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.     

Admissibility of estimators of the probability of unobserved outcomes

The problem of estimating the probability of unobserved outcomes or, as it is sometimes called, the conditional probability of a new species, is studied. Good's estimator, which is essentially the same as Robbins' estimator, namely the number of singleton species observed divided by the sample size, is studied from a decision theory point of view. The results obtained are as follows: (1) When the total number of different species is assumed bounded by some known number, Good's and Robbins' estimators are inadmissible for squared error loss. (2) If the number of different species can be infinite, Good's and Robbins' estimators are admissible for squared error loss. (3) Whereas Robbins' estimator is a UMVUE for theunconditional probability of a new species obtained in one extra sample point, Robbins' estimator is not a uniformly minimum mean squared error unbiased estimator of the conditional probability of a new species. This answers a question raised by Robbins. (4) It is shown that for Robbins' model and squared error loss, there are admissible Bayes estimators which do not depend only on a minimal sufficient statistic. A discussion of interpretations and significance of the results is offered. Research supported by NSF Grant DMS-88-22622.     

Admissible and minimax multiparameter estimation in exponential families

Consider p independent distributions each belonging to the one parameter exponential family with distribution functions absolutely continuous with respect to Lebesgue measure. For estimating the natural parameter vector with p ≥ p₀ (p₀ is typically 2 or 3), a general class of estimators dominating the minimum variance unbiased estimator (MVUE) or an estimator which is a known constant multiple of the MVUE is produced under different weighted squared error losses. Included as special cases are some results of Hudson [13] and Berger [5]. Also, for a subfamily of the general exponential family, a class of estimators dominating the MVUE of the mean vector or an estimator which is a known constant multiple of the MVUE is produced. The major tool is to obtain a general solution to a basic differential inequality.     

Sobolev-Hermite versus Sobolev nonparametric density estimation on $${\mathbb {R}}$$

In this paper, our aim is to revisit the nonparametric estimation of a square integrable density f on \({\mathbb {R}}\), by using projection estimators on a Hermite basis. These estimators are studied from the point of view of their mean integrated squared error on \({\mathbb {R}}\). A model selection method is described and proved to perform an automatic bias variance compromise. Then, we present another collection of estimators, of deconvolution type, for which we define another model selection strategy. Although the minimax asymptotic rates of these two types of estimators are mainly equivalent, the complexity of the Hermite estimators is usually much lower than the complexity of their deconvolution (or kernel) counterparts. These results are illustrated through a small simulation study.

     

Integrated squared error of kernel-type estimator of distribution function

Let X ₁ ,...,X _n be a random sample drawn from distribution function F(x) with density function f(x) and suppose we want to estimate X(x). It is already shown that kernel estimator of F(x) is better than usual empirical distribution function in the sense of mean integrated squared error. In this paper we derive integrated squared error of kernel estimator and compare the error with that of the empirical distribution function. It is shown that the superiority of kernel estimators is not necessarily true in the sense of integrated squared error.     

Double k-Class Estimators in Regression Models with Non-spherical Disturbances

In this paper, we consider a family of feasible generalised double k-class estimators in a linear regression model with non-spherical disturbances. We derive the large sample asymptotic distribution of the proposed family of estimators and compare its performance with the feasible generalized least squares and Stein-rule estimators using the mean squared error matrix and risk under quadratic loss criteria. A Monte-Carlo experiment investigates the finite sample behaviour of the proposed family of estimators.     

<Emphasis Type="Italic">L</Emphasis><Subscript>1</Subscript> linear interpolator for missing values in time series

We propose a minimum mean absolute error linear interpolator (MMAELI), based on theL ₁ approach. A linear functional of the observed time series due to non-normal innovations is derived. The solution equation for the coefficients of this linear functional is established in terms of the innovation series. It is found that information implied in the innovation series is useful for the interpolation of missing values. The MMAELIs of the AR(1) model with innovations following mixed normal andt distributions are studied in detail. The MMAELI also approximates the minimum mean squared error linear interpolator (MMSELI) well in mean squared error but outperforms the MMSELI in mean absolute error. An application to a real series is presented. Extensions to the general ARMA model and other time series models are discussed. This research was supported by a CityU Research Grant and Natural Science Foundation of China.     

On adaptive wavelet estimation of a quadratic functional from a deconvolution problem

We observe a stochastic process where a convolution product of an unknown function and a known function is corrupted by Gaussian noise. We wish to estimate the squared \mathbbL²{\mathbb{L}^2} -norm of the unknown function from the observations. To reach this goal, we develop adaptive estimators based on wavelet and thresholding. We prove that they achieve (near) optimal rates of convergence under the mean squared error over a wide range of smoothness classes.     

一类独立随机变量均值中参数的估计

本文给出了独立随机变量均值中参数ａ＝（ａ１，…，ａｐ）′的估计量（ａ１Ｘ１，…，ａｐＸｐ）′在加权平方和损失下为可容许估计的充要条件及在一般损失Ｌｖ下为可容许估计的充分条件（在线性估计类中）。     

Estimation of the Scale Matrix and its Eigenvalues in the Wishart and the Multivariate F Distributions

In this paper, the problem of estimating the scale matrix and their eigenvalues in a Wishart distribution and in a multivariate F distribution (which arise naturally from a two-sample setting) are considered. A new class of estimators which shrink the eigenvalues towards their arithmetic mean are proposed. It is shown that the new estimator which dominates the usual unbiased estimator under the squared error loss function. A simulation study was carried out to study the performance of these estimators.     

平衡随机模型中方差分量的非负估计

众所周知, 对于平衡随机模型, 方差分量的方差分析估计为一致最小方差无偏估计. 本文基于方差分量的方差分析估计, 构造了一个二次不变估计类, 它包含了一些常用重要估计. 证明了该估计类在一定条件下在均方误差意义下一致优于方差分析估计, 并在此估计类基础上, 给出了方差分量的两种非负估计, 它们在均方误差意义下分别一致优于方差分析估计和限制极大似然估计, 且有显式解、容易计算.     

Admissible Estimation for Finite Population When the Parameter Space is Restricted

This paper considers the admissibility of the estimators for finite population when the parameter space is restricted. We obtain all admissible linear estimators of an arbitrary linear function of characteristic values of a finite population in the class of linear estimators under the criterion of the expectation of mean squared error. Received February 12, 1999, Revised October 8, 1999, Accepted January 14, 2000     

Optimal kernel estimation of densities

Precise asymptotic behavior for mean integrated squared error (MISE) is determined for sequences of kernel estimators of a density in a broad class, including discontinuous and possibly unbounded densities. The paper shows that the sequence using the kernel optimal at each fixed sample size is asymptotically more efficient than a sequence generated by changing the bandwidth of a fixed kernel shape, regardless of the kernel shape. The class of densities considered are those whose characteristic functions behave at large arguments like the product of a Fourier series and a regularly varying function. This condition may be related to the smoothness of an m-th derivative of the density.Partially supported by National Science Foundation Grant DMS-8711924.     

On the role of the shrinkage parameter in local linear smoothing

Summary. It has been shown that local linear smoothing possesses a variety of very attractive properties, not least being its mean square performance. However, such results typically refer only to asymptotic mean squared error, meaning the mean squared error of the asymptotic distribution, and in fact, the actual mean squared error is often infinite. See Seifert and Gasser (1996). This difficulty may be overcome by shrinking the local linear estimator towards another estimator with bounded mean square. However, that approach requires information about the size of the shrinkage parameter. From at least a theoretical viewpoint, very little is known about the effects of shrinkage. In particular, it is not clear how small the shrinkage parameter may be chosen without affecting first-order properties, or whether infinitely supported kernels such as the Gaussian require shrinkage in order to achieve first-order optimal performance. In the present paper we provide concise and definitive answers to such questions, in the context of general ridged and shrunken local linear estimators. We produce necessary and sufficient conditions on the size of the shrinkage parameter that ensure the traditional mean squared error formula. We show that a wide variety of infinitely-supported kernels, with tails even lighter than those of the Gaussian kernel, do not require any shrinkage at all in order to achieve traditional first-order optimal mean square performance. Received: 22 May 1995 / In revised form: 23 January 1997     

Second derivative estimation using harmonic analysis

Simulation sensitivity analysis is an important problem for simulation practitioners analyzing complex systems. The significance of this problem has resulted in the development of various gradient estimators that can be used to address this issue. Although higher derivative estimators have been discussed concurrently, less attention has been given to assess the efficiency and feasibility of computing such estimators. In this paper, two second derivative estimators are presented. The first estimators, called the HFD estimators, combine harmonic gradient estimators with finite differences second derivative estimators. The resulting hybrid estimators requireO(p) fewer simulation runs to implement compared to the straightforward finite differences approach, wherep is the number of input parameters in the simulation model. The second estimators, called the HA estimators, incorporate harmonic analysis directly, requiring one or two simulation runs to implement, depending on whether a control variate simulation run is made. Expressions for the bias and the variance of the HFD and the HA estimators (with and without variance reduction techniques) are derived. Optimal mean squared error convergence rates are also discussed. In particular, the convergence rates for both these estimators are shown to be the same, though the computational performance of the HFD estimators is better than that for the HA estimators on anM/M/1 queue simulation model. Computational results for the HFD estimators on an (s, S) inventory system simulation model are also included.     

Minimum contrast estimation of stationary processes based on the squared periodogram?

We present a class of minimum contrast estimators based on the objective function that is composed using the squared periodogram. We prove the consistency and asymptotic normality of the proposed estimators.     

On the Accuracy of Binned Kernel Density Estimators

The accuracy of the binned kernel density estimator is studied for general binning rules. We derive mean squared error results for the closeness of this estimator to both the true density and the unbinned kernel estimator. The binning rule and smoothness of the kernel function are shown to influence the accuracy of the binned kernel estimators. Our results are used to compare commonly used binning rules, and to determine the minimum grid size required to obtain a given level of accuracy.     

A class of estimators with estimated optimum values in sample surveys

Srivastava and Jhajj (1981) proposed a class of estimators for population mean of a character using auxiliary information and optimum values involving unknown parameters. From the practical point of view, their results have very little utility. In view of practical utility, we propose a class of estimators with estimated optimum values. Further, it is shown that the proposed class with estimated optimum values attains the same minimum mean square error of the class of estimators based on optimum values.     

Simultaneous estimation of parameters in exponential families

Summary The paper considers estimation of the natural parameter vector or the mean vector from independent distributions each belonging to the one-parameter discrete or absolutely continuous exponential family. The usual estimators (maximum likelihood, minimum variance unbiased or best invariant) are improved simultaneously under various weighted squared error losses. Research supported by the NSF Grant Number MCS-8202116.     

Estimating a density and its derivatives via the minimum distance method

Summary This paper deals with minimum distance (MD) estimators and minimum penalized distance (MPD) estimators which are based on the L _p distance. Rates of strong consistency of MPD density estimators are established within the family of density functions which have a bounded m-th derivative. For the case p=2, it is also proved that the MPD density estimator achieves the optimum rate of decrease of the mean integrated square error and the L ₁ error. Estimation of derivatives of the density is considered as well.In a class parametrized by entire functions, it is proved that the rate of convergence of the MD density estimator (and its derivatives) to the unknown density (its derivatives) is of order in expected L ₁ and L ₂ distances. In the same class of distributions, MD estimators of unknown density and its derivatives are proved to achieve an extraordinary rate (log log n/n)^1/2 of strong consistency.