Risk and Pitman closeness properties of feasible generalized double k-class estimators in linear regression models with non-spherical disturbances under balanced loss function

In this article, a family of feasible generalized double k-class estimator in a linear regression model with non-spherical disturbances is considered. The performance of this estimator is judged with feasible generalized least-squares and feasible generalized Stein-rule estimators under balanced loss function using the criteria of quadratic risk and general Pitman closeness. A Monte-Carlo study investigates the finite sample properties of several estimators arising from the family of feasible double k-class estimators.     

Robust statistical inference based on the C-divergence family

This paper describes a family of divergences, named herein as the C-divergence family, which is a generalized version of the power divergence family and also includes the density power divergence family as a particular member of this class. We explore the connection of this family with other divergence families and establish several characteristics of the corresponding minimum distance estimator including its asymptotic distribution under both discrete and continuous models; we also explore the use of the C-divergence family in parametric tests of hypothesis. We study the influence function of these minimum distance estimators, in both the first and second order, and indicate the possible limitations of the first-order influence function in this case. We also briefly study the breakdown results of the corresponding estimators. Some simulation results and real data examples demonstrate the small sample efficiency and robustness properties of the estimators.

     

Weighted L quantile distance estimators for randomly censored data

The asymptotic properties of a family of minimum quantile distance estimators for randomly censored data sets are considered. These procedures produce an estimator of the parameter vector that minimizes a weighted L² distance measure between the Kaplan-Meier quantile function and an assumed parametric family of quantile functions. Regularity conditions are provided which insure that these estimators are consistent and asymptotically normal. An optimal weight function is derived for single parameter families, which, for location/scale families, results in censored sample analogs of estimators such as those suggested by Parzen.     

Statistical estimation in partial linear models with covariate data missing at random

In this paper, we consider the partial linear model with the covariables missing at random. A model calibration approach and a weighting approach are developed to define the estimators of the parametric and nonparametric parts in the partial linear model, respectively. It is shown that the estimators for the parametric part are asymptotically normal and the estimators of g(·) converge to g(·) with an optimal convergent rate. Also, a comparison between the proposed estimators and the complete case estimator is made. A simulation study is conducted to compare the finite sample behaviors of these estimators based on bias and standard error.     

Efficient computation of generalized median estimators

The generalized median (GM) estimator is a family of robust estimators that balances the competing demands of statistical efficiency and robustness. By choosing a kernel that is efficient for the parameter, the GM estimator gains robustness by computing the median of the kernel evaluated at all possible subsets from the sample. The GM estimator is often computationally infeasible because the number of subsets can be large for even modest sample sizes. Writing the estimator in terms of the quantile function facilitates an approximation using a sample of all possible subsets. While both sampling with and without replacement are feasible, sampling without replacement is preferred because of the reduction in variance from the sampling fraction. The proposed algorithm uses sequential sampling to compute an approximation within a user-chosen margin of error.     

One-step jackknife for M-estimators computed using Newton's method

To estimate the dispersion of an M-estimator computed using Newton's iterative method, the jackknife method usually requires to repeat the iterative process n times, where n is the sample size. To simplify the computation, one-step jackknife estimators, which require no iteration, are proposed in this paper. Asymptotic properties of the one-step jackknife estimators are obtained under some regularity conditions in the i.i.d. case and in a linear or nonlinear model. All the one-step jackknife estimators are shown to be asymptotically equivalent and they are also asymptotically equivalent to the original jackknife estimator. Hence one may use a dispersion estimator whose computation is the simplest. Finite sample properties of several one-step jackknife estimators are examined in a simulation study.The research was supported by Natural Sciences and Engineering Research Council of Canada.     

A Family of Estimators of Finite-Population Distribution Function Using Auxiliary Information

This paper considers the problem of estimating the finite-population distribution function and quantiles with the use of auxiliary information at the estimation stage of a survey. We propose the families of estimators of the distribution function of the study variate y using the knowledge of the distribution function of the auxiliary variate x. In addition to ratio, product and difference type estimators, many other estimators are identified as members of the proposed families. For these families the approximate variances are derived, and in addition, the optimum estimator is identified along with its approximate variance. Estimators based on the estimated optimum values of the unknown parameters used to minimize the variance are also given with their properties. Further, the family of estimators of a finite-population distribution function using two-phase sampling is given, and its properties are investigated.      

Minimum f-divergence estimators and quasi-likelihood functions

Maximum quasi-likelihood estimators have several nice asymptotic properties. We show that, in many situations, a family of estimators, called the minimum f-divergence estimators, can be defined such that each estimator has the same asymptotic properties as the maximum quasi-likelihood estimator. The family of minimum f-divergence estimators include the maximum quasi-likelihood estimators as a special case. When a quasi-likelihood is the log likelihood from some exponential family, Amari's dual geometries can be used to study the maximum likelihood estimator. A dual geometric structure can also be defined for more general quasi-likelihood functions as well as for the larger family of minimum f-divergence estimators. The relationship between the f-divergence and the quasi-likelihood function and the relationship between the f-divergence and the power divergence is discussed.This work was supported by National Science Foundation grant DMS 88-03584.     

A Class of Robust Principal Component Vectors

This paper is concerned with a study of robust estimation in principal component analysis. A class of robust estimators which are characterized as eigenvectors of weighted sample covariance matrices is proposed, where the weight functions recursively depend on the eigenvectors themselves. Also, a feasible algorithm based on iterative reweighting of the covariance matrices is suggested for obtaining these estimators in practice. Statistical properties of the proposed estimators are investigated in terms of sensitivity to outliers and relative efficiency via their influence functions, which are derived with the help of Stein's lemma. We give a simple condition on the weight functions which ensures robustness of the estimators. The class includes, as a typical example, a method by the self-organizing rule in the neural computation. A numerical experiment is conducted to confirm a rapid convergence of the suggested algorithm.     

The change-of-variance function for dependent data

Summary The infinitesimal stability of the asymptotic variance is considered forM-estimators of a location parameter when the nominal sample with i.i.d. data is contaminated by a possibly dependent process. It is shown that the resulting change-of-variance function can be expressed as a sum of two terms, one corresponding contamination of the univariate distribution, and one to contamination of the bivariate distributions. A change-of-variance sensitivity is introduced, the form of which is closely related to the average patch length of the outliers. Finally, optimalV-robust and mostV-robust score functions are derived. The resulting family of estimators is the same as for independent data in the general case, but the truncation point approaches zero when dependency is accounted for. For redescending score-functions, the family of estimators is changed.This paper was written under support by the Swedish Board for Technical Development, contract 712-89-1073     

<Emphasis Type="Italic">K</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>-nearest neighbor estimators of entropy

For estimating the entropy of an absolutely continuous multivariate distribution, we propose nonparametric estimators based on the Euclidean distances between the n sample points and their k _n -nearest neighbors, where {k _n : n = 1, 2, …} is a sequence of positive integers varying with n. The proposed estimators are shown to be asymptotically unbiased and consistent.      

A New Class of Minimum Power Divergence Estimators with Applications to Cancer Surveillance

The annual percent change (APC) has been adopted as a useful measure for analyzing the changing trends of cancer mortality and incidence rates by the NCI SEER program. Difficulties, however, arise when comparing the sample APCs between two overlapping regions because of induced dependence (e.g., comparing the cancer mortality change rate of California with that of the national level). This paper deals with a new perspective for understanding the sample distribution of the test-statistics for comparing the APCs between overlapping regions. Our proposal allows for computational readiness and easy interpretability. We further propose a more general family of estimators, namely, the so-called minimum power divergence estimators, including the maximum likelihood estimators as a special case. Our simulation experiments support the superiority of the proposed estimator to the conventional maximum likelihood estimator. The proposed method is illustrated by the analysis of the SEER cancer mortality rates observed from 1991 to 2006.     

Two-stage local M-estimation of additive models

This paper studies local M-estimation of the nonparametric components of additive models.A two-stage local M-estimation procedure is proposed for estimating the additive components and their derivatives.Under very mild conditions,the proposed estimators of each additive component and its derivative are jointly asymptotically normal and share the same asymptotic distributions as they would be if the other components were known.The established asymptotic results also hold for two particular local M-estimations:the local least squares and least absolute deviation estimations.However,for general two-stage local M-estimation with continuous and nonlinear ψ-functions,its implementation is time-consuming.To reduce the computational burden,one-step approximations to the two-stage local M-estimators are developed.The one-step estimators are shown to achieve the same effciency as the fully iterative two-stage local M-estimators,which makes the two-stage local M-estimation more feasible in practice.The proposed estimators inherit the advantages and at the same time overcome the disadvantages of the local least-squares based smoothers.In addition,the practical implementation of the proposed estimation is considered in details.Simulations demonstrate the merits of the two-stage local M-estimation,and a real example illustrates the performance of the methodology.     

Smoothed perturbation analysis for queues with finite buffers

Applying the technique of smoothed perturbation analysis (SPA) to theGI/G/1/K queue, we derive gradient estimators for two performance measures: the mean steady-state system time of a served customer and the probability that an arriving customer is rejected. Unbiasedness of the estimators follows from results of a previous general framework on SPA estimators. However, in that framework, the estimators often require the simulation of numerous additional sample subpaths, possibly making the technique practically infeasible in applications. We exploit some of the special structure of theGI/G/1/K queue to come up with an estimator which requires at most the simulation of a single additional sample subpath. By establishing certain regenerative properties, we provide a strong consistency proof for the estimator.     

Weighted L2 quantile distance estimators for randomly censored data

The asymptotic properties of a family of minimum quantile distance estimators for randomly censored data sets are considered. These procedures produce an estimator of the parameter vector that minimizes a weighted L² distance measure between the Kaplan-Meier quantile function and an assumed parametric family of quantile functions. Regularity conditions are provided which insure that these estimators are consistent and asymptotically normal. An optimal weight function is derived for single parameter families, which, for location/scale families, results in censored sample analogs of estimators such as those suggested by Parzen.     

Folded overlapping variance estimators for simulation

We propose and analyze a new class of estimators for the variance parameter of a steady-state simulation output process. The new estimators are computed by averaging individual estimators from “folded” standardized time series based on overlapping batches composed of consecutive observations. The folding transformation on each batch can be applied more than once to produce an entire set of estimators. We establish the limiting distributions of the proposed estimators as the sample size tends to infinity while the ratio of the sample size to the batch size remains constant. We give analytical and Monte Carlo results showing that, compared to their counterparts computed from nonoverlapping batches, the new estimators have roughly the same bias but smaller variance. In addition, these estimators can be computed with order-of-sample-size work.     

Optimally thresholded realized power variations for Lévy jump diffusion models

Thresholded Realized Power Variations (TPVs) are one of the most popular nonparametric estimators for general continuous-time processes with a wide range of applications. In spite of their popularity, a common drawback lies in the necessity of choosing a suitable threshold for the estimator, an issue which so far has mostly been addressed by heuristic selection methods. To address this important issue, we propose an objective selection method based on desirable optimality properties of the estimators. Concretely, we develop a well-posed optimization problem which, for a fixed sample size and time horizon, selects a threshold that minimizes the expected total number of jump misclassifications committed by the thresholding mechanism associated with these estimators. We analytically solve the optimization problem under mild regularity conditions on the density of the underlying jump distribution, allowing us to provide an explicit infill asymptotic characterization of the resulting optimal thresholding sequence at a fixed time horizon. The leading term of the optimal threshold sequence is shown to be proportional to Lévy’s modulus of continuity of the underlying Brownian motion, hence theoretically justifying and sharpening selection methods previously proposed in the literature based on power functions or multiple testing procedures. Furthermore, building on the aforementioned asymptotic characterization, we develop an estimation algorithm, which allows for a feasible implementation of the newfound optimal sequence. Simulations demonstrate the improved finite sample performance offered by optimal TPV estimators in comparison to other popular non-optimal alternatives.     

Nonparametric density estimation for linear processes with infinite variance

We consider nonparametric estimation of marginal density functions of linear processes by using kernel density estimators. We assume that the innovation processes are i.i.d. and have infinite-variance. We present the asymptotic distributions of the kernel density estimators with the order of bandwidths fixed as h =  cn ^−1/5, where n is the sample size. The asymptotic distributions depend on both the coefficients of linear processes and the tail behavior of the innovations. In some cases, the kernel estimators have the same asymptotic distributions as for i.i.d. observations. In other cases, the normalized kernel density estimators converge in distribution to stable distributions. A simulation study is also carried out to examine small sample properties.     

Probability density estimation with surrogate data and validation sample

The probability density estimation problem with surrogate data and validation sample is considered. A regression calibration kernel density estimator is defined to incorporate the information contained in both surrogate variates and validation sample. Also, we define two weighted estimators which have less asymptotic variances but have bigger biases than the regression calibration kernel density estimator. All the proposed estimators are proved to be asymptotically normal. And the asymptotic representations for the mean squared error and mean integrated square error of the proposed estimators are established, respectively. A simulation study is conducted to compare the finite sample behaviors of the proposed estimators.     

Ordinary and penalized minimum power-divergence estimators in two-way contingency tables

Basu and Basu (Statistica Sinica 8:841–860, 1998) have proposed an empty cell penalty for the minimum power-divergence estimators which can lead to improvements in the small sample properties of these estimators. In this paper, we study the small and moderate sample performances of the ordinary and penalized minimum power-divergence estimators in terms of efficiency and robustness for the log-linear models in two-way contingency tables under the assumptions of multinomial sampling. Calculations made by enumerating all possible sample combinations show that the penalized estimators are competitive with the ordinary estimators for the moderate samples and definitely better for the smallest sample considered for both efficiency and robustness under the considered models. The results also reveal that the bigger the main effects the more need for penalization.