Symmetrical Nonlinear Regression Models Research Articles

Heteroscedastic nonlinear regression models using asymmetric and heavy tailed two-piece distributions

In this paper, heteroscedastic nonlinear regression (HNLR) models under the flexible class of two–piece distributions based on the scale mixtures of normal (TP–SMN) family were examined. This novel class of nonlinear regression (NLR) models is a generalization of the well-known heteroscedastic symmetrical nonlinear regression models. The TP–SMN is a rich class of distributions that covers symmetric and asymmetric as well as heavy-tailed distributions. Using the suitable hierarchical representation of the family, the researchers first derived an EM–type algorithm for iteratively computing maximum likelihood (ML) estimates of the parameters. Then, in order to examine the performance of the proposed models and methods, some simulation studies were presented to show the robust aspect of this flexible class against outlying and also atypical data. As the last step, a natural real dataset was fitted under the proposed HNLR models.

Improved heteroskedasticity likelihood ratio tests in symmetric nonlinear regression models

In this paper we address the issue of testing inference of the dispersion parameter in heteroscedastic symmetric nonlinear regression models considering small samples. We derive Bartlett corrections to improve the likelihood ratio as well modified profile likelihood ratio tests. Our results extend some of those obtained in Cordeiro (J Stat Comput Simul 74:609–620, 2004) and Ferrari et al. (J Stat Plan Inference 124:423–437, 2004), who consider a symmetric nonlinear regression model and normal linear regression model, respectively. We also present the bootstrap and bootstrap Bartlett corrected likelihood ratio tests. Monte Carlo simulations are carried out to compare the finite sample performances of the three corrected tests and their uncorrected versions. The numerical evidence shows that the corrected modified profile likelihood ratio test, the bootstrap and bootstrap Bartlett corrected likelihood ratio test perform better than the other ones. We also present an empirical application.

Nonlinear regression models under skew scale mixtures of normal distributions

Normal nonlinear regression models are applied in some areas of the sciences and engineering to explain or describe the phenomena under study. However, it is well known that several phenomena are not always represented by the normal model due to lack of symmetry or the presence of heavy- and light-tailed distributions related to the normal law in the data. This paper proposes an extension of nonlinear regression models using the skew-scale mixtures of normal (SSMN) distributions proposed by Ferreira et al. (2011). This class of models provides a useful generalization of the symmetrical nonlinear regression models since the random term distributions cover both asymmetric and heavy-tailed distributions, such as the skew-t-normal, skew-slash and skew-contaminated normal, among others. An expectation–maximization (EM) algorithm for maximum likelihood (ML) estimates is presented and the observed information matrix is derived analytically. Some simulation studies are presented to examine the performance of the proposed methods, with relation to robustness and asymptotic properties of the ML estimates. Finally, an illustration of the method is presented considering a dataset previously analyzed under normal and skew-normal (SN) nonlinear regression models. The main conclusion is that the ML estimates from the heavy tails SSMN nonlinear models are more robust against outlying observations compared to the corresponding SN estimates.

Assessing Robustness of Inference in Symmetrical Nonlinear Regression Models

This article describes how diagnostic procedures were derived for symmetrical nonlinear regression models, continuing the work carried out by Cysneiros and Vanegas (2008) and Vanegas and Cysneiros (2010), who showed that the parameters estimates in nonlinear models are more robust with heavy-tailed than with normal errors. In this article, we focus on assessing if the robustness of this kind of models is also observed in the inference process (i.e., partial F-test). Symmetrical nonlinear regression models includes all symmetric continuous distributions for errors covering both light- and heavy-tailed distributions such as Student-t, logistic-I and -II, power exponential, generalized Student-t, generalized logistic, and contaminated normal. Firstly, a statistical test is shown to evaluating the assumption that the error terms all have equal variance. The results of simulation studies which describe the behavior of the test for heteroscedasticity proposed in the presence of outliers are then given. To assess the robustness of inference process, we present the results of a simulation study which described the behavior of partial F-test in the presence of outliers. Also, some diagnostic procedures are derived to identify influential observations on the partial F-test. As ilustration, a dataset described in Venables and Ripley (2002), is also analyzed.

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.

Estimation and diagnostics for heteroscedastic nonlinear regression models based on scale mixtures of skew-normal distributions

An extension of some standard likelihood based procedures to heteroscedastic nonlinear regression models under scale mixtures of skew-normal (SMSN) distributions is developed. This novel class of models provides a useful generalization of the heteroscedastic symmetrical nonlinear regression models (Cysneiros et al., 2010), since the random term distributions cover both symmetric as well as asymmetric and heavy-tailed distributions such as skew-t, skew-slash, skew-contaminated normal, among others. A simple EM-type algorithm for iteratively computing maximum likelihood estimates of the parameters is presented and the observed information matrix is derived analytically. In order to examine the performance of the proposed methods, some simulation studies are presented to show the robust aspect of this flexible class against outlying and influential observations and that the maximum likelihood estimates based on the EM-type algorithm do provide good asymptotic properties. Furthermore, local influence measures and the one-step approximations of the estimates in the case-deletion model are obtained. Finally, an illustration of the methodology is given considering a data set previously analyzed under the homoscedastic skew-t nonlinear regression model.

Nonlinear regression models based on scale mixtures of skew-normal distributions

An extension of some standard likelihood based procedures to nonlinear regression models under scale mixtures of skew-normal (SMSN) distributions is developed. This novel class of models provides a useful generalization of the symmetrical nonlinear regression models since the random terms distributions cover both symmetric as well as asymmetric and heavy-tailed distributions such as skew-t, skew-slash, skew-contaminated normal, among others. A simple EM-type algorithm for iteratively computing maximum likelihood estimates is presented and the observed information matrix is derived analytically. In order to examine the robust aspect of this flexible class against outlying and influential observations, some simulation studies have also been presented. Finally, an illustration of the methodology is given considering a data set previously analyzed under normal and skew-normal nonlinear regression models.

Bayesian nonlinear regression models with scale mixtures of skew-normal distributions: Estimation and case influence diagnostics

The purpose of this paper is to develop a Bayesian analysis for nonlinear regression models under scale mixtures of skew-normal distributions. This novel class of models provides a useful generalization of the symmetrical nonlinear regression models since the error distributions cover both skewness and heavy-tailed distributions such as the skew-t, skew-slash and the skew-contaminated normal distributions. The main advantage of these class of distributions is that they have a nice hierarchical representation that allows the implementation of Markov chain Monte Carlo (MCMC) methods to simulate samples from the joint posterior distribution. In order to examine the robust aspects of this flexible class, against outlying and influential observations, we present a Bayesian case deletion influence diagnostics based on the Kullback–Leibler divergence. Further, some discussions on the model selection criteria are given. The newly developed procedures are illustrated considering two simulations study, and a real data previously analyzed under normal and skew-normal nonlinear regression models.

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.

Assessment of diagnostic procedures in symmetrical nonlinear regression models

The aim of this paper is to derive diagnostic procedures based on case-deletion model for symmetrical nonlinear regression models, which complements Galea et al. (2005) that developed local influence diagnostics under some perturbation schemes. This class of models includes all symmetric continuous distributions for errors covering both light- and heavy-tailed distributions such as Student- t , logistic-I and -II, power exponential, generalized Student- t , generalized logistic and contaminated normal, among others. Thus, these models can be checked for robustness to outliers in the response variable and diagnostic methods may be a useful tool for an appropriate choice. First, an iterative process for the parameter estimation as well as some inferential results are presented. Besides, we present the results of a simulation study in which the characteristics of heavy-tailed models are evaluated in the presence of outliers. Then, we derive some diagnostic measures such as Cook distance, W–K statistic, one-step approach and likelihood displacement, generalizing results obtained for normal nonlinear regression models. Also, we present simulation studies that illustrate the behavior of diagnostic measures proposed. Finally, we consider two real data sets previously analyzed under normal nonlinear regression models. The diagnostic analysis indicates that a Student- t nonlinear regression model seems to fit the data better than the normal nonlinear regression model as well as other symmetrical nonlinear models in the sense of robustness against extreme observations.

Three Bartlett-type corrections for score statistics in symmetric nonlinear regression models

We present simple matrix formulae for corrected score statistics in symmetric nonlinear regression models. The corrected score statistics follow more closely a χ2 distribution than the classical score statistic. Our simulation results indicate that the corrected score tests display smaller size distortions than the original score test. We also compare the sizes and the powers of the corrected score tests with bootstrap-based score tests.

Residuals and their statistical properties in symmetrical nonlinear models

Abstract In this work we present theoretical details of a general residual for symmetric nonlinear regression models. This class of models includes all symmetric continuous distributions such as normal, Student- t , Pearson VII, power exponential and logistic. Such regression models are used for the analysis of data sets containing influential or outlying observations, that can significantly influence inferential conclusions. On the basis of expansions of Cox and Snell [Cox, D.R., Snell, E.J., 1968. A general definition of residuals. Journal of the Royal Statistical Society B 30, 248–275], we calculate first and second moments of a general definition of a residual for symmetrical nonlinear regression models. Also, the statistical properties of some proposed residuals are studied using Monte Carlo simulations for the Michaelis–Menten model, frequently used in chemical and biological experiments.

On diagnostics in symmetrical nonlinear models

In this paper, we discuss the extension of some diagnostic procedures to symmetrical nonlinear regression models. This class provides a useful generalization of normal nonlinear regression models since the location parameter continues to be the mean (when it exists) and the error distributions cover both light- and heavy-tailed distributions such as Student- t , logistic, power exponential, generalized Student- t , generalized logistic, contaminated normal, among others. Thus, the models can be checked for robustness and diagnostic methods may be useful tools for an appropriate choice. First, an iterative process for the parameter estimation as well as some inferential results are given. Then, we discuss the calculation of generalized leverage, local influence curvatures and standardized residuals in the symmetrical class extending results obtained for normal nonlinear models. Finally, as illustration of the proposed methods, we consider a data set previously analyzed under normal nonlinear regression models. A diagnostic analysis indicates that a Student- t nonlinear regression model with 4 degrees of freedom seems to fit the data better than the normal nonlinear regression model as well as other symmetrical nonlinear models in the sense of robustness against outlying and influential observations.

Corrected maximum-likelihood estimation in a class of symmetric nonlinear regression models

In this paper we derive general formulae for second-order biases of maximum-likelihood estimates in a class of symmetric nonlinear regression models. This class of models is commonly used for the analysis of data containing extreme or outlying observations in samples from a supposedly normal distribution. The formulae of the biases can be computed by means of an ordinary linear regression. They generalize some previous results by Cook et al., Biometrika 73 (1986) 615–623, Cordeiro and Vasconcellos, Statist. Probab. Lett. 35 (1997) 155–164 and Cordeiro et al., J. Statist. Comput. Simulation 60 (1998) 363–378. We derive simple closed-form expressions for these biases in special models. Simulation results are presented assessing the performance of the bias corrected estimates which indicate that they have smaller biases than the corresponding unadjusted estimates.