a < 1, where n is the numer f accumulated past servatins at hand. Key wrds: Asympttically ptimal, Bayes risk, cnvergence rates, empirical Bayes decisin rule, prir distriutin. AMS (MOS) suject classificatins: 62C INTRODUCTION In situatins invlving sequences f similar ut independent statistical decisin prlems, it is reasnale t frmulate the cmpnent prlem in the sequence as a Bayes statistical decisin prlem with respect t an unknwn prir distriutin ver the parameter space, and then use the accumulated servatins frm the previus decisin.prlems t imprve the decisin rule at each stage. This apprach was first develped y Rins [5] and was later studied in estimatin and hypthesis testing prlems y many authrs. Fr example, Rins [6] and Samuel [7] exhiit empirical Bayes rules fr the tw-actin prlems in which the distriutins f the servatins elng t a certain expnential family f praility distriutins. Jhns and Van l:tyzin [1] study the cnvergence rates f a sequence f empirical Bayes rules which they prpse fr the tw-actin prlems where the servatins are memers f sme cntinuus expnential families f distriutins. Als, Susarla and O Bryan [8] and Ngami [4] cnsider estimatin prlems in which the servatins are unifrmly distriuted ver the interval (0,a). These authrs shw that their empirical Bayes rules are asympttically 1Received: Septemer, 1991, Revised" Decemer, Printed in the U.S.A. (C) 1992 The Sciety f Applied Mathematics, Mdeling and Simulatin 167

2 168 MOHAMED TAHIR ptimal, in the sense that the assciated Bayes risks f these rules cnverge t the minimum Bayes risk which wuld have een tained if the prir distriutin were cmpletely knwn and the Bayes rule with respect t this prir distriutin were used. The jective f this paper is t investigate the cnvergence rate f a sequence f empirical Bayes decisin rules fr the tw-actin prlems in which the servatins are unifrmly distriuted ver the interval (0,), where is a value f a randm variale having an unknwn prir distriutin. Let X dente the randm servatin f interest in the cmpnent decisin prlem and suppse that X has density functin f0(x ) 1 where 8 is an unknwn numer such that 0 < 8 < _<, and I(.) dentes the indicatr functin. It is desired t determine a decisin rule fr chsing etween the hyptheses H: 0 _< 0 and H 1" 0 > 0, where 0 0 is a given numer such that 0 < 0 <. Let a 0 and a 1 e the tw pssile actins, f which a is apprpriate when H is true, i = 0,1, and suppse that the lss functin is f the frm = + (0) = (0 0 0) + (1.1) fr 0 < 0 <, where fr = O, 1, Li(O measures the lss incurred when actin a is taken and 0 is the true value f the parameter, and where (u) + = max{u, 0}. Suppse that is a realizatin f a randm variale O having an unknwn prir distriutin functin G. Thus, the frmulatin f the prlem assumes that the randm servatin X is cnditinally unifrmly distriuted ver (0,), given =, where is a randm variale having an unknwn prir distriutin G; that, ased n X, an actin a E {a0,a1} is t e taken; and that if actin a is taken, then the lss incurred is f the frm (i.i). Let 6(x) = P{accepting HlX = e a randmized decisin rule fr the ave decisin prlem. Then the Bayes risk incurred y

3 Cnvergence Rates fr Empirical Bayes Tw.Actin Prlems 169 using the decisin rule 8 with respect t the prir distriutin G is given y where = fa(z)6(zldz + C(G), (1.2/ a(x) = f Of(x)dG(O Of(x (1.3) f(z) = f f(z)dg(o) fr 0 < z < and f C(G) = L(OldG(O). Nte that C(G) is a cnstant which is independent f the decisin rule 6. Hence, it fllws frm (1.2) ha a Bayes decisin rule, 6a, is clearly given y The Bayes decisin rule a cannt e applied the dedsin prlem under sudy since it depends n the unknwn prir distriutin G. In view f this remark, an empiriem Bayes apprach is used wih prir disgriugins G fr which fodg(o)<, insure heg the Bayes 0 risk is always finite. Sectin 2 prvides a sequence f empirical Bayes rules fr the decisin prlem descried ave. Sectin 3 estalishes the asympttic ptimality f the prpsed rules and investigates the cnvergence rates f these rules. f this paper. Sectin 4 cntains an example which illustrates the results 2. THE PROPOSED EMPIRICAL BAYES RULES The cnstructin f a sequence f empirical Bayes decisin rules fr the decisin prlem descried in the previus sectin is mtivated y the fllwing servatins. First, a(z) in (1.3) can e rewritten as y using the definitin f f(z). O = 0, is given y ---- a(z) 1 G(z) Of(z ), Furthermre, the cnditinal distriutin functin f X, given

4 170 MOHAMED TAHIR fr 0 < x < ; fr 0 < z <. F(x = / f(t)dt = xf(x + I(,)(x) s that, the marginal distriutin functin f X is given y Therefre, fr 0 < a: <, y (2.1). F(z) = f F(z)dG(O a(a:) = 1 = zf(z) + G(z) F(z) + (z- O)f(a: An empirical Bayes decisin rule fr the (n + 1)s decisin prlem may e tained y first estimating a(a:), fr each z, using the n accumulated servatins frm the previus prlem, and then adpting an empirical Bayes decisin rule which is similar, in a sense, t the Bayes rule a, ut which des nt require the knwledge f the prir distriutin G. Specifically, let X1,...,X n e the servatins frm n past experiences f the cmpnent decisin prlem, where fr each = 1,...,n, X is cnditinally unifrmly distriuted ver the interval (0,9i) given i = tgi, and 1,O2,"" are independent and identically distriuted (i.i.d.) randm variales with cmmn distriutin G. Hence, X,X2,... are i.i.d, with cmmn distriutin F. Let X n + = X dente the current servatin and let dente the natural estimatr f F(). can e estimated y Fn(a:) = E I{xi <- Als, since y definitin f a density functin, f(x) lira h--o F(x + h)- F(x) h Fn(a: + hn) Fn(a:) f n(a:) = h. fr n >_ 1, where hn, n >_ 1, is a sequence f psitive real numers such that hn---,o and nhn--. as n. In view f (2.2), a(x) can e estimated y fr eachx, 0<z<. an(x = 1 Fn(z + (z- O0)fn(Z In view f (1.4) and the ave servatins, define the empirical Bayes decisin rule fr the (n + 1)st decisin prlem y n(a:) = 1 if an(x <_ 0 0 if an(a: > O. (2.4)

5 Cnvergence Rates fr Empirical Bayes Tw-Actin Prlems 171 fr each z, 0 < z <. Then clearly, 6 n des nt depend n the unknwn prir distriutin G. 3. ASYMPTOTIC OPTIMALITY Let A dente the class f all decisin rules, and let R*(G) dente the minimum Bayes risk with respect t the prir distriutin G. Then, R (G) = inf R(G, 5) = /a(z)6g(z)dz + C(G), as in (1.2). Als, let R(G) dente the Bayes risk incurred y using the empirical Bayes rule 8n, defined y (2.4), with respect t the prir distriutin G. Then, (3.1) 0 where E dentes expectatin with respect t the marginal jint distriutin f X1,...,Xn. Then, R(G)>_R*(G) fr all n_> 1, since R*(G) is the minimum risk, and hence R(G)- R*(G) may e used as a measure f the ptimality f the empirical Bayes rule di n. Lemma 3.1: Fr n > 1, Prf: It R;(G) R (G) < a(z) IP(I an(x a()i > la()i}d. An applicatin f (3.1) and (3.2), fllwed y (1.4) and (2.4), yields where R:(G)- R, (G) : /a(z)[p{an(z <_ O}-diG(X)]dz 0 =/la(z) B(n,z)dz, P{an(Z > 0} if B(n,z) = P{an(z) < 0} if a(z) > 0 (3.3)

6 172 MOHAMED TAHIR fr 0 < z <. The lemma nw fllws since [an(z )- ()[ > [a(z)[ is implied y an(z > 0 when a(z) _< 0, and y an(z <_ 0 when a(z) > 0. Definitin 3.1: Let vn, n >_ 1 e a sequence f psitive numers such that vno as n--*. A sequence f Bayes empirical estimatrs 5n, n >_ 1, is said t e asympttically ptimal at least f rder v The main result is presented next. Therem 3.1: Let 5n e as in (2.4) and suppse that f, the derivative f f, ezists and is cntinuus. If () f a() <, and if fr sme r, 0 < r < 2, ir i1_ (iv) fl:--o0 la(z) [f;(z)]"dz <, where f(x) stp < < ef( + t) and f() stp < < f ( + ) l, fr" sme e > O, then 1 chsing h n = n 2 + 1, fr sme, 0 < < 1, yields R(G)- R (G) <_ O(n- a) r Thus, the sequence 6n, n > 1 is asympttically ptimal at least f as ncx, where = 2[3:+:" rder n- a with respect t the prir distriutin G. that Prf: Let r > 0 e given. It fllws frm Lemma 3.1 and Markv s inequality n;,(a)- n (a) _< fl a( )I -ran(z)-a(x)[rdx fr all n _> 1. Furthermre, y (2.2), (2.3), and the cr-inequality (Lve, [3]) (3.3) + 1 O I" (; n) f(z)i" fr each z > 0, where c r = 1 r 2 r- accrding as 0 < r _< 1 r 1 _< r < 2 and F(z H(z; n) + hn)- f(z) (3.4)

7 Cnvergence Rates fr Empidcal Bayes Tw-Actin Prlems 173 fr : > 0. Next, since E[Fn(a:)] = F(z) and Varn()] = 1- F(z)]F(:), H(z; n)- f(z) = 1/2hnf (z +.), 1 where 0 < z. < h n. Letting h n = n 2-i-:i and cmining (3.3)-(3.7) yields as n---+, where ci = 2 / i R(G) R*(G) K1 K 2 and = O(n -< ) K1 = cr Ii a(z)11 -r [F(z)(1 r(z))]rl2dz = K 3 2-" i 2 r 0 are all finite y assumptins (ii)-(iv) f the therem. can e applied. 4. EXAMPLE The fllwing example prvides a class f prir distriutins G t which Therem 3.1 Suppse that G is the gamma distriutin with density functin s2oe-so if O>O () = f e <_, where s > 0 is a given numer. Then f(z) ] se- sx if z > 0 0 if z_<o,

8 174 MOHAMED TAHIR and a( if J (1 + sx SO)e q if z>0 z<_o, if z>0 z<0. Als, f(x) = f(x) and f(z) = sf(x). Clearly, Assumptin (i) f Therem 3.1 is satisfied. T verify Cnditin (ii), let 0 > 0 e a given numer and serve that say. Mrever, fl ()11 c 00 r [F(x)( 1 F(x))l r/2 dx <_ fl I + -O p-d + i (1 + sx 8/0) 1 r e 0 = I , z, _<0,O - + %(:,)--0, -(1 -)SXdx y a simple integratin, where e r = 1 r 2 r- 1 accrding as 0 < r _< 1 r 1 < r < 2; and i: <_ f (1 + sx)l-re-(1-)sxdx < 0 when 0 < r < 1. When 1 < r < 2, (1 + sx- $00) 1-r _< 1 fr all x > 0 0 and thus, -(i-)s:dz I:_< e < Therefre, Cnditin (ii)f Therem 3.1 is satisfied when 0 < r < T verify Cnditin (iii) f Therem 3.1, nte that c 0 /i:-oi" la()l -" [.f2,()]"/d -< +sz-sol1- dz say. Furthermre, y the cr-inequality + s r12 xq l s + sx e 0 = s/(. + g), -(I (4.1)

9 Cnvergence Rates fr Empirical Bayes Tw-Actin Prlems 175 J1 < cro SOl 1-r + Or(2- r)- lsl to2 0 s _< + -,Ol-" (4.2) Finally, t verify Cnditin (iv) f the therem, serve that where J1 is as in (4.1) and 0 as in (4.2). 0 [1] [2] [3] [4] [] [61 [7] [8] REFERENCES,M.V. Jhns and J. Van Ryzin, "Cnvergence rates fr empirical Bayes tw-actin prlems II. Cntinuus case", Ann. Math. Statist. 43, (1972), pp R.L. Lehman, "Testing Statistical Hyptheses", Wiley, New Yrk, M. L$ve, "Praility Thery", 3rd ed. Van Nstrand, Princetn, Y. Ngami, "Cnvergence rates fr empirical Bayes estimatin in the unifrm U(0,#) distriutin", Ann. Statist. 16, (1988), pp H. Rins, "An empirical Bayes apprach t statistics", Prc. Third Berkeley Symp. Math. Statist. 1, (1955), pp H. Rins, "The empirical Bayes apprach t statistical decisin prlems", Ann. Math. Statist. 35, (1964), pp E. Samuel, "An empirical Bayes apprach t the testing f certain parametric hyptheses, Ann. Math. Statist. 34, (1963), pp V. Susarla and T. O Bryan, "Empirical Bayes interval estimates invlving unifrm distriutins", Cmm. Statist. A-Thery Methds 8, (1979), pp

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