Generalized additive mixed models

Following the extension from linear mixed models to additive mixed models, extension from generalized linear mixed models to generalized additive mixed models is made, Algorithms are developed to compute the MLE's of the nonlinear effects and the covariance structures based on the penalized marginal likelihood. Convergence of the algorithms and selection of the smooth param¬eters are discussed.     

Generalized partially linear varying-coefficient models

Generalized varying-coefficient models are useful extensions of generalized linear models. They arise naturally when investigating how regression coefficients change over different groups characterized by certain covariates such as age. In this paper, we extend these models to generalized partially linear varying-coefficient models, in which some coefficients are constants and the others are functions of certain covariates. Procedures for estimating the linear and non-parametric parts are developed and their associated statistical properties are studied. The methods proposed are illustrated using some simulations and real data analysis.     

Partially linear models and their applications to change point detection of chemical process data

In many chemical data sets, the amount of radiation absorbed (absorbance) is related to the concentration of the element in the sample by Lambert–Beer's law. However, this relation changes abruptly when the variable concentration reaches an unknown threshold level, the so-called change point. In the context of analytical chemistry, there are many methods that describe the relationship between absorbance and concentration, but none of them provide inferential procedures to detect change points. In this paper, we propose partially linear models with a change point separating the parametric and nonparametric components. The Schwarz information criterion is used to locate a change point. A back-fitting algorithm is presented to obtain parameter estimates and the penalized Fisher information matrix is obtained to calculate the standard errors of the parameter estimates. To examine the proposed method, we present a simulation study. Finally, we apply the method to data sets from the chemistry area. The partially linear models with a change point developed in this paper are useful supplements to other methods of absorbance–concentration analysis in chemical studies, for example, and in many other practical applications.     

Generalized skew-normal models: properties and inference

In this article, we introduce a new family of asymmetric distributions, which depends on two parameters namely, α and β, and in the special case where β = 0, the skew-normal (SN) distribution considered by Azzallini [Azzalini, A., 1985, A class of distributions which includes the normal ones. Scandinavian Journal of Statistics, 12, 171–178.] is obtained. Basic properties such as a stochastic representation and the derivation of maximum likelihood and moment estimators are studied. The asymptotic behaviour of both types of estimators is also investigated. Results of a small-scale simulation study is provided illustrating the usefulness of the new model. An application to a real data set is reported showing that it can present better fit than the SN distribution.     

Generalized modified slash distribution with applications

Abstract

In this article a generalization of the modified slash distribution is introduced. This model is based on the quotient of two independent random variables, whose distributions are a normal and a one-parameter gamma, respectively. The resulting distribution is a new model whose kurtosis is greater than other slash distributions. The probability density function, its properties, moments, and kurtosis coefficient are obtained. Inference based on moment and maximum likelihood methods is carried out. The multivariate version is also introduced. Two real data sets are considered in which it is shown that the new model fits better to symmetric data with heavy tails than other slash extensions previously introduced in literature.     

Generalized additive models for longitudinal data

We introduce a class of models for longitudinal data by extending the generalized estimating equations approach of Liang and Zeger (1986) to incorporate the flexibility of nonparametric smoothing. The algorithm provides a unified estimation procedure for marginal distributions from the exponential family. We propose pointwise standard-error bands and approximate likelihood-ratio and score tests for inference. The algorithm is formally derived by using the penalized quasilikelihood framework. Convergence of the estimating equations and consistency of the resulting solutions are discussed. We illustrate the algorithm with data on the population dynamics of Colorado potato beetles on potato plants.     

Generalized linear models with functional predictors

Summary. We present a technique for extending generalized linear models to the situation where some of the predictor variables are observations from a curve or function. The technique is particularly useful when only fragments of each curve have been observed. We demonstrate, on both simulated and real data sets, how this approach can be used to perform linear, logistic and censored regression with functional predictors. In addition, we show how functional principal components can be used to gain insight into the relationship between the response and functional predictors. Finally, we extend the methodology to apply generalized linear models and principal components to standard missing data problems.     

Generalized Log-Lindley distribution and its applications in stochastic comparison

In this paper, a new distribution, generalized Log-Lindley distribution, which supports on (0, 1) is introduced as an alternative to the beta distribution. We show the new proposed distribution is also an extension of Log-Lindley distribution. Stochastic comparison for two parallel systems with generalized Log-Lindley distributed components is considered. Our results extend some existing results in the literature.     

Symmetric quantiles and their applications

To develop estimators with stronger efficiencies than the trimmed means which use the empirical quantile, Kim (1992) Kim, S. J. 1992. The metrically trimmed means as a robust estimator of location. Annals of Statistics, 20: 1534–1547. [Crossref], [Web of Science ®] [Google Scholar] and Chen & Chiang (1996) Chen, L. A. and Chiang, Y. C. 1996. Symmetric type quantile and trimmed means for location and linear regression model. Journal of Nonparametric Statistics, 7: 171–185. [Taylor & Francis Online] [Google Scholar], implicitly or explicitly used the symmetric quantile, and thus introduced new trimmed means for location and linear regression models, respectively. This study further investigates the properties of the symmetric quantile and extends its application in several aspects. (a) The symmetric quantile is more efficient than the empirical quantiles in asymptotic variances when quantile percentage α is either small or large. This reveals that for any proposal involving the α th quantile of small or large α s, the symmetric quantile is the right choice; (b) a trimmed mean based on it has asymptotic variance achieving a Cramer-Rao lower bound in one heavy tail distribution; (c) an improvement of the quantiles-based control chart by Grimshaw & Alt (1997) Grimshaw, S. D. and Alt, F. B. 1997. Control charts for quantile function values. Journal of Quality Technology, 29: 1–7. [Taylor & Francis Online] [Google Scholar] is discussed; (d) Monte Carlo simulations of two new scale estimators based on symmetric quantiles also support this new quantile.     

Generalized discrete discrepancy and its applications in experimental designs

In recent years, there has been increasing interest in the study of discrete discrepancy. In this paper, the popular discrete discrepancy is extended to the so-called generalized discrete discrepancy. Connections among generalized discrete discrepancy and other optimality criteria, such as orthogonality, generalized minimum aberration and minimum moment aberration, are investigated. These connections provide strong statistical justification of generalized discrete discrepancy. A lower bound of generalized discrete discrepancy is also obtained, which serves as an important benchmark of design uniformity.     

Precedence probabilities and their applications

In precedence tests, the test powers are given by the probabilities P(Y_(i) > X_(j)) for some i and j, where X_(j) and Y_(i) are the order statistics of two independent random samples. Such probabilities can also arise in reliability theory. In this paper, we investigate the properties of these probabilities and their applications.     

Generalized choice models for categorical time series

The NDARMA models of Jacobs and Lewis (1983) allow the modeling of categorical processes with an ARMA-like serial dependence structure. These models can be represented through a backshift mechanism, and we analyze marginal and bivariate properties of the resulting backshift process. Motivated by this backshift mechanism, we define the new class of generalized choice (GC) models, which include the usual NDARMA models as a special case, and we derive results describing the marginal and bivariate distribution of the GC model. We discuss implications concerning DMA(∞) models and the serial dependence structure of NDARMA models. Examples show that the family of GC models allows creating sparsely parametrized models for categorical processes with different types of serial dependence structure.     

Generalized log-gamma regression models with cure fraction

In this paper, the generalized log-gamma regression model is modified to allow the possibility that long-term survivors may be present in the data. This modification leads to a generalized log-gamma regression model with a cure rate, encompassing, as special cases, the log-exponential, log-Weibull and log-normal regression models with a cure rate typically used to model such data. The models attempt to simultaneously estimate the effects of explanatory variables on the timing acceleration/deceleration of a given event and the surviving fraction, that is, the proportion of the population for which the event never occurs. The normal curvatures of local influence are derived under some usual perturbation schemes and two martingale-type residuals are proposed to assess departures from the generalized log-gamma error assumption as well as to detect outlying observations. Finally, a data set from the medical area is analyzed.     

Generalized Markov-Polya urn models with predetermined strategies

By using combinatorial methods involving lattice path combinatorics, three generalized probability models dependent on predetermined strategies have been obtained with the help of urn models.The models have been developed with the help of a sampling scheme which unifies both, the binomial and the inverse binomial sampling schemes. These models generate a number of important discrete probability distributions both as particular cases and as limiting cases. Recurrence relations among the moments of the models have also been obtained.     

Generalized Tobit models: diagnostics and application in econometrics

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.     

Generalized additive models for location, scale and shape

Summary.  A general class of statistical models for a univariate response variable is presented which we call the generalized additive model for location, scale and shape (GAMLSS). The model assumes independent observations of the response variable y given the parameters, the explanatory variables and the values of the random effects. The distribution for the response variable in the GAMLSS can be selected from a very general family of distributions including highly skew or kurtotic continuous and discrete distributions. The systematic part of the model is expanded to allow modelling not only of the mean (or location) but also of the other parameters of the distribution of y , as parametric and/or additive nonparametric (smooth) functions of explanatory variables and/or random-effects terms. Maximum (penalized) likelihood estimation is used to fit the (non)parametric models. A Newton–Raphson or Fisher scoring algorithm is used to maximize the (penalized) likelihood. The additive terms in the model are fitted by using a backfitting algorithm. Censored data are easily incorporated into the framework. Five data sets from different fields of application are analysed to emphasize the generality of the GAMLSS class of models.