Comparisons of asymptotic biases and variances of m-estimators of scale under asymmetric contamination

We study robustness properties of two types of M-estimators of scale when both location and scale parameters are unknown: (i) the scale estimator arising from simultaneous M-estimation of location and scale; and (ii) its symmetrization about the sample median. The robustness criteria considered are maximal asymptotic bias and maximal asymptotic variance when the known symmetric unimodal error distribution is subject to unknown, possibly asymmetric, £-con-tamination. Influence functions and asymptotic variance functionals are derived, and computations of asymptotic biases and variances, under the normal distribution with ε-contamination at oo, are presented for the special subclass arising from Huber's Proposal 2 and its symmetrized version. Symmetrization is seen to reduce both asymptotic bias and variance. Some complementary theoretical results are obtained, and the tradeoff between asymptotic bias and variance is discussed.     

A robust approach to longitudinal data analysis

The author introduces robust techniques for estimation, inference and variable selection in the analysis of longitudinal data. She first addresses the problem of the robust estimation of the regression and nuisance parameters, for which she derives the asymptotic distribution. She uses weighted estimating equations to build robust quasi‐likelihood functions. These functions are then used to construct a class of test statistics for variable selection. She derives the limiting distribution of these tests and shows its robustness properties in terms of stability of the asymptotic level and power under contamination. An application to a real data set allows her to illustrate the benefits of a robust analysis.     

Accounting for Uncertainty in Heteroscedasticity in Nonlinear Regression

Toxicologists and pharmacologists often describe toxicity of a chemical using parameters of a nonlinear regression model. Thus estimation of parameters of a nonlinear regression model is an important problem. The estimates of the parameters and their uncertainty estimates depend upon the underlying error variance structure in the model. Typically, a priori the researcher would not know if the error variances are homoscedastic (i.e., constant across dose) or if they are heteroscedastic (i.e., the variance is a function of dose). Motivated by this concern, in this paper we introduce an estimation procedure based on preliminary test which selects an appropriate estimation procedure accounting for the underlying error variance structure. Since outliers and influential observations are common in toxicological data, the proposed methodology uses M-estimators. The asymptotic properties of the preliminary test estimator are investigated; in particular its asymptotic covariance matrix is derived. The performance of the proposed estimator is compared with several standard estimators using simulation studies. The proposed methodology is also illustrated using a data set obtained from the National Toxicology Program.     

Local composite quantile regression estimation of time-varying parameter vector for multidimensional time-inhomogeneous diffusion models

This paper is dedicated to the study of the composite quantile regression (CQR) estimations of time-varying parameter vectors for multidimensional diffusion models. Based on the local linear fitting for parameter vectors, we propose the local linear CQR estimations of the drift parameter vectors, and verify their asymptotic biases, asymptotic variances and asymptotic normality. Moreover, we discuss the asymptotic relative efficiency (ARE) of the local linear CQR estimations with respect to the local linear least-squares estimations. We obtain that the local estimations that we proposed are much more efficient than the local linear least-squares estimations. Simulation studies are constructed to show the performance of the estimations proposed.     

Asymptotic distributions of functions of a sample covariance matrix under the elliptical distribution

This paper is concerned with asymptotic distributions of functions of a sample covariance matrix under the elliptical model. Simple but useful formulae for calculating asymptotic variances and covariances of the functions are derived. Also, an asymptotic expansion formula for the expectation of a function of a sample covariance matrix is derived; it is given up to the second-order term with respect to the inverse of the sample size. Two examples are given: one of calculating the asymptotic variances and covariances of the stepdown multiple correlation coefficients, and the other of obtaining the asymptotic expansion formula for the moments of sample generalized variance.     

Semiparametric mixture of additive regression models

In this article, we propose a semiparametric mixture of additive regression models, in which the regression functions are additive and non parametric while the mixing proportions and variances are constant. Compared with the mixture of linear regression models, the proposed methodology is more flexible in modeling the non linear relationship between the response and covariate. A two-step procedure based on the spline-backfitted kernel method is derived for computation. Moreover, we establish the asymptotic normality of the resultant estimators and examine their good performance through a numerical example.     

MODELING AND ESTIMATING VARIANCES IN REGRESSION

ABSTRACT

It is well known that ignoring heteroscedasticity in regression analysis adversely affects the efficiency of estimation and renders the usual procedure for constructing prediction intervals inappropriate. In some applications, such as off-line quality control, knowledge of the variance function is also of considerable interest in its own right. Thus the modeling of variance constitutes an important part of regression analysis. A common practice in modeling variance is to assume that a certain function of the variance can be closely approximated by a function of a known parametric form. The logarithm link function is often used even if it does not fit the observed variation satisfactorily, as other alternatives may yield negative estimated variances. In this paper we propose a rich class of link functions for more flexible variance modeling which alleviates the major difficulty of negative variances. We suggest also an alternative analysis for heteroscedastic regression models that exploits the principle of “separation” discussed in Box (Signal-to-Noise Ratios, Performance Criteria and Transformation. Technometrics 1988, 30, 1–31). The proposed method does not require any distributional assumptions once an appropriate link function for modeling variance has been chosen. Unlike the analysis in Box (Signal-to-Noise Ratios, Performance Criteria and Transformation. Technometrics 1988, 30, 1–31), the estimated variances and their associated asymptotic variances are found in the original metric (although a transformation has been applied to achieve separation in a different scale), making interpretation of results considerably easier.     

Rank tests in the two-sample scale problem with unequal and unknown locations

The two-sample scale problem is studied in the case of unequal and unknown location parameters. The method proposed is based on the idea of Moses (1963) and it is distribution-free. The two samples are separated into random subgroups of the same sizek. It is proposed to choosek=4 and to apply the Wilconxon test or the Savage test to the ranges or sample variances of the subgroups. The asymptotic power functions of the tests are compared. For small and moderate sample sizes simulations are carried out. Relations to some other procedures, especially to the method of Compagnone and Denker (1996) are briefly discussed.     

A resampling method by perturbing the estimating functions for quantile regression with missing data

In this article, we propose a resampling method based on perturbing the estimating functions to compute the asymptotic variances of quantile regression estimators under missing at random condition. We prove that the conditional distributions of the resampling estimators are asymptotically equivalent to the distributions of quantile regression estimators. Our method can deal with complex situations, where the response and part of covariates are missing. Numerical results based on simulated and real data are provided under several designs.     

On two-step estimation for varying coefficient models

In this note we discuss two-step kernel estimation of varying coefficient regression models that have a common smoothing variable. The method allows one to use different bandwidths for different coefficient functions. We consider local polynomial fitting and present explicit formulas for the asymptotic biases and variances of the estimators.     

Locally most powerful two-sample rank tests for Lévy distributions

For two independent samples of independent random variables which follow a Lévy distribution, the scores for the locally most powerful rank tests for the location and scale problem are obtained. To carry the asymptotic normality of the rank statistics into practice the null means and variances are calculated. Research supported by Deutsche Forschungsgemeinschaft (DFG).     

Asymptotic Properties and Variance Estimators of the M-quantile Regression Coefficients Estimators

M-quantile regression is defined as a “quantile-like” generalization of robust regression based on influence functions. This article outlines asymptotic properties for the M-quantile regression coefficients estimators in the case of i.i.d. data with stochastic regressors, paying attention to adjustments due to the first-step scale estimation. A variance estimator of the M-quantile regression coefficients based on the sandwich approach is proposed. Empirical results show that this estimator appears to perform well under different simulated scenarios. The sandwich estimator is applied in the small area estimation context for the estimation of the mean squared error of an estimator for the small area means. The results obtained improve previous findings, especially in the case of heteroskedastic data.     

Influence functions of two families of robust estimators under proportional scatter matrices

In this paper, under a proportional model, two families of robust estimates for the proportionality constants, the common principal axes and their size are discussed. The first approach is obtained by plugging robust scatter matrices on the maximum likelihood equations for normal data. A projection- pursuit and a modified projection-pursuit approach, adapted to the proportional setting, are also considered. For all families of estimates, partial influence functions are obtained and asymptotic variances are derived from them. The performance of the estimates is compared through a Monte Carlo study.     

Complete ranking procedures with appropriate loss functions

This paper treats the problem of comparing different evaluations of procedures which rank the variances of k normal populations. Procedures are evaluated on the basis of appropriate loss functions for a particular goal. The goal considered involves ranking the variances of k independent normal populations when the corresponding population means are unknown. The variances are ranked by selecting samples of size n from each population and using the sample variances to obtain the ranking. Our results extend those of various authors who looked at the narrower problem of evaluating the standard proceduv 3 associated with selecting the smallest of the population variances (see e.g.,P. Somerville (1975)).

Different loss functions (both parametric and non-parametric) appropriate to the particular goal under consideration are proposed. Procedures are evaluated by the performance of their risk over a particular preference zone. The sample size n, the least favorable parametric configuration, and the maximum value of the risk are three quantities studied for each procedure. When k is small these quantities, calculated by numerical simulation, show which loss functions respond better and which respond worse to increases in sample size. Loss functions are compared with one another according to the extent of this response. Theoretical results are given for the case of asymptotically large k. It is shown that for certain cases the error incurred by using these asymptotic results is small when k is only moderately large.

This work is an outgrowth of and extends that of J. Reeves and M.J. Sobel (1987) by comparing procedures on the basis of the sample size (perpopulation) required to obtain various bounds on the associated risk functions. New methodologies are developed to evaluate complete ranking procedures in different settings.     

Statistical inference based on generalized Lindley record values

This paper addresses the problems of frequentist and Bayesian estimation for the unknown parameters of generalized Lindley distribution based on lower record values. We first derive the exact explicit expressions for the single and product moments of lower record values, and then use these results to compute the means, variances and covariance between two lower record values. We next obtain the maximum likelihood estimators and associated asymptotic confidence intervals. Furthermore, we obtain Bayes estimators under the assumption of gamma priors on both the shape and the scale parameters of the generalized Lindley distribution, and associated the highest posterior density interval estimates. The Bayesian estimation is studied with respect to both symmetric (squared error) and asymmetric (linear-exponential (LINEX)) loss functions. Finally, we compute Bayesian predictive estimates and predictive interval estimates for the future record values. To illustrate the findings, one real data set is analyzed, and Monte Carlo simulations are performed to compare the performances of the proposed methods of estimation and prediction.     

Simultaneous Confidence Bands and Hypothesis Testing in Varying-coefficient Models

Regression analysis is one of the most commonly used techniques in statistics. When the dimension of independent variables is high, it is difficult to conduct efficient non-parametric analysis straightforwardly from the data. As an important alternative to the additive and other non-parametric models, varying-coefficient models can reduce the modelling bias and avoid the "curse of dimensionality" significantly. In addition, the coefficient functions can easily be estimated via a simple local regression. Based on local polynomial techniques, we provide the asymptotic distribution for the maximum of the normalized deviations of the estimated coefficient functions away from the true coefficient functions. Using this result and the pre-asymptotic substitution idea for estimating biases and variances, simultaneous confidence bands for the underlying coefficient functions are constructed. An important question in the varying coefficient models is whether an estimated coefficient function is statistically significantly different from zero or a constant. Based on newly derived asymptotic theory, a formal procedure is proposed for testing whether a particular parametric form fits a given data set. Simulated and real-data examples are used to illustrate our techniques.     

An Extension of the Normal/Independent Family for Studies of Robust Estimators

The family of normal scale mixture distributions, also called the Normal/Independent family, has been used for efficient Monte Carlo studies of robust estimators. The distributions in this family are unimodal. The Normal/Independent family is extended by introducing a location mixing in addition to the scale mixing. Distributions in this extension may be nonunimodal. The asymptotic variances of robust estimators of location are compared using the distributions from the extension. A Monte Carlo swindle similar to the one used in the Princeton study is given for the extended family. A small simulation study demonstrates the efficiency of the swindle. The swindle is compared with other swindle methods based on Fisher's score function and regression.     

ASYMPTOTIC DISTRIBUTIONS OF SEMIPARAMETRIC MAXIMUM LIKELIHOOD ESTIMATORS WITH ESTIMATING EQUATIONS FOR GROUP-CENSORED DATA

Semiparametric maximum likelihood estimation with estimating equations (SMLE) is more flexible than traditional methods; it has fewer restrictions on distributions and regression models. The required information about distribution and regression structures is incorporated in estimating equations of the SMLE to improve the estimation quality of non‐parametric methods. The likelihood of SMLE for censored data involves complicated implicit functions without closed‐form expressions, and the first derivatives of the log‐profile‐likelihood cannot be expressed as summations of independent and identically distributed random variables; it is challenging to derive asymptotic properties of the SMLE for censored data. For group‐censored data, the paper shows that all the implicit functions are well defined and obtains the asymptotic distributions of the SMLE for model parameters and lifetime distributions. With several examples the paper compares the SMLE, the regular non‐parametric likelihood estimation method and the parametric MLEs in terms of their asymptotic efficiencies, and illustrates application of SMLE. Various asymptotic distributions of the likelihood ratio statistics are derived for testing the adequacy of estimating equations and a partial set of parameters equal to some known values.     

Estimation for Location-Scale Models with Censored Data

This article considers a class of estimators for the location and scale parameters in the location-scale model based on ‘synthetic data’ when the observations are randomly censored on the right. The asymptotic normality of the estimators is established using counting process and martingale techniques when the censoring distribution is known and unknown, respectively. In the case when the censoring distribution is known, we show that the asymptotic variances of this class of estimators depend on the data transformation and have a lower bound which is not achievable by this class of estimators. However, in the case that the censoring distribution is unknown and estimated by the Kaplan–Meier estimator, this class of estimators has the same asymptotic variance and attains the lower bound for variance for the case of known censoring distribution. This is different from censored regression analysis, where asymptotic variances depend on the data transformation. Our method has three valuable advantages over the method of maximum likelihood estimation. First, our estimators are available in a closed form and do not require an iterative algorithm. Second, simulation studies show that our estimators being moment-based are comparable to maximum likelihood estimators and outperform them when sample size is small and censoring rate is high. Third, our estimators are more robust to model misspecification than maximum likelihood estimators. Therefore, our method can serve as a competitive alternative to the method of maximum likelihood in estimation for location-scale models with censored data. A numerical example is presented to illustrate the proposed method.     

LOGISTIC REGRESSION COMPARED TO NORMAL DISCRIMINATION FOR NON-NORMAL POPULATIONS'

An investigation is undertaken of the logistic regression procedure for estimating the posterior probability of an object belonging to one of two populations. The asymptotic bias and mean square error associated with the procedure are derived for univariate populations whose distributions satisfy the general Day-Kerridge model for which the logistic form is valid for the posterior probability. These properties are compared with those of the normal discrimination method based on the classical assumption of normal populations with common variances. The asymptotic relative efficiency of logistic regression is considered on the basis of asymptotic mean square error.