多參數(shù)指數(shù)族參數(shù)經驗貝葉斯估計的收斂速度

1、COMUNICATIONS IN STATISTICS 1996，25(5)，1089 1098CONVERGENCE RATES FOR EMPIRICAL BAYES ESTIMATORS OF PARAMETERS IN MULTI-PARAMETER EXPONENTIAL FAMILIESHengqing Tong Department of Mathematics and Physics, Wuhan University of TechnologyKey Words and Phrases: multi-parameter exponential family; orthogon

2、al polynomial; kernel estimation of density; empirical Bayes ; convergence rate.ABSTRACTThis paper obtains the convergence rates of the empirical Bayes estimators of parameters in the multi-parameter exponential families. The rates can approximate to arbitrarily. The paper presents the multivariate

3、orthogonal polynomials which are continuous on the total space .1. Introduction Lin(1975)investigated the convergence rates in empirical Bayes (EB) estimation problems. He gave the univariate orthogonal polynomials which are not continuous on the total real axis. Singh(1977,1979)obtained the rates o

4、f convergence of the EB estimators in one-parameter exponential families. Gajek (1986) improved the univariate density kernel estimators which are not bona fide functions. Now we consider the convergence rates of the EB estimators of the parameters in the multi-parameter exponential families. The ra

5、tes can approximate to arbitrarily and we give a concise proof .We give the multi-variate orthogonal polynomials, which are continuous on the total space.2. Convergence of the kernel estimator of density and its partial derivative in multi-parameter exponential familiesLet have the distribution of m

6、ulti-parameter exponential family. Its density and its mixed partial derivatives are written as： （2.1）Where and .First，we give the kernel estimators of density and its mixed partial derivatives .By using multivariate kernel function, the estimators of and its partial derivatives are （2.2）Where with

7、being the dimension of an orthogonal polynomial space in which the kernel function are constructed. is determined not only by , but also by .It satisfies （2.3）Where and are constants and each may be differ. also satisfies： （2.4）The multivariate kernel function is constructed as follows: （2.5）Where a

8、re common univariate kernel functions, they satisfying （2.6）and （2.7）The construction of this univariate kernel function is as follows. In the numerical determinant ，all the elements of the last row are 1,and the elements from the first row to the row are.Then, the elements of therow inare replaced

9、by 上，and the variety determinant is obtained。It is a degree polynomial. Let （2.8）Note that（2.8） satisfies（2.6）and（2.7）.Therefore, they are orthogonal polynomials. At the truncated points, （2.9）are continuous. Therefore, is continuous on the total space.Second, we investigate the convergence to .Supp

10、ose the partial derivative functions are locally bounded, i. e., when for all, there exists and ，and when and ，,（ is the sample space of ），there is （2.10）THEOREM 1,Suppose are locally bounded，then （2.11） （2.12）PROOF Let ，Since ， are i. i. d,， （2.13）From the multivariate Taylor formula and the orthog

11、onal conditions of the kernel functions, （2.14）Where ，Because is locally bounded and the kernel functions as well as the integral field are bounded,（2.11) is obtained. Moreover, from （2.15）and （2.16）We know (2.12) is correct. Third, we consider the convergence of the vector of the kernel estimator o

12、f the partial derivatives. When, all are denoted .Arrange the elements into a vector . Similarly may denote kinds of kernel estimators . We also arrange them into a vector. From （2.7) ,we know their constructions of kernel functions are different and the orthogonal condition are different. From Theo

13、rem 1, (2.17)Moreover, (2.18)Suppose ，According to Jensen and Holder inequalities, we have （2.19）COROLLARY 1 Suppose ，then (2.20)and (2.21)3. Convergence rate of estimators of parameter in multi-parameter exponential familiesFirst, we give the construction of estimators of parameter. Sample comes fr

14、om the multi-parameter exponential families. Its conditional density with parameters is (3.1)If the prior distribution of is ，the unconditional density of is (3.2)Where is the parameter space of. Consider the quadratic loss function （3.3）Where is a positive definite matrix ， is an estimator of . The

15、 Bayes estimator of is its posterior expectation is then: (3.4)Its risk is (3.5)If prior distribution is unknown, we construct the estimator of by using the historic samples and the present sample. According to （3.4）,the estimator of is： (3.6) Where . The truncated function of vector is defined as t

16、he truncation of each component: (3.7)The truncation of a numerical value is as usual: (3.8)Obviously Second, we give two lemmas with regard to vectors.LEMMA 1 Suppose is finite, then (3.9)The proof is easy and is omitted.LEMMA 2 Let and be random variables, and be real values, and be vectors, ，then

17、The proof can be completed by application of Jensen and triangle inequalities.THEOREM 2 Suppose ，and (3.11) (3.12) (3.13)then (3.14)PROOF From Lemma 1 (3.15)Let ，we have (3.16)Where the characteristic function if and if .Obviously, in the field, . Therefore the inequality is correct. Moreover, (3.17

18、)From Lemma 2 and Theorem 1, we obtain (3.18)From the conditions of Theorem 2 (3.19)From Holder, Jensen and triangle inequalities, (3.20)For we get (3.21)From the condition of Theorem 2, we know (3.22)Synthesizing（3.15）,（3.16），（3.19）and（3.22）we know Theorem 2 is correct.When approximates to infinite, the rates are.ACKNOWLEDGEMENTSpresentation of the pap This work was supported by the Natural Science Foundation of China. The author wishes to Professor Yaoting Zhang and Associate Professor Yunxia Ma. Gratitude expressed also to the Editors and the Referees for their comments which co