Audrey H.M.A. Cysneiros

On Birnbaum-Saunders inference

The Birnbaum-Saunders distribution, also known as the fatigue-life distribution, is frequently used in reliability studies. We obtain adjustments to the Birnbaum-Saunders profile likelihood function. The modified versions of the likelihood function were obtained for both the shape and scale parameters, i.e., we take the shape parameter to be of interest and the scale parameter to be of nuisance, and then consider the situation in which the interest lies in performing inference on the scale parameter with the shape parameter entering the modeling in nuisance fashion. Modified profile maximum likelihood estimators are obtained by maximizing the corresponding adjusted likelihood functions. We present numerical evidence on the finite sample behavior of the different estimators and associated likelihood ratio tests. The results favor the adjusted estimators and tests we propose. A novel aspect of the profile likelihood adjustments obtained in this paper is that they yield improved point estimators and tests. The two profile likelihood adjustments work well when inference is made on the shape parameter, and one of them displays superior behavior when it comes to performing hypothesis testing inference on the scale parameter. Two empirical applications are briefly presented.

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.

Improved Birnbaum-Saunders inference under type II censoring

The Birnbaum-Saunders distribution is useful for modeling reliability data. In this paper we obtain adjusted profile maximum likelihood estimators for the Birnbaum-Saunders distribution shape parameter under type II data censoring. We consider the adjustments to the profile likelihood function proposed by Barndorff-Nielsen (1983), Cox and Reid (1987), Fraser and Reid (1995), Fraser et?al. (1999) and Severini (1998, 1999). We obtain modified profile likelihood ratio tests and also consider bootstrap-based inference. Interval estimation is addressed as well. Several bootstrap confidence intervals are considered. Monte Carlo simulation results on point estimation, interval estimation and hypothesis testing inference are reported. Finally, we present two applications that use real (not simulated) data.

ON IMPROVING THE χ2 APPROXIMATION OF SCORE TESTS IN LOCATION-SCALE NONLINEAR MODELS

ABSTRACT This paper considers the issue of testing parameters based on score tests in location-scale nonlinear models assuming known scale parameter, which encompasses the elliptical family of distributions and also asymmetric distributions such as the extreme value distributions. Significance levels derived from the score statistic can be misleading, particularly in small samples. We obtain, in matrix notation, a Bartlett-type correction formula to improve score tests in this class of models, thus generalizing results by Ferrari and Cordeiro (Ferrari, S.L.P.; Cordeiro, G.M. Corrected Score Tests for Exponential Family Nonlinear Models. Statist. Probab. Lett. 1996, 26, 7–12) and Ferrari and Arellano-Valle (Ferrari, S.L.P.; Arellano-Valle, R.B. Modified Likelihood Ratio and Score Tests in Linear Regression Models Using the t Distribution. Braz. J. Probab. Statist. 1996, 10, 15–33.). Our results are used to obtain a corrected score statistic for testing that a subset of the nonlinear regression coefficients equals a given vector of constants. The corrected score statistic is distributed as chi-squared with an error of order , n being the sample size, whereas the original score statistic has a chi-squared distribution with error of order . We show that the formulae derived for the Bartlett-type corrections generalize a number of previously published results. We present simulation results comparing the sizes of the usual score tests and their modified versions for linear and nonlinear regression models when the scale parameter is known or it is replaced by a consistent estimate. The paper also provides a numerical comparison of the sizes of analytical corrections for score and likelihood ratio tests and bootstrap tests.

An improved test for heteroskedasticity using adjusted modified profile likelihood inference

This paper addresses the issue of testing for heteroskedasticity in linear regression models. We derive a Bartlett adjustment to the modified profile likelihood ratio test (J. Roy. Statist. Soc. B 49 (1987) 1) for heteroskedasticity in the normal linear regression model. Our results generalize those in Ferrari and Cribari-Neto (Statist. Probab. Lett. 57 (2002) 353), since they allow for a vector-valued structure for the parameter that defines the skedastic function. Monte Carlo evidence shows that the proposed test displays reliable finite-sample behavior, outperforming the original likelihood ratio test, the Bartlett-corrected likelihood ratio test, and the modified profile likelihood ratio test.

An extended Birnbaum–Saunders distribution: Theory, estimation, and applications

Abstract Birnbaum and Saunders (1969a) pioneered a lifetime model which is commonly used in reliability studies. Based on this distribution, a new model called the gamma Birnbaum–Saunders distribution is proposed for describing fatigue life data. Several properties of the new distribution including explicit expressions for the ordinary and incomplete moments, generating and quantile functions, mean deviations, density function of the order statistics, and their moments are derived. We discuss the method of maximum likelihood and a Bayesian approach to estimate the model parameters. The superiority of the new model is illustrated by means of three failure real data sets. We also propose a new extended regression model based on the logarithm of the new distribution. The last model can be very useful to the analysis of real data and provide more realistic fits than other special regression models.

Gaussian Markov random field spatial models in GAMLSS

ABSTRACT This paper describes the modelling and fitting of Gaussian Markov random field spatial components within a Generalized AdditiveModel for Location, Scale and Shape (GAMLSS) model. This allows modelling of any or all the parameters of the distribution for the response variable using explanatory variables and spatial effects. The response variable distribution is allowed to be a non-exponential family distribution. A new package developed in R to achieve this is presented. We use Gaussian Markov random fields to model the spatial effect in Munich rent data and explore some features and characteristics of the data. The potential of using spatial analysis within GAMLSS is discussed. We argue that the flexibility of parametric distributions, ability to model all the parameters of the distribution and diagnostic tools of GAMLSS provide an ideal environment for modelling spatial features of data.

Detecting influential observations in Watson data

ABSTRACT A method for detecting outliers in axial data has been proposed by Best and Fisher (1986). For extending that work, we propose four new methods. Two of them are suitable for outlier detection and they depend on the classic geodesic distance and a modified version of this distance. The other two procedures, which are designed for influential observation detection, are based on the Kullback–Leibler and Cook’s distances. Some simulation experiments are performed to compare all considered methods. Detection and error rates are used as comparison criteria. Numerical results provide evidence in favor of the KL distance.

Bias-Corrected Maximum Likelihood Estimators in Nonlinear Heteroscedastic Models

Nonlinear heteroscedastic models are widely used in econometrics and statistical applications. We derive matrix formulae for the second-order biases of the maximum likelihood estimators of the parameters in the mean and variance response which generalize previous results by Cook et al. (1986) and Cordeiro (1993). The biases of the estimators are easily obtained as vectors of regression coefficients from suitable weighted linear regressions. The practical use of such biases is illustrated in a simulation study and in an application to a real data set.

Bartlett Adjustments for Overdispersed Generalized Linear Models

Heteroscedastic regression models have recently gained popularity in industrial applications for analyzing unreplicated experiments, experiments for robust design, and the analysis of process data. Many authors have also considered dispersion modeling to obtain correct standard errors and confidence intervals for mean parameters in regression analysis. The popularity of overdispersed generalized linear models is growing steadily to explore and model many kinds of data, especially counts and proportions. In this article, Bartlett corrections for overdispersed generalized linear models are derived. Our formulae cover many important and commonly used models, thus generalizing results by Botter and Cordeiro (1998) for double generalized linear models and by Cordeiro (1983) for generalized linear models. By simulation, the practical use of such corrections is illustrated.