Mônica C. Sandoval

Diagnostic techniques in generalized estimating equations

We consider herein diagnostic methods for the quasi-likelihood regression models developed by Zeger and Liang [Zeger, S. L., Liang, K.-Y., 1986, Longitudinal data analysis for discrete and conti-nuous outcomes. Biometrics, 42, 121–130.] to analyse discrete and continuous longitudinal data. Our proposal generalises well-known measures (projection matrix, Cooks distance and standardised resi-duals) developed for independent responses. Moreover, half-normal probability plots with simulated envelopes were developed for assessing the adequacy of the fitted model when the marginal distributions belong to the exponential family. To obtain such a plot, correlated outcomes were generated by simulation using algorithms described in the literature. Finally, two applications were presented to illustrate the techniques.We consider herein diagnostic methods for the quasi-likelihood regression models developed by Zeger and Liang [Zeger, S. L., Liang, K.-Y., 1986, Longitudinal data analysis for discrete and conti-nuous outcomes. Biometrics, 42, 121–130.] to analyse discrete and continuous longitudinal data. Our proposal generalises well-known measures (projection matrix, Cooks distance and standardised resi-duals) developed for independent responses. Moreover, half-normal probability plots with simulated envelopes were developed for assessing the adequacy of the fitted model when the marginal distributions belong to the exponential family. To obtain such a plot, correlated outcomes were generated by simulation using algorithms described in the literature. Finally, two applications were presented to illustrate the techniques.

A regression model for special proportions

Credit cards are a financial product with special characteristics. Dividing the amount paid by the customer in a given month by the total bill results in a variable that is partly discrete and partly continuous, which we call the relative payment amount (RPA). This variable is discrete at 0, c and 1, and it is continuous in the open interval (c, 1). The 0<c<1 value is known and is given by the ratio between the value of the minimum payment and the full amount, and this value is not fixed for all customers. Thus, in practice, the RPA is a variable whose support of its distribution is non-constant across population units. In this work, we propose a regression model for the RPA. The model allows all of the unknown parameters of the conditional distribution of the response variable to be modelled as a function of the explanatory variables, and it also accounts for the non-constant known parameter c. The estimation of the parameters of this model is discussed, diagnostic analysis is addressed and closed-form expressions for the score function and for the Fisher’s information matrix are provided. Moreover, some results related to the non-constant nature of c are obtained, simulation studies are performed and an application using real credit card data is presented.

Adjusted Pearson residuals in beta regression models

In this paper, matrix formulae of order n−1, where n is the sample size, for the first two moments of Pearson residuals are obtained in beta regression models. Adjusted Pearson residuals are also obtained, having, to this order, expected value zero and variance one. Monte Carlo simulation results are presented illustrating the behaviour of both adjusted and unadjusted residuals.

Linear calibration in functional regression models

This paper discusses calibration in functional regression models. Classical and inverse type estimators are considered. First order approximation to the bias and to the mean squared error (MSE) of the estimators are considered. Numerical comparisons seem to indicate that the classical estimator obtained via maximum likelihood estimation performs better than the other estimators considered.

On the skew-normal calibration model

In this article, we present the EM-algorithm for performing maximum likelihood estimation of an asymmetric linear calibration model with the assumption of skew-normally distributed error. A simulation study is conducted for evaluating the performance of the calibration estimator with interpolation and extrapolation situations. As one application in a real data set, we fitted the model studied in a dimensional measurement method used for calculating the testicular volume through a caliper and its calibration by using ultrasonography as the standard method. By applying this methodology, we do not need to transform the variables to have symmetrical errors. Another interesting aspect of the approach is that the developed transformation to make the information matrix nonsingular, when the skewness parameter is near zero, leaves the parameter of interest unchanged. Model fitting is implemented and the best choice between the usual calibration model and the model proposed in this article was evaluated by developing the Akaike information criterion, Schwarz’s Bayesian information criterion and Hannan–Quinn criterion.

Local influence in estimating equations

Local influence diagnostics based on estimating equations as the role of a gradient vector derived from any fit function are developed for repeated measures regression analysis. Our proposal generalizes tools used in other studies (Cook, 1986; Cadigan and Farrell, 2002), considering herein local influence diagnostics for a statistical model where estimation involves an estimating equation in which all observations are not necessarily independent of each other. Moreover, the measures of local influence are illustrated with some simulated data sets to assess influential observations. Applications using real data are presented.

The Truncated Inflated Beta Distribution

The study of proportions is a common topic in many fields of study. The standard beta distribution or the inflated beta distribution may be a reasonable choice to fit a proportion in most situations. However, they do not fit well variables that do not assume values in the open interval (0, c), 0 < c < 1. For these variables, the authors introduce the truncated inflated beta distribution (TBEINF). This proposed distribution is a mixture of the beta distribution bounded in the open interval (c, 1) and the trinomial distribution. The authors present the moments of the distribution, its scoring vector, and Fisher information matrix, and discuss estimation of its parameters. The properties of the suggested estimators are studied using Monte Carlo simulation. In addition, the authors present an application of the TBEINF distribution for unemployment insurance data.

Skewness of maximum likelihood estimators in the varying dispersion beta regression model

Abstract Beta regression models have been widely used to model rates and proportions. We obtain a matrix formula of order where n is the sample size, for the skewness coefficient of the distribution of the maximum likelihood estimators of the linear parameters in varying dispersion beta regression models. The formula can be used to verify whether inference based on the asymptotic distribution of the maximum likelihood estimators should be performed. A simulation study and two applications are presented to illustrate the results.

Corrigendum to: “Covariance matrix formula for generalized linear models with unknown dispersion” by G. M. Cordeiro, L. P. Barroso, and D. A. Botter [Communications in Statistics—Theory and Methods (2006) 35(1), 113–120]

then φβ = T βφ and the expression of the second-order covariance matrix is asymmetric. The result of Peers and Iqbal, instead of Shenton and Bowman, was widely applied in the literature to obtain the second-order covariance matrix in several regression models. Thus, we suggest a review of these papers’ results and future works about second-order covariance matrix to be based on Shenton and Bowman (1977, p. 67). Additionally, expression (3.6), in Cordeiro et al. (2006), should be corrected. The correction is given by

On predicting the finite population distribution function

In this article, we consider the optimal prediction of the finite population distribution function under general linear regression models with normally distributed errors. Emphasis is placed on the case where the error variance is unknown. Large sample approximations to the prediction variance of the optimal predictors are also derived.