CONVERGENCE RATES IN EMPIRICAL BAYES PROBLEMS WITH A WEIGHTED SQUARED-ERROR LOSS. THE PARETO DISTRIBUTION CASE

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1 CONVERGENCE RATES IN EMPIRICAL BAYES PROBLEMS WITH A WEIGHTED SQUARED-ERROR LOSS. THE PARETO DISTRIBUTION CASE VASILE PREDA ad ROXANA CIUMARA We study the problem of estimatig the scale parameter θ for a Pareto distributio uder a weighted squared-error loss through the empirical Bayes approach. A empirical Bayes estimator is proposed ad some asymptotic optimality properties are give. Also, uder certai coditios, the empirical Bayes estimator proposed is asymptotically optimal with rate of covergece of order 2 3. AMS 2 Subject Classificatio: 62P5. Key words: empirical Bayes, weighted squared-error loss, asymptotic optimality, rate of covergece. 1. INTRODUCTION Robbis [12] argued that, for some estimatio problems, the iformatio obtaied at each step could be used to improve the ext step decisio. These procedures, kow as empirical Bayes or adaptive methods, were studied by may authors ad amog them we remid the work of Johs [4], Samuel [13], Berger ad Berlier [2], Preda [9, 11], Tiwari ad Zalkikar [16], Liag [5], ad Sigh [15]. The usefuless of empirical Bayes estimatio i practical statistical applicatios depeds o the overall risks rate of covergece to optimal risk. The problem of covergece rates for empirical Bayes estimates was studied by Li [6], Preda [8] ad by Tiwari ad Zalkikar [16] ad Liag [5], cosiderig a squared-error loss. Tiwari ad Zalkikar [16] foud that, uder certai coditios, the empirical Bayes estimator for the scale parameter i the Pareto distributio is asymptotically optimal ad the rate of covergece is of order 1 2. Liag [5] used the same squared-error loss, but relaxed the coditios stated i Tiwari ad Zalkikar [16] ad proved that the empirical Bayes estimator proposed of the same parameter of the Pareto distributio is asymptotically optimal with associated rate of covergece of order 2 3. REV. ROUMAINE MATH. PURES APPL., 52 27), 6,

2 674 Vasile Preda ad Roxaa Ciumara 2 I this paper, we cosider a weighted squared-error loss ad propose a empirical Bayes estimator for the scale parameter of the Pareto distributio. We assume that the weights are give by a fuctio which satisfies certai properties. I Sectio 2, we describe the Pareto distributio with a kow shape parameter α ad ukow scale parameter θ. Furthermore, the coditios that have to be satisfied i order to obtai the results from Sectios 3 ad 4 are stated. We defie the Bayes risk for a weighted squared-error loss ad the overall Bayes risk for a sequece of empirical Bayes estimators. Next, asymptotic optimality ad rate of covergece for a sequece of empirical Bayes estimators are defied. I Sectio 3, we cosider the coditios imposed i Sectio 2 ad propose a empirical Bayes estimator for the ukow scale parameter of the Pareto distributio for a class of prior distributios. A useful study of the Pareto distributio could be foud i Arold [1] ad Preda [1]. I Sectio 4, we study asymptotic optimality ad prove that uder the coditios assumed the rate of covergece is of order SOME PRELIMINARIES Let X be a radom variable havig a Pareto distributio with probability desity fuctio f x θ) = αθα x α+1, where x>θ, α>adθ>. The shape parameter α is kow ad the scale parameter θ is ukow. We suppose that the parameter θ represets a value of a radom variable Θ, which has a prior distributio fuctio G :, ) [, 1]. I this case, the margial desity of X is give by fx) = mix,m) f x θ)dgθ) = mix,m) f x θ) gθ)dθ, where dgθ) =gθ)dθ. Deotig f x θ) =ϕθ)u x), where ϕθ) =αθ α ad ux) = 1 x,weget α+1 or fx) =ux) x α+1 fx) = mix,m) mix,m) αθ α dgθ) αθ α dgθ). As for the prior distributio G, we impose the coditios below. Coditios o G Liag [5]):

3 3 Covergece rates i empirical Bayes problems 675 A1) Gm) = 1 for some kow positive real umber m. A2) If a = sup{θ Gθ) =} the f is a decreasig fuctio i x o a,m]. We cosider the problem of estimatig the parameter θ uder a weighted squared-error loss, L : R 2 + R + defied as L x, θ) =wθ)x θ) 2, with a weight fuctio w : R + R + cotiuous ad differetiable. The robustess of loss fuctio of this type was studied by Makov [7]. Next, we suppose that w satisfies the coditios below. Coditios o w: A3) c 1 R + such that w θ) c 1, θ R +. A4) c 2 R + such that w θ)+θw θ) c 2, θ R + ad ε> such that wθ)+θw θ) >εo,m]. A5) ε > such that ε <qx) =E wθ) X = x), x,m]. Example 2.1. If w : R + R +, wθ) = 1 1+θ,weget w θ) 1ad wθ) +θw θ) 1, thus, coditios A3) ad A4) are satisfied. If we cosider a uiform o [, 1] prior distributio, ad m = 1, the coditio A5) also holds sice 1 2 <qx) 1. The Bayes estimator of θ give X = x is E ΘwΘ) X = x) 2.1) ϕ G x) =argmie LX, Θ) X = x) = E wθ) X = x) assumig that all posterior expectatios ivolved i the above expressio exist ad E wθ) X = x). The Bayes risk of ϕ G is R G, ϕ G )=E L ϕ G X), Θ)) = E wθ) ϕ G X) Θ) 2), where the expectatio is take with respect to X, Θ). Let X 1,X 2,...,X be the past data, idepedet ad idetically distributed radom variables with probability desity fuctio fx). Deote by X =X 1,X 2,...,X )adϕ X) =ϕ X, X ) the empirical Bayes estimator of the parameter θ basedopastdatax ad the preset observatio X. The coditioal Bayes risk of ϕ give X is ) R G, ϕ X )=E w Θ) ϕ X) Θ) 2 X ad R G, ϕ )=E R G, ϕ X ))

4 676 Vasile Preda ad Roxaa Ciumara 4 is the overall Bayes risk of ϕ. Here the expectatio is take with respect to X. We ote that because ϕ G is the Bayes estimator, that is, ϕ G x) =argmie LX, Θ) X = x) we have R G, ϕ G ) R G, ϕ X ) X vector of past data ad N.Moreover, 2.2) R G, ϕ G ) R G, ϕ ) N. Thus, R G, ϕ ) R G, ϕ G ) is oegative ad could be used as a measure of performace of the empirical Bayes estimator ϕ. Defiitio 2.1 Robbis [12], Preda [11], Liag [5]). A sequece ϕ ) 1 of empirical Bayes estimators is said to be asymptotically optimal if R G, ϕ ) R G, ϕ G ). Moreover, if R G, ϕ ) R G, ϕ G )=Oα ), where α ) 1 is a sequece of real umbers α > adα, the ϕ ) 1 is said to be asymptotically optimal with covergece rate of order α. 3. THE EMPIRICAL BAYES ESTIMATOR I order to propose a empirical Bayes estimator, we first have to derive the Bayes estimator. Theorem 3.1. Uder coditios A1) A5), the Bayes estimator of the scale parameter for the Pareto distributio is give by x wx) f Mx) x 3.1) ϕ G x) = α+1 fx) if <x m m wm) if x>m, f Mm) m α+1 fm) where wx) = wx) qx), Mx) = Mx) qx) ad Mx) = x θα+1 wθ)+θw θ)) df θ). Proof. O accout of 2.1), we evaluate the umerator E ΘwΘ) X = x) i the expressio of the Bayes estimator. For <x m we have E ΘwΘ) X = x) = xwx) fx) fx) 1 x x α+1 θ α+1 wθ)+θw θ) ) df θ) fx) that is, 1 3.2) E ΘwΘ) X = x) = xwx) x α+1 fx) Mx),

5 5 Covergece rates i empirical Bayes problems 677 where Mx) = x θα+1 wθ)+θw θ)) df θ). It follows from coditio A4), that Mx). Sice E ΘwΘ) X = x), we get 1 3.3) x α+1 Mx) xw x) fx) ad 3.4) E ΘwΘ) X = x) xwx). For x>mwe have E ΘwΘ) X = x) = mα+2 wm) x α+1 fx) fm) Mm) x α+1 fx). Sice 3.5) x α+1 fx) =m α+1 fm) for x>m,weget 3.6) E ΘwΘ) X = x) =mwm) Mm) m α+1 fm), for all x>m. Because E wθ) X = x) =qx) >, wx) = wx) qx) for <x m we obviously have ϕ G x) =x wx) Sice qx) = qm), for x>mwe have Mx) x α+1 fx). ad Mx) = Mx) qx), ϕ G x) =m wm) Mm) m α+1 fm). Thus, for x>m, ϕ G x) =ϕ G m). 1 We proved before that M x) xwx) for<x m. Sice we x α+1 fx) imposed coditio A5) ad qx) >, the previous iequality implies 3.7) Mx) x α+1 fx) x wx). Now, we ca express the Bayes estimator of θ as x wx) f Mx) x ϕ G x) = α+1 fx) if <x m m wm) f Mm) if x>m. m α+1 fm)

6 678 Vasile Preda ad Roxaa Ciumara 6 Let b ) 1 be a sequece of strictly positive real umbers such that b adb. We defie f x) = F x + b ) F x) b, where F x) is the empirical distributio fuctio based o X 1,X 2,...,X. We ote that f x) ca be expressed as 3.8) f x) = 1 b Moreover, E f x)) = F x+b) F x) b I x,x+b] X j ). j=1 fx). Thus, f x) isacosistet estimator of fx) Iosifescu, Mihoc ad Theodorescu [3]). Next, defie 3.9) M x) = 1 j=1 Xj α+1 w Xj )+X j w X j ) ) I,x) X j ). We ca easily see that E M x)) = Mx) sicex 1,X 2,...,X are idepedet ad idetically distributed radom variables: E M x)) = 1 E X α+1 w Xj )+X j w X j ) ) I,x) X j )) = = x j θ α+1 wθ)+θw θ) ) df θ) =Mx). Thus, M x) is a cosistet estimator of Mx) Iosifescu, Mihoc ad Theodorescu [3]). The empirical Bayes estimator for the scale parameter θ that we propose is give by [ 3.1) ϕ X) = X wx) M ) ] X) X α+1 I f X),m] X) + where M = M q [ + m wm) M ) ] m) m α+1 I f m) m, ) X), ad a b =maxa, b).

7 7 Covergece rates i empirical Bayes problems ASYMPTOTIC OPTIMALITY OF THE EMPIRICAL BAYES ESTIMATOR PROPOSED I this sectio we study the asymptotic optimality of empirical Bayes estimator. Our aalysis is based o coditios A1) A5). The mai result is as follows. Theorem 4.1. If b ) 1 is a sequece of strictly positive real umbers such that b ad b, ϕ ) is the sequece of empirical Bayes estimators 3.1) ad ϕ G is the Bayes estimator 3.1), the ) ) 1 1 R G, ϕ ) R G, ϕ G )=O + O + O b 2 b ). Proof. SiceR G, ϕ G ) R G, ϕ ) ad coditio A3) holds, we have R G, ϕ ) R G, ϕ G )= = E R G, ϕ X )) E wθ) ϕ G X) Θ) 2) c 1 E ϕ X) ϕ G X)) 2). Moreover, R G, ϕ ) R G, ϕ G ) c 1 E ϕ X) ϕ G X)) 2) = [ m = c 1 E ϕ x) ϕ G x)) 2) fx)dx+ E ϕ x) ϕ G x)) 2) ] fx)dx. m For x>mwe have ϕ G x) =ϕ G m) adϕ x) =ϕ m). Therefore, ϕ x) ϕ G x) =ϕ m) ϕ G m) ad, cosequetly, E ϕ x) ϕ G x)) 2) fx)dx = E ϕ m) ϕ G m)) 2) 1 F m)). Assume ow that <x m. I this case, o accout of coditios A4) we have ϕ G x) x wx) =x wx) qx) ad ϕ x) x wx) =x wx) qx) because M x). We thus obtai ϕ x) ϕ G x)) x wx) =x w x) qx). So, cosiderig the expressios of ϕ x) adϕ G x) ad followig the same reasoig as i Sigh [14], we get E ϕ x) ϕ G x)) 2) = E m M x) q x) x α+1 f x) ) ) Mx) 2 qx) x α+1 fx)

8 68 Vasile Preda ad Roxaa Ciumara 8 M ) ) 8 f 2 x)q 2 x) E x) Mx) 2 x α+1 + ) ) 8 Mx) 2 + f 2 x)q 2 x) x α+1 + x2 w 2 x) E f x) fx)) 2). fx) 2 Sice E M x)) = Mx), we have M ) ) x) Mx) 2 ) M x) E =Var = =Var 1 j=1 X α+1 j x α+1 x α+1 w Xj x α+1 )+X j w X j ) ) I,x) X j ) 1 c2 2 from coditio A4). Moreover, E f x) fx)) 2) =Varf x)) + E f x)) fx)) 2 with Var f x)) = Var 1 b I x,x+b] X j ) 1 j=1 b 2 x+b x fy)dy fx) b, where the last iequality holds because of coditio A2) while, from Liag [5], E f x)) fx)) 2 f 2 x) α +1)2 b 2 4x 2. Fially, ) Mx) 2 x α+1 + x2 w 2 x) x 2 w 2 x)+ x2 w 2 x) 2x 2 w 2 x). fx) 2 2 Now, o accout of the above expressios, for <x m we have E ϕ x) ϕ G x)) 2) 8c 2 2 f 2 x)q 2 x) + 16x 2 c 2 1 b fx)q 2 x) + 4c2 1 α +1)2 b 2 q 2 x) ad E ϕ m) ϕ G m)) 2) 8c 2 2 f 2 m)q 2 m) + 16m 2 c 2 1 b fm)q 2 m) + 4c2 1 α+1)2 b 2 q 2 = m) ) ) 1 1 = O + O + O b 2 b ).

9 9 Covergece rates i empirical Bayes problems 681 m The, sice A2) ad A5) hold, we get E ϕ x) ϕ G x)) 2) fx)dx 1 8mc 2 2 ε 2 fm) m 3 c 2 1 b 3ε 2 ) ) 1 1 = O + O + O b 2 b ). Summaryzig the results obtaied by ow, we have R G, ϕ ) R G, ϕ G )=O +b 2 4c 2 1 α+1)2 ε 2 ) ) O + O b 2 b ). Remark 4.1. Uder the coditios of Theorem 3.1, if w θ) =1thewe get the Bayes estimator ad, respectively, the empirical Bayes estimator from Liag [5]. = REFERENCES [1] B.C. Arold, Pareto Distributios. Iteratioal Co-operative Publishig House, Fairlad, MD, [2] J. Berger ad L.M. Berlier, Robust Bayes empirical Bayes aalysis with ε-cotamiated priors. A. Statist ), 2, [3] M. Iosifescu, G. Mihoc ad R. Theodorescu, Teoria probabilităţilor şi statistica matematică. Ed. Tehică, Bucureşti, [4] M.V. Johs, Jr., Noparametric empirical Bayes procedures. A. Math. Statist ), [5] T.C. Liag, Covergece rates for empirical Bayes estimatio of the scale parameter i a Pareto distributio. Comput. Statist. Data Aal ), [6] P.E. Li, Rates of covergece i empirical Bayes estimatio problems: Cotiuous case. A. Statist ), [7] U. Makov, Loss robustess via Fisher-weighted squared-error loss fuctio. Isurace Math. Ecoom ), 1 6. [8] V. Preda, Etropia poderată şi problema de selecţie eparametrică. Stud. Cerc. Mat ), [9] V. Preda ad V. Craiu, Probleme de decizie multiplă. Tipografia Uiv. Bucureşti, 198. [1] V. Preda, Iformatioal characterizig of Pareto ad power distributios. Bull. Math. Soc. Sci. Math. R.S. Roumaie N.S.) 2876) 1984), [11] V. Preda, Teoria deciziilor statistice. Ed. Academiei Româe, [12] H. Robbis, A empirical Bayes approach to statistics. I: Proc. Third Berkeley Sympos. Math. Statist. Probab ), [13] E. Samuel, A empirical Bayes approach to the testig of certai parametric hypotheses. A. Math. Statist ), [14] R.S. Sigh, Applicatios of estimators of a desity ad its derivatives to certai statistical problems. J. Roy. Statist. Soc. Ser. B ),

10 682 Vasile Preda ad Roxaa Ciumara 1 [15] R.S. Sigh, Empirical Bayes estimatio i Lebesgue-expoetial families rates ear the best possible rate. A. Statist ), [16] R.C. Tiwari ad J.N. Zalkikar, Empirical Bayes estimatio of the scale parameter i a Pareto distributio. Comput. Statist. Data Aal ), Received 11 December 26 Uiversity of Bucharest Faculty of Mathematics ad Computer Sciece Str. Academiei Bucharest, Romaia preda@fmi.uibuc.ro ad Academy of Ecoomic Studies Departmet of Mathematics Calea Dorobatilor Bucharest, Romaia roxaa ciumara@yahoo.com