The convergent rate of empirical bayes estimators for parameters of normal distribution family

This paper is a continuance of [1]. In [1] the empirical Bayes estimator of the parameter vector of normal distribution family was introduced, and for the loss function (1) its asymptotically optimal property was proved with respect to the prior distribution family (2). In this paper its convergent rate is given under stronger conditions than (2) for the prior distributions.Institute of Systems Science, Academia Sinica     

Assessing the performance of empirical bayes estimators

Methods for deriving empirical Bayes estimators are generally available. Corresponding general techniques for assessing the performance of these estimators are not widely developed yet, however. In this paper we provide a general procedure for assessing and comparing the performance of the empirical Bayes estimators and other estimators in a given data set.     

A subclass of bayes linear estimators that are minimax

In the linear regression model with ellipsoidal parameter constraints, the problem of estimating the unknown parameter vector is studied. A well-described subclass of Bayes linear estimators is proposed in the paper. It is shown that for each member of this subclass, a generalized quadratic risk function exists so that the estimator is minimax. Moreover, some of the proposed Bayes linear estimators are admissible with respect to all possible generalized quadratic risks. Also, a necessary and sufficient condition is given to ensure that the considered Bayes linear estimator improves the least squares estimator over the whole ellipsoid whatever generalized risk function is chosen.     

Some uniform convergence results for kernel estimators

This paper derives some uniform convergence rates for kernel regression of some index functions that may depend on infinite dimensional parameter. The rates of convergence are computed for independent, strongly mixing and weakly dependent data respectively. These results extend the existing literature and are useful for the derivation of large sample properties of the estimators in some semiparametric and nonparametric models.     

On the lower bounds for mean square error of empirical bayes estimators

The usual empirical Bayes setting is considered with θ being a shift or a scale parameter. A class of empirical Bayes estimators of a function b(θ) is proposed. The properties of the estimates are studied and mean square errors are calculated. The lower bounds are constructed for mean square errors of the empirical Bayes estimators over the class of all empirical Bayes estimators of b(θ). The results are applied to the case b(θ)=θ. The examples of the upper and lower bounds for mean square error are presented for the most popular families of conditional distributions. Added to the English translaion.     

The superiority of empirical bayes estimation of parameters in partitioned normal linear model

In this article,the empirical Bayes（EB）estimators are constructed for the estimable functions of the parameters in partitioned normal linear model.The superiorities of the EB estimators over ordinary least-squares（LS）estimator are investigated under mean square error matrix（MSEM）criterion.     

Some bounds on best uniform linear approximation

Numerische Mathematik -     

Pointwise and uniform convergence of kernel density estimators using random bandwidths

We obtain the rates of pointwise and uniform convergence of kernel density estimators using random bandwidths under i.i.d. as well as strongly mixing dependence assumptions. Pointwise rates are faster and not affected by the tail of the density.     

The superiority of bayes estimators in a multivariate linear model with respect to normal-inverse Wishart prior

In this paper, the multivariate linear model Y = X B +e, e ～ Nm×k(0, Im ?Σ) is considered from the Bayes perspective. Under the normal-inverse Wishart prior for(B, Σ), the Bayes estimators are derived. The superiority of the Bayes estimators of B and Σ over the least squares estimators under the criteria of Bayes mean squared error(BMSE) and Bayes mean squared error matrix(BMSEM) is shown. In addition, the Pitman Closeness(PC) criterion is also included to investigate the superiority of the Bayes estimator of B.     

The central limit theorem for maximum likelihood estimators of vector parameters: Locally uniform convergence

Weak conditions are given which insure that the convergence in the central limit theorem for maximum likelihood estimators of vector parameters takes place uniformly on compact subsets of the parameter-space.     

A general law of moment convergence rates for uniform empirical process

Let {X _n ; n ≥ 1} be a sequence of independent and identically distributed U[0,1]-distributed random variables. Define the uniform empirical process $F_n (t) = n^{ - \tfrac{1} {2}} \sum\nolimits_{i = 1}^n {(I_{\{ X_i \leqslant t\} } - t),0} \leqslant t \leqslant 1,\left\| {F_n } \right\| = \sup _{0 \leqslant t \leqslant 1} \left| {F_n (t)} \right| $F_n (t) = n^{ - \tfrac{1} {2}} \sum\nolimits_{i = 1}^n {(I_{\{ X_i \leqslant t\} } - t),0} \leqslant t \leqslant 1,\left\| {F_n } \right\| = \sup _{0 \leqslant t \leqslant 1} \left| {F_n (t)} \right| . In this paper, the exact convergence rates of a general law of weighted infinite series of E {‖F _n ‖ − ɛg ^s (n)}₊ are obtained.     

The theoretical and empirical rate of convergence for geometric branch-and-bound methods

Geometric branch-and-bound solution methods, in particular the big square small square technique and its many generalizations, are popular solution approaches for non-convex global optimization problems. Most of these approaches differ in the lower bounds they use which have been compared empirically in a few studies. The aim of this paper is to introduce a general convergence theory which allows theoretical results about the different bounds used. To this end we introduce the concept of a bounding operation and propose a new definition of the rate of convergence for geometric branch-and-bound methods. We discuss the rate of convergence for some well-known bounding operations as well as for a new general bounding operation with an arbitrary rate of convergence. This comparison is done from a theoretical point of view. The results we present are justified by some numerical experiments using the Weber problem on the plane with some negative weights.     

The convergence rates of empirical Bayes estimation in a multiple linear regression model

Empirical Bayes (EB) estimation of the parameter vector =(,²) in a multiple linear regression modelY=X+ is considered, where is the vector of regression coefficient, N(0,² I) and ² is unknown. In this paper, we have constructed the EB estimators of by using the kernel estimation of multivariate density function and its partial derivatives. Under suitable conditions it is shown that the convergence rates of the EB estimators areO(n ^{-(k-1)(k-2)/k(2k+p+1)}), where the natural numberk3, 1/3<<1, andp is the dimension of vector .The project is supported by the National Natural Science Foundation of China.     

The weak convergence of uniform measures

This paper presents a necessary and sufficient condition for the weak convergence of uniform measures on an arbitrary Hausdorff uniform space in terms of their projections in metric spaces. This result was inspired by and extends a result of Bartoszynski which characterizes the weak convergence of countably additive measures on C[0,1] in terms of their projections in finite dimensional spaces.     

The superiorities of bayes linear unbiased estimator in multivariate linear models

In this article,the Bayes linear unbiased estimator (BALUE) of parameters is derived for the multivariate linear models.The superiorities of the BALUE over the least square estimator (LSE) is studied in terms of the mean square error matrix (MSEM) criterion and Bayesian Pitman closeness (PC) criterion.     

Estimates of rate of convergence of best approximations by local functions

Questions are considered on the rate of convergence (in some abstract space of functions) of approximations that are the best in another space. Under specific conditions it is shown that the best approximations by local functions in a weighted Sobolev space W _p,B ^r yield almost-best approximation in W _q,B ^r with q[p,+).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 58, pp. 22–37, 1976.