Statistical analysis of proficiency testing results under elliptical distributions

The use of inter-laboratory test comparisons to determine the performance of individual laboratories for specific tests (or for calibration) [ISO/IEC Guide 43-1, 1997. Proficiency testing by interlaboratory comparisons - Part 1: Development and operation of proficiency testing schemes] is called Proficiency Testing (PT). In this paper we propose the use of the generalized likelihood ratio test to compare the performance of the group of laboratories for specific tests relative to the assigned value and illustrate the procedure considering an actual data from the PT program in the area of volume. The proposed test extends the test criteria in use allowing to test for the consistency of the group of laboratories. Moreover, the class of elliptical distributions are considered for the obtained measurements. (C) 2008 Elsevier B.V. All rights reserved.

Influence diagnostics in nonlinear mixed-effects elliptical models

In this work we propose and analyze nonlinear elliptical models for longitudinal data, which represent an alternative to gaussian models in the cases of heavy tails, for instance. The elliptical distributions may help to control the influence of the observations in the parameter estimates by naturally attributing different weights for each case. We consider random effects to introduce the within-group correlation and work with the marginal model without requiring numerical integration. An iterative algorithm to obtain maximum likelihood estimates for the parameters is presented, as well as diagnostic results based on residual distances and local influence [Cook, D., 1986. Assessment of local influence. journal of the Royal Statistical Society - Series B 48 (2), 133-169; Cook D., 1987. Influence assessment. journal of Applied Statistics 14 (2),117-131; Escobar, L.A., Meeker, W.Q., 1992, Assessing influence in regression analysis with censored data, Biometrics 48, 507-528]. As numerical illustration, we apply the obtained results to a kinetics longitudinal data set presented in [Vonesh, E.F., Carter, R.L., 1992. Mixed-effects nonlinear regression for unbalanced repeated measures. Biometrics 48, 1-17], which was analyzed under the assumption of normality. (C) 2009 Elsevier B.V. All rights reserved.

A modified signed likelihood ratio test in elliptical structural models

In this paper we deal with the issue of performing accurate testing inference on a scalar parameter of interest in structural errors-in-variables models. The error terms are allowed to follow a multivariate distribution in the class of the elliptical distributions, which has the multivariate normal distribution as special case. We derive a modified signed likelihood ratio statistic that follows a standard normal distribution with a high degree of accuracy. Our Monte Carlo results show that the modified test is much less size distorted than its unmodified counterpart. An application is presented.

Assessment of variance components in nonlinear mixed-effects elliptical models

The issue of assessing variance components is essential in deciding on the inclusion of random effects in the context of mixed models. In this work we discuss this problem by supposing nonlinear elliptical models for correlated data by using the score-type test proposed in Silvapulle and Silvapulle (1995). Being asymptotically equivalent to the likelihood ratio test and only requiring the estimation under the null hypothesis, this test provides a fairly easy computable alternative for assessing one-sided hypotheses in the context of the marginal model. Taking into account the possible non-normal distribution, we assume that the joint distribution of the response variable and the random effects lies in the elliptical class, which includes light-tailed and heavy-tailed distributions such as Student-t, power exponential, logistic, generalized Student-t, generalized logistic, contaminated normal, and the normal itself, among others. We compare the sensitivity of the score-type test under normal, Student-t and power exponential models for the kinetics data set discussed in Vonesh and Carter (1992) and fitted using the model presented in Russo et al. (2009). Also, a simulation study is performed to analyze the consequences of the kurtosis misspecification.

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.

Influence diagnostics for elliptical semiparametric mixed models

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.

Random number generators for the generalized Birnbaum-Saunders distribution

The generalized Birnbaum-Saunders distribution pertains to a class of lifetime models including both lighter and heavier tailed distributions. This model adapts well to lifetime data, even when outliers exist, and has other good theoretical properties and application perspectives. However, statistical inference tools may not exist in closed form for this model. Hence, simulation and numerical studies are needed, which require a random number generator. Three different ways to generate observations from this model are considered here. These generators are compared by utilizing a goodness-of-fit procedure as well as their effectiveness in predicting the true parameter values by using Monte Carlo simulations. This goodness-of-fit procedure may also be used as an estimation method. The quality of this estimation method is studied here. Finally, through a real data set, the generalized and classical Birnbaum-Saunders models are compared by using this estimation method.

An Asymmetric Extension for the Family of Elliptical Slash Distributions

In this article, we introduce an asymmetric extension to the univariate slash-elliptical family of distributions studied in Gomez et al. (2007a). This new family results from a scale mixture between the epsilon-skew-symmetric family of distributions and the uniform distribution. A general expression is presented for the density with special cases such as the normal, Cauchy, Student-t, and Pearson type II distributions. Some special properties and moments are also investigated. Results of two real data sets applications are also reported, illustrating the fact that the family introduced can be useful in practice.

Bayesian estimation of regression parameters in elliptical measurement error models

The main object of this paper is to discuss the Bayes estimation of the regression coefficients in the elliptically distributed simple regression model with measurement errors. The posterior distribution for the line parameters is obtained in a closed form, considering the following: the ratio of the error variances is known, informative prior distribution for the error variance, and non-informative prior distributions for the regression coefficients and for the incidental parameters. We proved that the posterior distribution of the regression coefficients has at most two real modes. Situations with a single mode are more likely than those with two modes, especially in large samples. The precision of the modal estimators is studied by deriving the Hessian matrix, which although complicated can be computed numerically. The posterior mean is estimated by using the Gibbs sampling algorithm and approximations by normal distributions. The results are applied to a real data set and connections with results in the literature are reported. (C) 2011 Elsevier B.V. All rights reserved.

Influence diagnostics for linear models with first-order autoregressive elliptical errors

We introduce in this paper the class of linear models with first-order autoregressive elliptical errors. The score functions and the Fisher information matrices are derived for the parameters of interest and an iterative process is proposed for the parameter estimation. Some robustness aspects of the maximum likelihood estimates are discussed. The normal curvatures of local influence are also derived for some usual perturbation schemes whereas diagnostic graphics to assess the sensitivity of the maximum likelihood estimates are proposed. The methodology is applied to analyse the daily log excess return on the Microsoft whose empirical distributions appear to have AR(1) and heavy-tailed errors. (C) 2008 Elsevier B.V. All rights reserved.