Post-Processing Accept-Reject Samples: Recycling and Rescaling

Abstract

This article proposes alternative methods for constructing estimators from accept-reject samples by incorporating the variables rejected by the algorithm. The resulting estimators are quick to compute, and turn out to be variations of importance sampling estimators, although their derivations are quite different. We show that these estimators are superior asymptotically to the classical accept-reject estimator, which ignores the rejected variables. In addition, we consider the issue of rescaling of estimators, a topic that has implications beyond accept-reject and importance sampling. We show how rescaling can improve an estimator and illustrate the domination of the standard importance sampling techniques in different setups.     

随机删失数据下几种风险率函数估计的渐近性质

文中对于删失数据下几种不同的风险率函数估计进行了研究。使用与以往不同的方法，在较弱的条件下，改进并扩充了现有文献的结果，获得了这几种风险率函数估计的渐近正态性，一致强弱相合收敛速度以及重对数律且进行了数值模拟。     

Alternatives to a Semi-Parametric Estimator of Parameters of Rare Events—The Jackknife Methodology*

In this paper, and in a context of regularly varying tails, we propose different alternatives to a well-known estimator of the tail index—the Hill estimator (Hill, 1975). These alternatives have essentially in mind a reduction in bias, preferably without increasing Mean Square Error, by the use of suitable Generalized Jackknife methodologies (Gray and Schucany, 1972). The first estimate obtained through this methodolgy is the one introduced by Peng (1998), under a different context. Other Generalized Jackknife estimators are linear combinations of Hill estimators at different levels. This methodology of affine combinations of Hill estimators at different levels may be easily generalized to other semi-parametric estimators of the tail index, like Pickands' estimator (Pickands, 1975) or the Moment's estimator (Dekkers et al., 1989), and consequently to a general real tail index, seeming to be a promising field of research.     

不同自变量的变系数模型的估计

讨论具有不同自变量的变系数模型的函数系数的估计及其大样本性质。使用局部线性方法和积分方法,得到函数系数的积分估计;由于该估计有较大的方差,进一步使用回切法改进这一估计,获得了函数系数的改进估计;同时,研究了改进估计的渐近正态性。最后,用模拟例子说明提出的估计方法是有效的。     

On a class of almost unbiased ratio estimators

Summary Murthy and Nanjamma [4] studied the problem of construction of almost unbiased ratio estimators for any sampling design using the technique of interpenetrating subsamples. Subsequently, Rao [7], [8] has given a general method of constructing unbiased ratio estimators by considering linear combinations of the two simple estimators based on the ratio of means and the mean of ratios. However, it is difficult to choose an optimum weight (Rao [9]) which minimizes the variance of the combined estimator since the weights are random in certain cases. In this note, we consider a different method of combining these estimators and obtain a general class of almost unbiased ratio estimators of which Murthy and Nanjamma's is a particular case and derive an optimum in this class. The case of simple random sampling where a similar class of almost unbiased ratio estimators can be developed is briefly discussed. The results are illustrated by means of simple numerical examples.     

Double Shrinkage Estimation of Common Coefficients in Two Regression Equations with Heteroscedasticity

The problem of estimating the common regression coefficients is addressed in this paper for two regression equations with possibly different error variances. The feasible generalized least squares (FGLS) estimators have been believed to be admissible within the class of unbiased estimators. It is, nevertheless, established that the FGLS estimators are inadmissible in light of minimizing the covariance matrices if the dimension of the common regression coefficients is greater than or equal to three. Double shrinkage unbiased estimators are proposed as possible candidates of improved procedures.     

Error Process Indexed by Bandwidth Matrices in Multivariate Local Linear Smoothing

We focus on nonparametric multivariate regression function estimation by locally weighted least squares. The asymptotic behavior for a sequence of error processes indexed by bandwidth matrices is derived. We discuss feasible data-driven consistent estimators minimizing asymptotic mean squared error or efficient estimators reducing asymptotic bias at points where opposite sign curvatures of the regression function are present in different directions.     

系数函数光滑度互不相同的变系数模型的一步估计法

该文提出了一种一步估计方法用以估计变系数模型中具有互不相同光滑度的未知函数, 所有未知函数和它们的导数的估计量由 一次极小化得到. 给出了估计量的渐近性质, 包括渐近偏差、方差和渐近分布, 一步估计量被证明达到了最优收敛速度.     

Efficient estimation of functional-coefficient regression models with different smoothing variables

In this article,a procedure for estimating the coefficient functions on the functional-coefficient regression models with different smoothing variables in different coefficient functions is defined.First step,by the local linear technique and the averaged method,the initial estimates of the coefficient functions are given.Second step,based on the initial estimates,the efficient estimates of the coefficient functions are proposed by a one-step back-fitting procedure.The efficient estimators share the same asymptotic normalities as the local linear estimators for the functional-coefficient models with a single smoothing variable in different functions.Two simulated examples show that the procedure is effective.     

Possibility distributions: A unified representation of usual direct-probability-based parameter estimation methods

The paper presents a possibility theory based formulation of one-parameter estimation that unifies some usual direct probability formulations. Point and confidence interval estimation are expressed in a single theoretical formulation and incorporated into estimators of a generic form: a possibility distribution. New relationships between continuous possibility distribution and probability concepts are established. The notion of specificity ordering of a possibility distribution, corresponding to fuzzy subsets inclusion, is then used for comparing the efficiency of different estimators for the case of data points coming from a symmetric probability distribution. The usefulness of the approach is illustrated on common mean and median estimators from identical independent data sample of different size and of different common symmetric continuous probability distributions.     

The efficiency of jack-knifed and usual ridge type estimators: A comparison

Ordinary and jack-knifed ridge type estimators are compared for different measures of goodness. Although jack-knifing reduces bias considerably the jack-knifed ridge estimators have larger variance and may have a larger Mean Square Error than the usual ridge estimators.     

Approximate Bayes estimators applied to the Bilal model

This paper develops approximate Bayes estimators of the parameter of the Bilal failure model by using the method of Tierney and Kadane [Accurate approximations for posterior moments and marginal densities, J. Amer. Statist. Assoc. 81 (1986) 82–86.] based on Type-2 censored sample and four different loss functions. Existence and uniqueness theorem for the maximum likelihood estimate are established. Based on Monte Carlo simulation study, comparisons are made between those estimators and their corresponding Bayes estimators obtained by using Gibb’s sampling approach.     

Optimizing in the class of Fuller modified limited information maximum likelihood estimators

A general class of Fuller modified maximum likelihood estimators are considered. It is shown that this class possesses finite moments. Asymptotic bias and asymptotic mean squared error are derived using small-σ expansions. A simulation study is carried out to compare different estimators in this class with standard estimators.     

Multivariate numerical differentiation

We present an innovative method for multivariate numerical differentiation i.e. the estimation of partial derivatives of multidimensional noisy signals. Starting from a local model of the signal consisting of a truncated Taylor expansion, we express, through adequate differential algebraic manipulations, the desired partial derivative as a function of iterated integrals of the noisy signal. Iterated integrals provide noise filtering. The presented method leads to a family of estimators for each partial derivative of any order. We present a detailed study of some structural properties given in terms of recurrence relations between elements of a same family. These properties are next used to study the performance of the estimators. We show that some differential algebraic manipulations corresponding to a particular family of estimators lead implicitly to an orthogonal projection of the desired derivative in a Jacobi polynomial basis functions, yielding an interpretation in terms of the popular least squares. This interpretation allows one to (1) explain the presence of a spatial delay inherent to the estimators and (2) derive an explicit formula for the delay. We also show how one can devise, by a proper combination of different elementary estimators of a given order derivative, an estimator giving a delay of any prescribed value. The simulation results show that delay-free estimators are sensitive to noise. Robustness with respect to noise can be highly increased by utilizing voluntary-delayed estimators. A numerical implementation scheme is given in the form of finite impulse response digital filters. The effectiveness of our derivative estimators is attested by several numerical simulations.     

Almost surely consistent nonparametric regression from recursive partitioning schemes

Presented here are results on almost sure convergence of estimators of regression functions subject to certain moment restrictions. Two somewhat different notions of almost sure convergence are studied: unconditional and conditional given a training sample. The estimators are local means derived from certain recursive partitioning schemes.     

Multivariate numerical differentiation

We present an innovative method for multivariate numerical differentiation i.e. the estimation of partial derivatives of multidimensional noisy signals. Starting from a local model of the signal consisting of a truncated Taylor expansion, we express, through adequate differential algebraic manipulations, the desired partial derivative as a function of iterated integrals of the noisy signal. Iterated integrals provide noise filtering. The presented method leads to a family of estimators for each partial derivative of any order. We present a detailed study of some structural properties given in terms of recurrence relations between elements of a same family. These properties are next used to study the performance of the estimators. We show that some differential algebraic manipulations corresponding to a particular family of estimators lead implicitly to an orthogonal projection of the desired derivative in a Jacobi polynomial basis functions, yielding an interpretation in terms of the popular least squares. This interpretation allows one to (1) explain the presence of a spatial delay inherent to the estimators and (2) derive an explicit formula for the delay. We also show how one can devise, by a proper combination of different elementary estimators of a given order derivative, an estimator giving a delay of any prescribed value. The simulation results show that delay-free estimators are sensitive to noise. Robustness with respect to noise can be highly increased by utilizing voluntary-delayed estimators. A numerical implementation scheme is given in the form of finite impulse response digital filters. The effectiveness of our derivative estimators is attested by several numerical simulations.     

Variable window width kernel estimates of probability densities

Kernel density estimators which allow different amounts of smoothing at different locations are studied. Modifications of estimators proposed by Breiman, Meisel and Purcell (1977) and Abramson (1982a), which have variable window widths, are seen to have very fast rates of convergence. These rates have traditionally been obtained using a less natural higher order kernel, which has the disadvantage of allowing an estimator which takes on negative values.Research partially supported by NSF Grants DMS-8400602 and DMS-8701201     

On optimal estimators of shift population density functions and conditions of their consistency

A new technique of optimal estimation of density functions for exponential shift families on a homogeneous space of a Lie group is proposed. In contrast to traditional methods, the approach considered is essentially based on the algebraic properties of shift families. Here we give a universal formula for consistent estimators of density functions covering different classes of estimators such as unbiased estimators with uniformly minimum variance and Bayesian estimators under two popular loss functions. The representations of some maximal invariant density functions are derived and simultaneously a close connection between the estimators and these density functions is established. Proceedings of the XVII Seminar on Stability Problems for Stochastic Models, Kazan, Russia, 1995, Part III.     

基于纵向数据的半参数变系数部分线性回归模型

纵向数据是数理统计研究中的复杂数据类型之一0,在生物、医学和经济学中具有广泛的应用．在实际中经常需要对纵向数据进行统计分析和建模．文章讨论了纵向数据下的半参数变系数部分线性回归模型,这里的纵向数据的在纵向观察在时间上可以是不均等的,也可看成是按某一随机过程来发生．所研究的半参数变系数模型包括了许多半参数模型,比如部分线性模型和变系数模型等．利用计数过程理论和局部线性回归方法,对于纵向数据下半参数变系数进行了统计推断,给出了参数分量和非参数分量的profile最小二乘估计,研究了这些估计的渐近性质,获得这些估计的相合性和渐近正态性．     

A bootstrap approach to model checking for linear models under length-biased data

In this paper, we propose two bootstrap-based model checking tests for a parametric linear model when data are affected by length-bias. These tests are based on the measure of the discrepancy between nonparametric and parametric estimators for the regression function when the data are drawn under a length-biased mechanism. We consider two different discrepancy measures: the supremum and the integral of the quadratic difference between the parametric and nonparametric estimators.