The complementary exponential geometric distribution: Model, properties, and a comparison with its counterpart

In this paper, we proposed a new two-parameter lifetime distribution with increasing failure rate, the complementary exponential geometric distribution, which is complementary to the exponential geometric model proposed by Adamidis and Loukas (1998). The new distribution arises on a latent complementary risks scenario, in which the lifetime associated with a particular risk is not observable; rather, we observe only the maximum lifetime value among all risks. The properties of the proposed distribution are discussed, including a formal proof of its probability density function and explicit algebraic formulas for its reliability and failure rate functions, moments, including the mean and variance, variation coefficient, and modal value. The parameter estimation is based on the usual maximum likelihood approach. We report the results of a misspecification simulation study performed in order to assess the extent of misspecification errors when testing the exponential geometric distribution against our complementary one in the presence of different sample size and censoring percentage. The methodology is illustrated on four real datasets; we also make a comparison between both modeling approaches. (C) 2011 Elsevier B.V. All rights reserved.

MODELING THE EVOLUTION OF COMPLEX NETWORKS THROUGH THE PATH-STAR TRANSFORMATION AND OPTIMAL MULTIVARIATE METHODS

The topology of real-world complex networks, such as in transportation and communication, is always changing with time. Such changes can arise not only as a natural consequence of their growth, but also due to major modi. cations in their intrinsic organization. For instance, the network of transportation routes between cities and towns ( hence locations) of a given country undergo a major change with the progressive implementation of commercial air transportation. While the locations could be originally interconnected through highways ( paths, giving rise to geographical networks), transportation between those sites progressively shifted or was complemented by air transportation, with scale free characteristics. In the present work we introduce the path-star transformation ( in its uniform and preferential versions) as a means to model such network transformations where paths give rise to stars of connectivity. It is also shown, through optimal multivariate statistical methods (i.e. canonical projections and maximum likelihood classification) that while the US highways network adheres closely to a geographical network model, its path-star transformation yields a network whose topological properties closely resembles those of the respective airport transportation network.

Modeling connectivity in terms of network activity

A new complex network model is proposed which is founded on growth, with new connections being established proportionally to the current dynamical activity of each node, which can be understood as a generalization of the Barabasi-Albert static model. By using several topological measurements, as well as optimal multivariate methods (canonical analysis and maximum likelihood decision), we show that this new model provides, among several other theoretical kinds of networks including Watts-Strogatz small-world networks, the greatest compatibility with three real-world cortical networks.

On beta regression residuals

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.

Inference and local influence assessment in skew-normal null intercept measurement error model

In this article, we discuss inferential aspects of the measurement error regression models with null intercepts when the unknown quantity x (latent variable) follows a skew normal distribution. We examine first the maximum-likelihood approach to estimation via the EM algorithm by exploring statistical properties of the model considered. Then, the marginal likelihood, the score function and the observed information matrix of the observed quantities are presented allowing direct inference implementation. In order to discuss some diagnostics techniques in this type of models, we derive the appropriate matrices to assessing the local influence on the parameter estimates under different perturbation schemes. The results and methods developed in this paper are illustrated considering part of a real data set used by Hadgu and Koch [1999, Application of generalized estimating equations to a dental randomized clinical trial. Journal of Biopharmaceutical Statistics, 9, 161-178].

Local influence for Student-t partially linear models

In this paper we extend partial linear models with normal errors to Student-t errors Penalized likelihood equations are applied to derive the maximum likelihood estimates which appear to be robust against outlying observations in the sense of the Mahalanobis distance In order to study the sensitivity of the penalized estimates under some usual perturbation schemes in the model or data the local influence curvatures are derived and some diagnostic graphics are proposed A motivating example preliminary analyzed under normal errors is reanalyzed under Student-t errors The local influence approach is used to compare the sensitivity of the model estimates (C) 2010 Elsevier B V All rights reserved

Bimodal extension based on the skew-normal distribution with application to pollen data

This paper considers an extension to the skew-normal model through the inclusion of an additional parameter which can lead to both uni- and bi-modal distributions. The paper presents various basic properties of this family of distributions and provides a stochastic representation which is useful for obtaining theoretical properties and to simulate from the distribution. Moreover, the singularity of the Fisher information matrix is investigated and maximum likelihood estimation for a random sample with no covariates is considered. The main motivation is thus to avoid using mixtures in fitting bimodal data as these are well known to be complicated to deal with, particularly because of identifiability problems. Data-based illustrations show that such model can be useful. Copyright (C) 2009 John Wiley & Sons, Ltd.

Comparing diagnostic tests with missing data

When missing data occur in studies designed to compare the accuracy of diagnostic tests, a common, though naive, practice is to base the comparison of sensitivity, specificity, as well as of positive and negative predictive values on some subset of the data that fits into methods implemented in standard statistical packages. Such methods are usually valid only under the strong missing completely at random (MCAR) assumption and may generate biased and less precise estimates. We review some models that use the dependence structure of the completely observed cases to incorporate the information of the partially categorized observations into the analysis and show how they may be fitted via a two-stage hybrid process involving maximum likelihood in the first stage and weighted least squares in the second. We indicate how computational subroutines written in R may be used to fit the proposed models and illustrate the different analysis strategies with observational data collected to compare the accuracy of three distinct non-invasive diagnostic methods for endometriosis. The results indicate that even when the MCAR assumption is plausible, the naive partial analyses should be avoided.

Bayesian inference for a skew-normal IRT model under the centred parameterization

Item response theory (IRT) comprises a set of statistical models which are useful in many fields, especially when there is interest in studying latent variables. These latent variables are directly considered in the Item Response Models (IRM) and they are usually called latent traits. A usual assumption for parameter estimation of the IRM, considering one group of examinees, is to assume that the latent traits are random variables which follow a standard normal distribution. However, many works suggest that this assumption does not apply in many cases. Furthermore, when this assumption does not hold, the parameter estimates tend to be biased and misleading inference can be obtained. Therefore, it is important to model the distribution of the latent traits properly. In this paper we present an alternative latent traits modeling based on the so-called skew-normal distribution; see Genton (2004). We used the centred parameterization, which was proposed by Azzalini (1985). This approach ensures the model identifiability as pointed out by Azevedo et al. (2009b). Also, a Metropolis Hastings within Gibbs sampling (MHWGS) algorithm was built for parameter estimation by using an augmented data approach. A simulation study was performed in order to assess the parameter recovery in the proposed model and the estimation method, and the effect of the asymmetry level of the latent traits distribution on the parameter estimation. Also, a comparison of our approach with other estimation methods (which consider the assumption of symmetric normality for the latent traits distribution) was considered. The results indicated that our proposed algorithm recovers properly all parameters. Specifically, the greater the asymmetry level, the better the performance of our approach compared with other approaches, mainly in the presence of small sample sizes (number of examinees). Furthermore, we analyzed a real data set which presents indication of asymmetry concerning the latent traits distribution. The results obtained by using our approach confirmed the presence of strong negative asymmetry of the latent traits distribution. (C) 2010 Elsevier B.V. All rights reserved.

Missing data mechanisms and their implications on the analysis of categorical data

We review some issues related to the implications of different missing data mechanisms on statistical inference for contingency tables and consider simulation studies to compare the results obtained under such models to those where the units with missing data are disregarded. We confirm that although, in general, analyses under the correct missing at random and missing completely at random models are more efficient even for small sample sizes, there are exceptions where they may not improve the results obtained by ignoring the partially classified data. We show that under the missing not at random (MNAR) model, estimates on the boundary of the parameter space as well as lack of identifiability of the parameters of saturated models may be associated with undesirable asymptotic properties of maximum likelihood estimators and likelihood ratio tests; even in standard cases the bias of the estimators may be low only for very large samples. We also show that the probability of a boundary solution obtained under the correct MNAR model may be large even for large samples and that, consequently, we may not always conclude that a MNAR model is misspecified because the estimate is on the boundary of the parameter space.

Inference for a skew extension of the Grubbs model

In this paper, we discuss inferential aspects for the Grubbs model when the unknown quantity x (latent response) follows a skew-normal distribution, extending early results given in Arellano-Valle et al. (J Multivar Anal 96:265-281, 2005b). Maximum likelihood parameter estimates are computed via the EM-algorithm. Wald and likelihood ratio type statistics are used for hypothesis testing and we explain the apparent failure of the Wald statistics in detecting skewness via the profile likelihood function. The results and methods developed in this paper are illustrated with a numerical example.

Inflated beta distributions

This paper considers the issue of modeling fractional data observed on [0,1), (0,1] or [0,1]. Mixed continuous-discrete distributions are proposed. The beta distribution is used to describe the continuous component of the model since its density can have quite different shapes depending on the values of the two parameters that index the distribution. Properties of the proposed distributions are examined. Also, estimation based on maximum likelihood and conditional moments is discussed. Finally, practical applications that employ real data are presented.

Multivariate measurement error models based on scale mixtures of the skew-normal distribution

Scale mixtures of the skew-normal (SMSN) distribution is a class of asymmetric thick-tailed distributions that includes the skew-normal (SN) distribution as a special case. The main advantage of these classes of distributions is that they are easy to simulate and have a nice hierarchical representation facilitating easy implementation of the expectation-maximization algorithm for the maximum-likelihood estimation. In this paper, we assume an SMSN distribution for the unobserved value of the covariates and a symmetric scale mixtures of the normal distribution for the error term of the model. This provides a robust alternative to parameter estimation in multivariate measurement error models. Specific distributions examined include univariate and multivariate versions of the SN, skew-t, skew-slash and skew-contaminated normal distributions. The results and methods are applied to a real data set.

Diagnostic tools in beta regression with varying dispersion

We consider the issue of performing residual and local influence analyses in beta regression models with varying dispersion, which are useful for modelling random variables that assume values in the standard unit interval. In such models, both the mean and the dispersion depend upon independent variables. We derive the appropriate matrices for assessing local influence on the parameter estimates under different perturbation schemes. An application using real data is presented and discussed.

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.