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In this paper, we develop statistical methodology for the analysis of data under nonnormal distributions, in the context of mixed effects models. Although the multivariate normal distribution is useful in many cases, it is not appropriate, for instance, when the data come from skewed and/or heavy‐tailed distributions. To analyse data with these characteristics, in this paper, we extend the standard linear mixed effects model, considering the family of generalized hyperbolic distributions. We propose methods for statistical inference based on the likelihood function, and due to its complexity, the EM algorithm is used to find the maximum likelihood estimates with the standard errors and the exact likelihood value as a by‐product. We use simulations to investigate the asymptotic properties of the expectation‐maximization algorithm (EM) estimates and prediction accuracy. A real example is analysed, illustrating the usefulness of the proposed methods. [ABSTRACT FROM AUTHOR]

Multivariate regression models are helpful in many fields. However, independent variables (covariates or predictors) could be measured with error. That implies the necessity of considering a new kind of model called Multivariate Regression Models with Measurement Error (MRMMEs). This paper aims to carry out a statistical analysis of these models. We include estimation, hypothesis testing, model assessment, and influence diagnostics. Furthermore, besides considering the classical assumption of the normal distribution, we use maximum likelihood for the whole inference process. Finally, we study the developed approach's performance through simulation experiments and re-analyze the human lung function dataset presented in the literature to illustrate the methodology. [ABSTRACT FROM AUTHOR]

In this paper the influence measures for the Negative Binomial regression model are presented. Based on the conditional expectation of the complete-data log-likelihood function we derive some influence measures, such as case deletion (global influence) and local influence analysis. For the implementation of the influence measures we present explicit expressions and discuss an appropriate perturbation scheme. To illustrate the results, simulations and real data applications are presented. Results show that both global and local influence methods are effective in detecting possible observations that influence the parameter estimation, or at least in focusing researchers attention on those observations. [ABSTRACT FROM AUTHOR]

In this paper, we consider a regression model under the generalized Waring distribution for modeling count data. We develop and implement local influence diagnostic techniques based on likelihood displacement. Also we develop case-deletion methods. The generalized Waring regression model is presented as a mixture of the Negative Binomial and the Beta II distributions, and it is compared to the Negative Binomial and Waring regression models. Estimation is performed by maximum likelihood function. The influence measures developed in this paper are applied to a Spanish football league data set. Empirical results show that the generalized Waring regression model performs better when compared to the Negative Binomial and Waring regression models. Technical details are presented in the Appendix. [ABSTRACT FROM AUTHOR]

In this paper, we deal with inference about the structural parameters in a heteroscedastic functional measurement error models under the normal distribution assumption. Given a minimal sufficient statistic for the incidental parameters, the conditional maximum likelihood (CML) approach is used. We show that CML estimators have explicit expressions and their sampling distribution is exact. We also show that the classical test statistics to test hypotheses of interest coincide and have exact distributions. We apply the statistical inference tools developed to a data set on comparison of measurement methods. • Inference on the structural parameters in a functional measurement error model. • Minimal sufficient statistic for the incidental parameters. • We apply the developed methodology to a data set on comparison of measurement methods. [ABSTRACT FROM AUTHOR]

The standard Tobit model is constructed under the assumption of a normal distribution and has been widely applied in econometrics. Atypical/extreme data have a harmful effect on the maximum likelihood estimates of the standard Tobit model parameters. Then, we need to count with diagnostic tools to evaluate the effect of extreme data. If they are detected, we must have available a Tobit model that is robust to this type of data. The family of elliptically contoured distributions has the Laplace, logistic, normal and Student-t cases as some of its members. This family has been largely used for providing generalizations of models based on the normal distribution, with excellent practical results. In particular, because the Student-t distribution has an additional parameter, we can adjust the kurtosis of the data, providing robust estimates against extreme data. We propose a methodology based on a generalization of the standard Tobit model with errors following elliptical distributions. Diagnostics in the Tobit model with elliptical errors are developed. We derive residuals and global/local influence methods considering several perturbation schemes. This is important because different diagnostic methods can detect different atypical data. We implement the proposed methodology in an R package. We illustrate the methodology with real-world econometrical data by using the R package, which shows its potential applications. The Tobit model based on the Student-t distribution with a small quantity of degrees of freedom displays an excellent performance reducing the influence of extreme cases in the maximum likelihood estimates in the application presented. It provides new empirical evidence on the capabilities of the Student-t distribution for accommodation of atypical data. [ABSTRACT FROM AUTHOR]

In this paper, we investigate model assessment, estimation and hypothesis testing in a linear functional relationship for replicated data when the distribution of the measurement errors is a multivariate Student t distribution. For statistical inference, we adopt the unbiased estimating equations approach. The resulting estimator is consistent and asymptotically normal; a closed form expression is also given for its asymptotic covariance matrix. A simple graphical device for model checking is proposed. We also describe how to test some hypotheses of interest on the parameter vector using the Wald statistic. A simulation study is performed to gauge the performance of the estimators and of the Wald statistic. The methodology developed in the paper is illustrated with a real data set. [ABSTRACT FROM AUTHOR]

Abstract: This paper deals with the local influence assessment of the effects of minor perturbations of data on corrected score estimators in the functional heteroscedastic measurement error models with known variances. By extending to the context of measurement error models the differential-geometrical framework proposed by Zhu et al. [H.T. Zhu, J.G. Ibrahim, S. Lee, H. Zhang, Perturbation selection and influence measures in local influence analysis, The Annals of Statistics 35 (2007) 2565–2588], an -dimensional Riemannian manifold is defined. The associated metric tensor is utilized for the selection of appropriate perturbation schemes. The Levi-Civita connection and first and second derivatives of the corrected score estimators are used to construct influence measures. Simple formulas are obtained under different perturbation schemes. A comparison with the slope and curvature based diagnostics defined from the surface of the corrected score estimators formed by perturbation is included. A real data application and a simulated example illustrate the performance of the proposed diagnostics. [Copyright &y& Elsevier]

The main goal of this article is to consider influence assessment in models with error-prone observations and variances of the measurement errors changing across observations. The techniques enable to identify potential influential elements and also to quantify the effects of perturbations in these elements on some results of interest. The approach is illustrated with data from the WHO MONICA Project on cardiovascular disease. [ABSTRACT FROM AUTHOR]

In this paper we consider Sharpe's single-index model or Sharpe's model, by assuming that the returns obtained follow a multivariate t elliptical distribution. Also, given that the returns of the market are not observable, the statistical analysis was made in the context of an errors-in-variables model. In order to analyze the sensibility to possible outliers and/or atypical returns of the maximum likelihood estimators the local influence method [10] was implemented. The results are illustrated by using a set of shares of companies belonging to the Chilean Stock Market. The main conclusion is that the t model with small degrees of freedom is able to incorporate possible outliers and influential returns in the data. [ABSTRACT FROM AUTHOR]

The purpose of this article is to investigate estimation and hypothesis testing by maximum likelihood and method of moments in functional models within the class of elliptical symmetric distributions. The main results encompass consistency and asymptotic normality of the method of moments estimators. Also, the asymptotic covariance matrix of the maximum likelihood estimator is derived, extending some existing results in elliptical distributions. A measure of asymptotic relative efficiency is reported. Wald-type statistics are considered and numerical results obtained by Monte Carlo simulation to investigate the performance of estimators and tests are provided for Student-t and contaminated normal distributions. An application to a real dataset is also included. [ABSTRACT FROM AUTHOR]

In many epidemiological studies it is common to resort to regression models relating incidence of a disease and its risk factors. The main goal of this paper is to consider inference on such models with error-prone observations and variances of the measurement errors changing across observations. We suppose that the observations follow a bivariate normal distribution and the measurement errors are normally distributed. Aggregate data allow the estimation of the error variances. Maximum likelihood estimates are computed numerically via the EM algorithm. Consistent estimation of the asymptotic variance of the maximum likelihood estimators is also discussed. Test statistics are proposed for testing hypotheses of interest. Further, we implement a simple graphical device that enables an assessment of the model's goodness of fit. Results of simulations concerning the properties of the test statistics are reported. The approach is illustrated with data from the WHO MONICA Project on cardiovascular disease. Copyright © 2008 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper we consider the Capital Asset Pricing Model under Elliptical (symmetric) Distributions. This class of distributions, which contains the normal distribution, t, contaminated normal and power exponential, among others, offers a more flexible framework for modelling asset prices or returns. In order to analyze the sensibility to possible outliers and/or atypical returns of the maximum likelihood estimators, the local influence method was implemented. The results are illustrated by using a set of shares from companies who trade in the Chilean Stock Market. Our main conclusion is that symmetric distributions having heavier tails than those of the normal distribution, especially the t distribution with small degrees of freedom, show a better fit and allow the reduction of the influence of atypical returns in the maximum likelihood estimators. [ABSTRACT FROM AUTHOR]

The purpose of this article is to investigate hypothesis testing in functional comparative calibration models. Wald type statistics are considered which are asymptotically distributed according to the chi-square distribution. The statistics are based on maximum likelihood, corrected score approach, and method of moment estimators of the model parameters, which are shown to be consistent and asymptotically normally distributed. Results of analytical and simulation studies seem to indicate that the Wald statistics based on the method of moment estimators and the corrected score estimators are, as expected, less efficient than the Wald type statistic based on the maximum likelihood estimators for small n. Wald statistic based on moment estimators are simpler to compute than the other Wald statistics tests and their performance improves significantly as n increases. Comparisons with an alternative F statistics proposed in the literature are also reported. [ABSTRACT FROM AUTHOR]

In this paper, we study a class of skew-normal distributions driven by the convolution of two independent random variables: a normal and a beta distributed random variables. This problem is motivated by the numerical simulation of the oviductal egg transport in mammals, expressed as a series of microsphere instant velocities regulated by ovarian hormones including estradiol. We propose a closed form convolution formula, represented in terms of the infinite series expanded using Hermite polynomials. We also analyse the convergence of such series and perform the numerical experiments to illustrate these formulae. [ABSTRACT FROM AUTHOR]

In this paper we present various diagnostic methods for a linear regression model under a logarithmic Birnbaum-Saunders distribution for the errors, which may be applied for accelerated life testing or to compare the median lives of several populations. Some influence methods, such as the local influence, total local influence of an individual and generalized leverage are derived, analysed and discussed. We also present a connection between the local influence and generalized leverage methods. A discussion of the computation of the likelihood displacement as well as the normal curvature in the local influence method are presented. Finally, an example with real data is given for illustration. [ABSTRACT FROM AUTHOR]

The influence of observations on the parameter estimates for the simple structural errors-in-variables model with no equation error, under the Student-t distribution, is investigated using the local influence approach. The main conclusion is that the Student-t model with small degrees of freedom is able to incorporate possible outliers and influential observations in the data. The likelihood displacement approach is useful for outlier detection, especially when a masking phenomenon is present and the degrees of freedom parameter is large. The diagnostics are illustrated with two examples. [ABSTRACT FROM AUTHOR]

Data with spatial dependence are often modeled by geoestatistical tools. In spatial regression, the mean response is described using explanatory variables with georeferenced data. This modeling frequently considers Gaussianity assuming the response follows a symmetric distribution. However, when this assumption is not satisfied, it is useful to suppose distributions with the same asymmetric behavior of the data. This is the case of the Birnbaum–Saunders (BS) distribution, which has been considered in different areas and particularly in environmental sciences due to its theoretical arguments. We propose a geostatistical model based on a new approach to quantile regression considering the BS distribution. Global and local diagnostic analytics are derived for this model. The estimation of model parameters and its local influence are conducted by the maximum likelihood method. Global influence is based on the Cook distance and it is compared to local influence, in both cases to detect influential observations, whose detection and removal can modify the conclusions of a study. We illustrate the proposed methodology applying it to environmental data, which shows this situation changing the conclusions after removing potentially influential observations. A comparison with Gaussian spatial regression is conducted. [ABSTRACT FROM AUTHOR]

This article proposes a methodology for diagnostics in a logistic regression with random intercept motivated by a biological study. The methodology includes local and global influence techniques allowing us to contrast the results of both types of influence. The proposed methodology is applied to a case study with real data to show its potential. This study corresponds to the reproduction of arachnids reporting how the local and global influence of atypical observations can modify the significance of parameters, and then the biological conclusions. The model fitting is evaluated through predictive indicators. The methodology is summarized in an algorithm and a demo example is implemented in R code to facilitate its application. To evaluate the performance of the methodology, Monte Carlo simulations are conducted. [ABSTRACT FROM AUTHOR]

A method is proposed in this paper to assess the local influence of minor perturbations for the Sharpe model when the normal distribution is replaced by normal/independent (NI) distributions. The family of NI distributions is an attractive class of symmetric heavy-tailed densities that includes as special cases the normal, t-Student, slash, and the contaminated normal distributions. Since the returns of the market are not observable, the statistical analysis is carried out in the context of an errors-in-variables model. An influence analysis for detecting influential observations (atypical returns) is developed to investigate the sensitivity of the maximum likelihood estimators. Diagnostic measures are obtained based on the conditional expectation of the complete-data log-likelihood function. The results are illustrated by using a set of shares of companies traded in the Chilean stock market. [ABSTRACT FROM AUTHOR]

An introduction is presented in which the editor discusses various reports within the issue on topics including Decision theory, Sampling, Distribution theory, and Regression analysis.

An introduction is presented in which the editor discusses various reports within the issue on topics including linear mixed models, distribution theory, garch models and quality control.

To obtain regression parameter estimates in generalized estimation equation modeling, whether in longitudinal or spatially correlated data, it is necessary to specify the structure of the working correlation matrix. The regression parameter estimates can be affected by the choice of this matrix. Within spatial statistics, the correlation matrix also influences how spatial variability is modeled. Therefore, this study proposes a new method for selecting a working matrix, based on conditioning the variance-covariance matrix naive. The method performance is evaluated by an extensive simulation study, using the marginal distributions of normal, Poisson, and gamma for spatially correlated data. The correlation structure specification is based on semivariogram models, using the Wendland, Matérn, and spherical model families. The results reveal that regarding the hit rates of the true spatial correlation structure of simulated data, the proposed criterion resulted in better performance than competing criteria: quasi-likelihood under the independence model criterion QIC, correlation information criterion CIC, and the Rotnizky–Jewell criterion RJC. The application of an appropriate spatial correlation structure selection was shown using the first-semester average rainfall data of 2021 in the state of Pernambuco, Brazil. [ABSTRACT FROM AUTHOR]

In this paper we propose a linear functional model with normal random effects and elliptical errors, thus extending the standard normal models considered previously. The corrected score approach (Nakamura, 1990) is used for parameter estimation and the resulting estimators are shown to be consistent and asymptotically normal. The local influence approach (Cook, 1986) is used for assessing influence of small perturbations on the parameter estimates. A simulation study is presented illustrating the good performance of the proposed approach, including the robustness property for the heavier tail models. [ABSTRACT FROM AUTHOR]

In this paper we extend semiparametric mixed linear models with normal errors to elliptical errors in order to permit distributions with heavier and lighter tails than the normal ones. Penalized likelihood equations are applied to derive the maximum penalized likelihood estimates (MPLEs) which appear to be robust against outlying observations in the sense of the Mahalanobis distance. A reweighed iterative process based on the back-fitting method is proposed for the parameter estimation and the local influence curvatures are derived under some usual perturbation schemes to study the sensitivity of the MPLEs. Two motivating examples preliminarily analyzed under normal errors are reanalyzed considering some appropriate elliptical errors. The local influence approach is used to compare the sensitivity of the model estimates. [ABSTRACT FROM AUTHOR]

Spatial linear models have been applied in numerous fields such as agriculture, geoscience and environmental sciences, among many others. Spatial dependence structure modelling, using a geostatistical approach, is an indispensable tool to estimate the parameters that define this structure. However, this estimation may be greatly affected by the presence of atypical observations in the sampled data. The purpose of this paper is to use diagnostic techniques to assess the sensitivity of the maximum-likelihood estimators, covariance functions and linear predictor to small perturbations in the data and/or the spatial linear model assumptions. The methodology is illustrated with two real data sets. The results allowed us to conclude that the presence of atypical values in the sample data have a strong influence on thematic maps, changing the spatial dependence structure. [ABSTRACT FROM AUTHOR]

In this article, we study the behavior of the coefficient of variation (CV) of a random variable that follows a symmetric distribution in the real line. Specifically, we estimate this coefficient using the maximum-likelihood (ML) method. In addition, we provide asymptotic inference for this parameter, which allows us to contrast hypothesis and construct confidence intervals. Furthermore, we produce influence diagnostics to evaluate the sensitivity of the ML estimate of this coefficient when atypical data are present. Moreover, we illustrate the obtained results by using financial real data. Finally, we carry out a simulation study to detect the potential influence of atypical observations on the ML estimator of the CV of a symmetric distribution. The illustration and simulation demonstrate the robustness of the ML estimation of this coefficient. [ABSTRACT FROM AUTHOR]

In this work we propose Bayesian measures to quantify the influence of observations on the structural parameters of the simple measurement error model (MEM). Different influence measures, like those based on q-divergence between posterior distributions and Bayes risk, are studied to evaluate the influence. A strategy based on the perturbation function and MCMC samples is used to compute these measures. The samples from the posterior distributions are obtained by using the Metropolis-Hastings algorithm and assuming specific proper prior distributions. The results are illustrated with an application to a real example modeled with MEM in the literature. [ABSTRACT FROM AUTHOR]

The paper present an explicit expression for the density of a n-dimensional random vector with a singular Elliptical distribution. Based on this, the densities of the generalized Chi-squared and generalized t distributions are derived, examining the Pearson Type VII distribution and Kotz Type distribution (as specific Elliptical distributions). Finally, the results are applied to the study of the distribution of the residuals of an Elliptical linear model and the distribution of the t-statistic, based on a sample from an Elliptical population. [ABSTRACT FROM AUTHOR]

In the present paper, a novel spatial quantile regression model based on the Birnbaum–Saunders distribution is formulated. This distribution has been widely studied and applied in many fields. To formulate such a spatial model, a parameterization of the multivariate Birnbaum–Saunders distribution, where one of its parameters is associated with the quantile of the respective marginal distribution, is established. The model parameters are estimated by the maximum likelihood method. Finally, a data set is applied for illustrating the formulated model. [ABSTRACT FROM AUTHOR]