The Influence of Random Effects on Univariate and Bivariate Discrete Proportional Hazards Models

The random effects survival model has been widely used in the recent literature as a generalization of the continuous proportional hazards model. When a random effect is present, it is known that the hazard rates are generally underestimated in the context of continuous proportional hazards models. This article establishes theorems for the influence of random effects on both univariate and bivariate discrete proportional hazards models.     

On influence diagnostic in univariate elliptical linear regression models

We discuss in this paper the assessment of local influence in univariate elliptical linear regression models. This class includes all symmetric continuous distributions, such as normal, Student-t, Pearson VII, exponential power and logistic, among others. We derive the appropriate matrices for assessing the local influence on the parameter estimates and on predictions by considering as influence measures the likelihood displacement and a distance based on the Pearson residual. Two examples with real data are given for illustration.     

高频数据驱动的连续时间模型估计

现代金融经济学中连续时间模型能够更方便地描述重要经济变量的动态过程如股价、汇率和利率等。为连续时间模型提出了一种高频数据驱动的二阶段估计方法,增强了连续时间扩展模型的弹性和可操作性。以Vasicek模型为例给出了该方法的应用实例,首先在第一阶段使用实现波动率方法估计出模型的扩散项参数,然后使用实际数据的稳态分布的前向方程估计漂移项参数。此方法对模型初始设定和优化算法依赖程度低,结果较为稳定可靠。     

Variance partitioning in multilevel logistic models that exhibit overdispersion

Summary.  A common application of multilevel models is to apportion the variance in the response according to the different levels of the data. Whereas partitioning variances is straightforward in models with a continuous response variable with a normal error distribution at each level, the extension of this partitioning to models with binary responses or to proportions or counts is less obvious. We describe methodology due to Goldstein and co-workers for apportioning variance that is attributable to higher levels in multilevel binomial logistic models. This partitioning they referred to as the variance partition coefficient. We consider extending the variance partition coefficient concept to data sets when the response is a proportion and where the binomial assumption may not be appropriate owing to overdispersion in the response variable. Using the literacy data from the 1991 Indian census we estimate simple and complex variance partition coefficients at multiple levels of geography in models with significant overdispersion and thereby establish the relative importance of different geographic levels that influence educational disparities in India.     

Diagnostic tools in generalized Weibull linear regression models

We propose some statistical tools for diagnosing the class of generalized Weibull linear regression models [A.A. Prudente and G.M. Cordeiro, Generalized Weibull linear models, Comm. Statist. Theory Methods 39 (2010), pp. 3739–3755]. This class of models is an alternative means of analysing positive, continuous and skewed data and, due to its statistical properties, is very competitive with gamma regression models. First, we show that the Weibull model induces ma-ximum likelihood estimators asymptotically more efficient than the gamma model. Standardized residuals are defined, and their statistical properties are examined empirically. Some measures are derived based on the case-deletion model, including the generalized Cook's distance and measures for identifying influential observations on partial F-tests. The results of a simulation study conducted to assess behaviour of the global influence approach are also presented. Further, we perform a local influence analysis under the case-weights, response and explanatory variables perturbation schemes. The Weibull, gamma and other Weibull-type regression models are fitted into three data sets to illustrate the proposed diagnostic tools. Statistical analyses indicate that the Weibull model fitted into these data yields better fits than other common alternative models.     

Multivariate survival mixed models for genetic analysis of longevity traits

A class of multivariate mixed survival models for continuous and discrete time with a complex covariance structure is introduced in a context of quantitative genetic applications. The methods introduced can be used in many applications in quantitative genetics although the discussion presented concentrates on longevity studies. The framework presented allows to combine models based on continuous time with models based on discrete time in a joint analysis. The continuous time models are approximations of the frailty model in which the baseline hazard function will be assumed to be piece-wise constant. The discrete time models used are multivariate variants of the discrete relative risk models. These models allow for regular parametric likelihood-based inference by exploring a coincidence of their likelihood functions and the likelihood functions of suitably defined multivariate generalized linear mixed models. The models include a dispersion parameter, which is essential for obtaining a decomposition of the variance of the trait of interest as a sum of parcels representing the additive genetic effects, environmental effects and unspecified sources of variability; as required in quantitative genetic applications. The methods presented are implemented in such a way that large and complex quantitative genetic data can be analyzed. Some key model control techniques are discussed in a supplementary online material.     

A note on non-negative continuous time processes

Summary.  Recently there has been much work on developing models that are suitable for analysing the volatility of a continuous time process. One general approach is to define a volatility process as the convolution of a kernel with a non-decreasing Lévy process, which is non-negative if the kernel is non-negative. Within the framework of time continuous autoregressive moving average (CARMA) processes, we derive a necessary and sufficient condition for the kernel to be non-negative. This condition is in terms of the Laplace transform of the CARMA kernel, which has a simple form. We discuss some useful consequences of this result and delineate the parametric region of stationarity and non-negative kernel for some lower order CARMA models.     

Effects of drift and noise on the optimal sliding window size for data stream regression models

The analysis of non stationary data streams requires a continuous adaption of the model to the relevant most recent data. This requires that changes in the data stream must be distinguished from noise. Many approaches are based on heuristic adaptation schemes. We analyze simple regression models to understand the joint effects of noise and concept drift and derive the optimal sliding window size for the regression models. Our theoretical analysis and simulations show that a near optimal window size can be crucial. Our models can be used as benchmarks for other models to see how they cope with noise and drift.     

The Kumaraswamy normal linear regression model with applications

ABSTRACT

For any continuous baseline G distribution, Cordeiro and Castro pioneered the Kumaraswamy-G family of distributions with two extra positive parameters, which generalizes both Lehmann types I and II classes. We study some mathematical properties of the Kumaraswamy-normal (KwN) distribution including ordinary and incomplete moments, mean deviations, quantile and generating functions, probability weighted moments, and two entropy measures. We propose a new linear regression model based on the KwN distribution, which extends the normal linear regression model. We obtain the maximum likelihood estimates of the model parameters and provide some diagnostic measures such as global influence, local influence, and residuals. We illustrate the potentiality of the introduced models by means of two applications to real datasets.     

Augmented-limited regression models with an application to the study of the risk perceived using continuous scales

Studies of risk perceived using continuous scales of [0,100] were recently introduced in psychometrics, which can be transformed to the unit interval, but the presence of zeros or ones are commonly observed. Motivated by this, we introduce a full inferential set of tools that allows for augmented and limited data modeling. We considered parameter estimation, residual analysis, influence diagnostic and model selection for zero-and/or-one augmented beta rectangular (ZOABR) regression models and their particular nested models, which is based on a new parameterization of the beta rectangular distribution. Different from other alternatives, we performed maximum-likelihood estimation using a combination of the EM algorithm (for the continuous part) and Fisher scoring algorithm (for the discrete part). Also, we perform an additional step, by considering other link functions, besides the usual logistic link, for modeling the response mean. By considering randomized quantile residuals, (local) influence diagnostics and model selection tools, we identified that the ZOABR regression model is the best one. We also conducted extensive simulations studies, which indicate that all developed tools work properly. Finally, we discuss the use of this type of models to treat psychometric data. It is worthwhile to mention that applications of the developed methods go beyond to Psychometric data. Indeed, they can be useful when the response variable in bounded, including or not the respective limits.     

Generalized linear mixed joint model for longitudinal and survival outcomes

Longitudinal studies often entail categorical outcomes as primary responses. When dropout occurs, non-ignorability is frequently accounted for through shared parameter models (SPMs). In this context, several extensions from Gaussian to non-Gaussian longitudinal processes have been proposed. In this paper, we formulate an approach for non-Gaussian longitudinal outcomes in the framework of joint models. As an extension of SPMs, based on shared latent effects, we assume that the history of the response up to current time may have an influence on the risk of dropout. This history is represented by the current, expected, value of the response. Since the time a subject spends in the study is continuous, we parametrize the dropout process through a proportional hazard model. The resulting model is referred to as Generalized Linear Mixed Joint Model (GLMJM). To estimate model parameters, we adopt a maximum likelihood approach via the EM algorithm. In this context, the maximization of the observed data log-likelihood requires numerical integration over the random effect posterior distribution, which is usually not straightforward; under the assumption of Gaussian random effects, we compare Gauss-Hermite and Pseudo-Adaptive Gaussian quadrature rules. We investigate in a simulation study the behaviour of parameter estimates in the case of Poisson and Binomial longitudinal responses, and apply the GLMJM to a benchmark dataset.     

Practical modeling strategies for unbalanced longitudinal data analysis

In practice, data are often measured repeatedly on the same individual at several points in time. Main interest often relies in characterizing the way the response changes in time, and the predictors of that change. Marginal, mixed and transition are frequently considered to be the main models for continuous longitudinal data analysis. These approaches are proposed primarily for balanced longitudinal design. However, in clinic studies, data are usually not balanced and some restrictions are necessary in order to use these models. This paper was motivated by a data set related to longitudinal height measurements in children of HIV-infected mothers that was recorded at the university hospital of the Federal University in Minas Gerais, Brazil. This data set is severely unbalanced. The goal of this paper is to assess the application of continuous longitudinal models for the analysis of unbalanced data set.     

Matrix-analytic Models and their Analysis

We survey phase-type distributions and Markovian point processes, aspects of how to use such models in applied probability calculations and how to fit them to observed data. A phase-type distribution is defined as the time to absorption in a finite continuous time Markov process with one absorbing state. This class of distributions is dense and contains many standard examples like all combinations of exponential in series/parallel. A Markovian point process is governed by a finite continuous time Markov process (typically ergodic), such that points are generated at a Poisson intensity depending on the underlying state and at transitions; a main special case is a Markov-modulated Poisson process. In both cases, the analytic formulas typically contain matrix-exponentials, and the matrix formalism carried over when the models are used in applied probability calculations as in problems in renewal theory, random walks and queueing. The statistical analysis is typically based upon the EM algorithm, viewing the whole sample path of the background Markov process as the latent variable.     

A distance-based rounding strategy for post-imputation ordinal data

Multiple imputation has emerged as a widely used model-based approach in dealing with incomplete data in many application areas. Gaussian and log-linear imputation models are fairly straightforward to implement for continuous and discrete data, respectively. However, in missing data settings which include a mix of continuous and discrete variables, correct specification of the imputation model could be a daunting task owing to the lack of flexible models for the joint distribution of variables of different nature. This complication, along with accessibility to software packages that are capable of carrying out multiple imputation under the assumption of joint multivariate normality, appears to encourage applied researchers for pragmatically treating the discrete variables as continuous for imputation purposes, and subsequently rounding the imputed values to the nearest observed category. In this article, I introduce a distance-based rounding approach for ordinal variables in the presence of continuous ones. The first step of the proposed rounding process is predicated upon creating indicator variables that correspond to the ordinal levels, followed by jointly imputing all variables under the assumption of multivariate normality. The imputed values are then converted to the ordinal scale based on their Euclidean distances to a set of indicators, with minimal distance corresponding to the closest match. I compare the performance of this technique to crude rounding via commonly accepted accuracy and precision measures with simulated data sets.     

Identifiable finite mixtures of location models for clustering mixed-mode data

For clustering mixed categorical and continuous data, Lawrence and Krzanowski (1996) proposed a finite mixture model in which component densities conform to the location model. In the graphical models literature the location model is known as the homogeneous Conditional Gaussian model. In this paper it is shown that their model is not identifiable without imposing additional restrictions. Specifically, for g groups and m locations, (g!)m–1 distinct sets of parameter values (not including permutations of the group mixing parameters) produce the same likelihood function. Excessive shrinkage of parameter estimates in a simulation experiment reported by Lawrence and Krzanowski (1996) is shown to be an artifact of the model's non-identifiability. Identifiable finite mixture models can be obtained by imposing restrictions on the conditional means of the continuous variables. These new identified models are assessed in simulation experiments. The conditional mean structure of the continuous variables in the restricted location mixture models is similar to that in the underlying variable mixture models proposed by Everitt (1988), but the restricted location mixture models are more computationally tractable.     

Fibre analysis and tangent models

Available methods for deriving tests and confidence regions for multiparameter models seem seriously limited in their feasibility. This paper gives preliminary survey to some developing techniques that use conditioning as a primary device to derive tests and confidence regions. The techniques usefibre analysis which involves the conditional determination of OLS (observed level of significance); the techniques are available initally for models with pivotal quantities and for continuous exponential models. The techniques can then be extended to more general models using the notion of tangent model.     

Influence diagnostics in nonlinear reproductive dispersion mixed models

The class of nonlinear reproductive dispersion mixed models (NRDMMs) is an extension of nonlinear reproductive dispersion models and generalized linear mixed models. This paper discusses the influence analysis of the model based on Laplace approximation. The equivalence of case-deletion models and mean-shift outlier models in NRDMMs is investigated, and some diagnostic measures are proposed via the case-deletion method. We also investigate the assessment of local influence of various perturbation schemes. The proposed method is illustrated with an example.     

Models for Dependent Extremes Using Stable Mixtures

Abstract.  This paper unifies and extends results on a class of multivariate extreme value (EV) models studied by Hougaard, Crowder and Tawn. In these models, both unconditional and conditional distributions are themselves EV distributions, and all lower-dimensional marginals and maxima belong to the class. One interpretation of the models is as size mixtures of EV distributions, where the mixing is by positive stable distributions. A second interpretation is as exponential-stable location mixtures (for Gumbel) or as power-stable scale mixtures (for non-Gumbel EV distributions). A third interpretation is through a peaks over thresholds model with a positive stable intensity. The mixing variables are used as a modelling tool and for better understanding and model checking. We study EV analogues of components of variance models, and new time series, spatial and continuous parameter models for extreme values. The results are applied to data from a pitting corrosion investigation.     

Specification and Identification of Stochastic Demand Models

This paper is concerned with stochastic demand systems for continuous choices that arise from structural random utility models. It examines under which nonparametric conditions on the structural random utility specification the implied reduced form model is nonsingular and invertible. For parametric members within this class of random utility models, the paper provides conditions for local identification from the reduced form under moment assumptions.     

Specification and Identification of Stochastic Demand Models

This paper is concerned with stochastic demand systems for continuous choices that arise from structural random utility models. It examines under which nonparametric conditions on the structural random utility specification the implied reduced form model is nonsingular and invertible. For parametric members within this class of random utility models, the paper provides conditions for local identification from the reduced form under moment assumptions.