Least absolute value estimation in regression models: an annotated bibliography

This paper presents a comprehensive listing of articles on least absolute value (LAV) estimation as applied to linear and non-linear regression models and in systems of equations. References to the LAV method as applied in approximation theory are also included. Annotations describing the content of each article follow each reference.     

Corrected maximum likelihood estimators in heteroscedastic symmetric nonlinear models

In this article, we derive general matrix formulae for second-order biases of maximum likelihood estimators (MLEs) in a class of heteroscedastic symmetric nonlinear regression models, thus generalizing some results in the literature. This class of regression models includes all symmetric continuous distributions, and has a wide range of practical applications in various fields such as engineering, biology, medicine and economics, among others. The variety of distributions with different kurtosis coefficients than the normal may give more flexibility in the choice of an appropriate distribution, particularly to accommodate outlying and influential observations. We derive a joint iterative process for estimating the mean and dispersion parameters. We also present simulation studies for the biases of the MLEs.     

Optimal designs for estimating the slope of a regression

In the common linear model with quantitative predictors we consider the problem of designing experiments for estimating the slope of the expected response in a regression. We discuss locally optimal designs, where the experimenter is only interested in the slope at a particular point, and standardized minimax optimal designs, which could be used if precise estimation of the slope over a given region is required. General results on the number of support points of locally optimal designs are derived if the regression functions form a Chebyshev system. For polynomial regression and Fourier regression models of arbitrary degree the optimal designs for estimating the slope of the regression are determined explicitly for many cases of practical interest.     

Influence diagnostics in heteroscedastic and/or autoregressive nonlinear elliptical models for correlated data

In this paper, we propose nonlinear elliptical models for correlated data with heteroscedastic and/or autoregressive structures. Our aim is to extend the models proposed by Russo et al. 22 by considering a more sophisticated scale structure to deal with variations in data dispersion and/or a possible autocorrelation among measurements taken throughout the same experimental unit. Moreover, to avoid the possible influence of outlying observations or to take into account the non-normal symmetric tails of the data, we assume elliptical contours for the joint distribution of random effects and errors, which allows us to attribute different weights to the observations. We propose an iterative algorithm to obtain the maximum-likelihood estimates for the parameters and derive the local influence curvatures for some specific perturbation schemes. The motivation for this work comes from a pharmacokinetic indomethacin data set, which was analysed previously by Bocheng and Xuping 1 under normality.     

Empirical likelihood-based inference in nonlinear regression models with missing responses at random

This paper investigates the estimations of regression parameters and response mean in nonlinear regression models in the presence of missing response variables that are missing with missingness probabilities depending on covariates. We propose four empirical likelihood (EL)-based estimators for the regression parameters and the response mean. The resulting estimators are shown to be consistent and asymptotically normal under some general assumptions. To construct the confidence regions for the regression parameters as well as the response mean, we develop four EL ratio statistics, which are proven to have the χ² distribution asymptotically. Simulation studies and an artificial data set are used to illustrate the proposed methodologies. Empirical results show that the EL method behaves better than the normal approximation method and that the coverage probabilities and average lengths depend on the selection probability function.     

Goodness-of-fit tests for parametric models in censored regression

The authors propose a goodness-of-fit test for parametric regression models when the response variable is right-censored. Their test compares an estimation of the error distribution based on parametric residuals to another estimation relying on nonparametric residuals. They call on a bootstrap mechanism in order to approximate the critical values of tests based on Kolmogorov-Smirnov and Cramér-von Mises type statistics. They also present the results of Monte Carlo simulations and use data from a study about quasars to illustrate their work.     

Statistical diagnostics for nonlinear regression models based on scale mixtures of skew-normal distributions

The purpose of this paper is to develop diagnostics analysis for nonlinear regression models (NLMs) under scale mixtures of skew-normal (SMSN) distributions introduced by Garay et al. [Nonlinear regression models based on SMSN distributions. J. Korean Statist. Soc. 2011;40:115–124]. This novel class of models provides a useful generalization of the symmetrical NLM [Vanegas LH, Cysneiros FJA. Assessment of diagnostic procedures in symmetrical nonlinear regression models. Comput. Statist. Data Anal. 2010;54:1002–1016] since the random terms distributions cover both symmetric as well as asymmetric and heavy-tailed distributions such as the skew-t, skew-slash, skew-contaminated normal distributions, among others. Motivated by the results given in Garay et al. [Nonlinear regression models based on SMSN distributions. J. Korean Statist. Soc. 2011;40:115–124], we presented a score test for testing the homogeneity of the scale parameter and its properties are investigated through Monte Carlo simulations studies. Furthermore, local influence measures and the one-step approximations of the estimates in the case-deletion model are obtained. The newly developed procedures are illustrated considering a real data set.     

Quantile regression for robust inference on varying coefficient partially nonlinear models

In this paper, we propose a robust statistical inference approach for the varying coefficient partially nonlinear models based on quantile regression. A three-stage estimation procedure is developed to estimate the parameter and coefficient functions involved in the model. Under some mild regularity conditions, the asymptotic properties of the resulted estimators are established. Some simulation studies are conducted to evaluate the finite performance as well as the robustness of our proposed quantile regression method versus the well known profile least squares estimation procedure. Moreover, the Boston housing price data is given to further illustrate the application of the new method.     

Approximate tests of hypotheses in regression models with grouped data

Techniques for testing hypotheses about parameters in the regression models under the situation of grouped data are provided. A test statistic similar to conventional F statistic is considered. A simulation study performed for a few cases shows that the proposed statistic has an approximate F distribution and is useful in applications.     

Restricted minimax robust designs for misspecified regression models

The authors propose and explore new regression designs. Within a particular parametric class, these designs are minimax robust against bias caused by model misspecification while attaining reasonable levels of efficiency as well. The introduction of this restricted class of designs is motivated by a desire to avoid the mathematical and numerical intractability found in the unrestricted minimax theory. Robustness is provided against a family of model departures sufficiently broad that the minimax design measures are necessarily absolutely continuous. Examples of implementation involve approximate polynomial and second order multiple regression.     

Optimal design for linear regression models in the presence of heteroscedasticity caused by random coefficients

Random coefficients may result in heteroscedasticity of observations. For particular situations, where only one observation is available per individual, we derive optimal designs based on the geometry of the design locus.     

D-optimal designs for beta regression models with single predictor

The problem of construction of D-optimal designs for beta regression models involving one predictor is considered for the mean-precision parameterization suggested by Ferrari and Cribari-Neto [Beta regression for modelling rates and proportions. J Appl Stat. 2004;31:799–815]. Here we use the logit link function for the mean sub-model. These designs are presented and discussed for unrestricted as well as restricted design regions by considering the precision parameter as (1) a known constant and (2) an unknown constant. Efficiency comparison of obtained designs with commonly used equi-weighted, equi-spaced designs is made to recommend designs for practical use. Real-life applications are given to show the usefulness of these designs.     

Third-order inference for autocorrelation in nonlinear regression models

We propose third-order likelihood-based methods to derive highly accurate p-value approximations for testing autocorrelated disturbances in nonlinear regression models. The proposed methods are particularly accurate for small- and medium-sized samples whereas commonly used first-order methods like the signed log-likelihood ratio test, the Kobayashi (1991) test, and the standardized test can be seriously misleading in these cases. Two Monte Carlo simulations are provided to show how the proposed methods outperform the above first-order methods. An empirical example applied to US population census data is also provided to illustrate the implementation of the proposed method and its usefulness in practice.     

Nonlinear regression models for heterogeneous data with massive outliers

The income or expenditure-related data sets are often nonlinear, heteroscedastic, skewed even after the transformation, and contain numerous outliers. We propose a class of robust nonlinear models that treat outlying observations effectively without removing them. For this purpose, case-specific parameters and a related penalty are employed to detect and modify the outliers systematically. We show how the existing nonlinear models such as smoothing splines and generalized additive models can be robustified by the case-specific parameters. Next, we extend the proposed methods to the heterogeneous models by incorporating unequal weights. The details of estimating the weights are provided. Two real data sets and simulated data sets show the potential of the proposed methods when the nature of the data is nonlinear with outlying observations.     

A comparison of two exact confidence regions for partially nonlinear regression models

Exact confidence regions for all the parameters in nonlinear regression models can be obtained by comparing the lengths of projections of the error vector into orthogonal subspaces of the sample space. In certain partially nonlinear models an alternative exact region is obtained by replacing the linear parameters by their conditional estimates in the projection matrices. An ellipsoidal approximation to the alternative region is obtained in terms of the tangent-plane coordinates, similar to one previously obtained for the more usual region. This ellipsoid can be converted to an approximate region for the original parameters and can be used to compare the two types of exact confidence regions.     

A-optimal designs for heteroscedastic multifactor regression models

Abstract

This paper searches for A-optimal designs for Kronecker product and additive regression models when the errors are heteroscedastic. Sufficient conditions are given so that A-optimal designs for the multifactor models can be built from A-optimal designs for their sub-models with a single factor. The results of an efficiency study carried out to check the adequacy of the products of optimal designs for uni-factor marginal models when these are used to estimate different multi-factor models are also reported.     

Locally and maximin optimal designs for multi-factor nonlinear models

This paper considers the search for locally and maximin optimal designs for multi-factor nonlinear models from optimal designs for sub-models of a lower dimension. In particular, sufficient conditions are given so that maximin D-optimal designs for additive multi-factor nonlinear models can be built from maximin D-optimal designs for their sub-models with a single factor. Some examples of application are models involving exponential decay in several variables.     

Bayesian and maximin optimal designs for heteroscedastic regression models

The authors consider the problem of constructing standardized maximin D‐optimal designs for weighted polynomial regression models. In particular they show that by following the approach to the construction of maximin designs introduced recently by Dette, Haines & Imhof (2003), such designs can be obtained as weak limits of the corresponding Bayesian _q‐optimal designs. They further demonstrate that the results are more broadly applicable to certain families of nonlinear models. The authors examine two specific weighted polynomial models in some detail and illustrate their results by means of a weighted quadratic regression model and the Bleasdale–Nelder model. They also present a capstone example involving a generalized exponential growth model.