Francisco Cribari-Neto

Beta Regression for Modelling Rates and Proportions

This paper proposes a regression model where the response is beta distributed using a parameterization of the beta law that is indexed by mean and dispersion parameters. The proposed model is useful for situations where the variable of interest is continuous and restricted to the interval (0, 1) and is related to other variables through a regression structure. The regression parameters of the beta regression model are interpretable in terms of the mean of the response and, when the logit link is used, of an odds ratio, unlike the parameters of a linear regression that employs a transformed response. Estimation is performed by maximum likelihood. We provide closed-form expressions for the score function, for Fishers information matrix and its inverse. Hypothesis testing is performed using approximations obtained from the asymptotic normality of the maximum likelihood estimator. Some diagnostic measures are introduced. Finally, practical applications that employ real data are presented and discussed.

Asymptotic inference under heteroskedasticity of unknown form

We focus on the finite-sample behavior of heteroskedasticity-consistent covariance matrix estimators and associated quasi-t tests. The estimator most commonly used is that proposed by Halbert White. Its finite-sample behavior under both homoskedasticity and heteroskedasticity is analyzed using Monte Carlo methods. We also consider two other consistent estimators, namely: the HC3 estimator, which is an approximation to the jackknife estimator, and the weighted bootstrap estimator. Additionally, we evaluate the finite-sample behavior of two bootstrap quasi-t tests: the test based on a single bootstrapping scheme and the test based on a double, nested bootstrapping scheme. The latter is very computer-intensive, but proves to work well in small samples. Finally, we propose a new estimator, which we call HC4; it is tailored to take into account the effect of leverage points in the design matrix on associated quasi-t tests.

On beta regression residuals

Abstract We propose two new residuals for the class of beta regression models, and numerically evaluate their behaviour relative to the residuals proposed by Ferrari and Cribari-Neto. Monte Carlo simulation results and empirical applications using real and simulated data are provided. The results favour one of the residuals we propose.

A generalization of the exponential-Poisson distribution

The two-parameter distribution known as exponential-Poisson (EP) distribution, which has decreasing failure rate, was introduced by Kus (2007). In this paper we generalize the EP distribution and show that the failure rate of the new distribution can be decreasing or increasing. The failure rate can also be upside-down bathtub shaped. A comprehensive mathematical treatment of the new distribution is provided. We provide closed-form expressions for the density, cumulative distribution, survival and failure rate functions; we also obtain the density of the ith order statistic. We derive the rth raw moment of the new distribution and also the moments of order statistics. Moreover, we discuss estimation by maximum likelihood and obtain an expression for Fishers information matrix. Furthermore, expressions for the Renyi and Shannon entropies are given and an application using a real data set is presented. Finally, simulation results on maximum likelihood estimation are presented.

Improved statistical inference for the two-parameter Birnbaum-Saunders distribution

We develop nearly unbiased estimators for the two-parameter Birnbaum-Saunders distribution [Birnbaum, Z.W., Saunders, S.C., 1969a. A new family of life distributions. J. Appl. Probab. 6, 319-327], which is commonly used in reliability studies. We derive modified maximum likelihood estimators that are bias-free to second order. We also consider bootstrap-based bias correction. The numerical evidence we present favors three bias-adjusted estimators. Different interval estimation strategies are evaluated. Additionally, we derive a Bartlett correction that improves the finite-sample performance of the likelihood ratio test in finite samples.

Bootstrap methods for heteroskedastic regression models: evidence on estimation and testing

This paper uses Monte Carlo simulation analysis to study the finite-sample behavior of bootstrap estimators and tests in the linear heteroskedastic model. We consider four different bootstrapping schemes, three of them specifically tailored to handle heteroskedasticity. Our results show that weighted bootstrap methods can be successfully used to estimate the variances of the least squares estimators of the linear parameters both under normality and under nonnormality. Simulation results are also given comparing the size and power of the bootstrapped Breusch-Pagan test with that of the original test and of Bartlett and Edgeworth-corrected tests. The bootstrap test was found to be robust against unfavorable regression designs.

On bartlett and bartlett-type corrections francisco cribari-neto

This paper reviews the literature on Bartlett and Bartlett-type corrections. It focuses on the corrections to the likelihood ratio, score and Wald test statistics. Three different Bartlett-type corrections which are equivalent to order n-1, n being the sample size, are compared through simulation. One of the forms displayed superior behavior both in terms of size and power. We also use Monte Carlo simulation to examine the effect of independent variables and the impact of the number of nuisance parameters on the finite-sample behavior of some asymptotic econometric criteria in regression models.

Influence diagnostics in beta regression

We consider the issue of assessing influence of observations in the class of beta regression models, which is useful for modelling random variables that assume values in the standard unit interval and are affected by independent variables. We propose a Cook-like distance and also measures of local influence under different perturbation schemes. Applications using real data are presented.

R: Yet Another Econometric Programming Environment

SUMMARY This article reviews R, an open-source S-like high-level matrix programming language that can be used for econometric simulations and data analysis. Copyright # 1999 John Wiley & Sons, Ltd.

Improved point and interval estimation for a beta regression model

In this paper we consider the beta regression model recently proposed by Ferrari and Cribari-Neto [2004. Beta regression for modeling rates and proportions. J. Appl. Statist. 31, 799-815], which is tailored to situations where the response is restricted to the standard unit interval and the regression structure involves regressors and unknown parameters. We derive the second order biases of the maximum likelihood estimators and use them to define bias-adjusted estimators. As an alternative to the two analytically bias-corrected estimators discussed, we consider a bias correction mechanism based on the parametric bootstrap. The numerical evidence favors the bootstrap-based estimator and also one of the analytically corrected estimators. Several different strategies for interval estimation are also proposed. We present an empirical application.