Distributions with Generalized Skewed Conditionals and Mixtures of Such Distributions

Mixtures of skewed distributions (univariate and bivariate) provide flexible models. An alternative modeling approach involves distributions with skewed conditional distributions and mixtures of such distributions. We consider the interrelationships between such models. Examples are provided to show that several skewed distributions already considered in the literature can be viewed as having been constructed via a combination of mixing and skewing.     

Identifiability of Finite Mixtures of Elliptical Distributions

Abstract.  We present general results on the identifiability of finite mixtures of elliptical distributions under conditions on the characteristic generators or density generators. Examples include the multivariate t -distribution, symmetric stable laws, exponential power and Kotz distributions. In each case, the shape parameter is allowed to vary in the mixture, in addition to the location vector and the scatter matrix. Furthermore, we discuss the identifiability of finite mixtures of elliptical densities with generators that correspond to scale mixtures of normal distributions.     

非参数可加ACD模型及其应用研究

非参数可加ACD模型对条件期望的函数形式与随机误差项的分布形式要求都没有参数ACD模型强,因此不会像参数ACD模型那样因模型形式设定错误而得出错误结论。非参数可加ACD模型估计出来的各个可加部分图形的形状对于正确设定参数ACD模型具有一定的指导作用。     

Residual Life Estimation Based on a Generalized Wiener Process with Skew-normal Random Effects

For reliability-critical and expensive products, it is necessary to estimate their residual lives based on available information, such as the degradation data, so that proper maintenance actions can be arranged to reduce or even avoid the occurrence of failures. In this work, by assuming that the product-to-product variability of the degradation is characterized by a skew-normal distribution, a generalized Wiener process-based degradation model is developed. Following that, the issue of residual life (RL) estimation of the target product is addressed in detail. The proposed degradation model provides greater flexibility to capture a variety of degradation processes, since several commonly used Wiener process-based degradation models can be seen as special cases. Through the EM algorism, the population-based degradation information is used to estimate the parameters of the model. Whenever new degradation measurement information of the target product becomes available, the degradation model is first updated based on the Bayesian method. In this way, the RL of the target product can be estimated in an adaptive manner. Finally, the developed methodology is demonstrated by a simulation study.     

Bayesian Analysis of Skew Gaussian Spatial Models Based on Censored Data

The customary approach to spatial data modeling in the presence of censored data, is to assume the underlying random field is Gaussian. However, in practice, we often faced data that the exploratory data analysis shows the skewness and consequently, it violates the normality assumption. In such setting, the skew Gaussian (SG) spatial model is used to overcome this issue. In this article, the SG model is fitted based on censored observations. For this purpose, we adopt the Bayesian approach and utilize the Markov chain Monte Carlo algorithms and data augmentations to carry out calculations. A numerical example illustrates the methodology.     

Zero-inflated Poisson model with clustered regression coefficients: Application to heterogeneity learning of field goal attempts of professional basketball players

Although basketball is a dynamic process sport, played between two sides of five players each, learning some static information is essential for professional players, coaches, and team managers. In order to have a deep understanding of field goal attempts among different players, we propose a zero-inflated Poisson model with clustered regression coefficients to learn the shooting habits of different players over the court and the heterogeneity among them. Specifically, the zero-inflated model captures a large portion of the court with zero field goal attempts, and the mixture of finite mixtures model captures the heterogeneity among different players based on clustered regression coefficients and inflated probabilities. Both theoretical and empirical justification through simulation studies validate our proposed method. We apply our proposed model to data from the National Basketball Association (NBA), for learning players' shooting habits and heterogeneity among different players over the 2017–2018 regular season. This illustrates our model as a way of providing insights from different aspects.     

A multimodal skewed extension of normal distribution: its properties and applications

A multimodal skewed extension of normal distribution is proposed by applying the general method as in [Huang WJ, Chen YH. Generalized skew-Cauchy distribution. Stat Probab Lett. 2007;77:1137–1147] for the construction of skew-symmetric distributions by using a trigonometric periodic skew function. Some of its distributional properties are investigated. Properties of maximum likelihood estimation of the parameters are studied numerically by simulation. The suitability of the proposed distribution in empirical data modelling is investigated by carrying out comparative fitting of two real-life data sets.     

Nonparametric Tail Copula Estimation: An Application to Stock and Volatility Index Returns

In this study, we measure asymmetric negative tail dependence and discuss their statistical properties. In a simulation study, we show the reliability of nonparametric estimators of tail copula to measure not only the common positive lower and upper tail dependence, but also the negative “lower–upper” and “upper–lower” tail dependence. The use of this new framework is illustrated in an application to financial data. We detect the existence of asymmetric negative tail dependence between stock and volatility indices. Many common parametric copula models used in finance fail to capture this characteristic.     

An Application of U-Statistics to Nonparametric Functional Data Analysis

Functional regression functions, with explanatory variables taking values in some abstract function space, have been studied extensively. In this article, we aim to investigate the multivariate functional regression function, and propose a nonparametric estimator for the multivariate case. By applying some properties of U-statistics, some asymptotic distributions of such estimator are obtained under different cases.     

Distributions generated by perturbation of symmetry with emphasis on a multivariate skew t-distribution

Summary . A fairly general procedure is studied to perturb a multivariate density satisfying a weak form of multivariate symmetry, and to generate a whole set of non-symmetric densities. The approach is sufficiently general to encompass some recent proposals in the literature, variously related to the skew normal distribution. The special case of skew elliptical densities is examined in detail, establishing connections with existing similar work. The final part of the paper specializes further to a form of multivariate skew t -density. Likelihood inference for this distribution is examined, and it is illustrated with numerical examples.     

Estimation of drug effect in radiographic progression for rheumatoid arthritis and psoriatic arthritis

To demonstrate the treatment effect on structural damage in rheumatoid arthritis (RA) and psoriatic arthritis (PsA), radiographic images of hands and feet are scored according to Sharp scoring systems in randomized clinical trials. However, the quantification of such an effect is challenging because the overall mean progression is lack of clinical interpretation. This article attempts to shed a light on the statistical challenges resulted from its scoring methods and heterogeneity of the study population and proposes a mixture distribution model approach to fit radiographic progression data. With such a model, the drug effect is fully captured by the mean progression of those patients who would progress in the study period under the control treatment. The resulting regression model also lends a tool in examining prognostic factors for radiographic progression. Simulations have been carried out to evaluate the precision of the parameter estimation procedure. Using the data examples from RA and PsA, we will show that the mixture distribution approach provides a better goodness of fit and leads to a casual inference of the study drug, hence a clinically meaningful interpretation.     

Comparison of dichotomized and distributional approaches in rare event clinical trial design: a fixed Bayesian design

This research was motivated by our goal to design an efficient clinical trial to compare two doses of docosahexaenoic acid supplementation for reducing the rate of earliest preterm births (ePTB) and/or preterm births (PTB). Dichotomizing continuous gestational age (GA) data using a classic binomial distribution will result in a loss of information and reduced power. A distributional approach is an improved strategy to retain statistical power from the continuous distribution. However, appropriate distributions that fit the data properly, particularly in the tails, must be chosen, especially when the data are skewed. A recent study proposed a skew-normal method. We propose a three-component normal mixture model and introduce separate treatment effects at different components of GA. We evaluate operating characteristics of mixture model, beta-binomial model, and skew-normal model through simulation. We also apply these three methods to data from two completed clinical trials from the USA and Australia. Finite mixture models are shown to have favorable properties in PTB analysis but minimal benefit for ePTB analysis. Normal models on log-transformed data have the largest bias. Therefore we recommend finite mixture model for PTB study. Either finite mixture model or beta-binomial model is acceptable for ePTB study.     

Convergence of Known Distributions to Limiting Normal or Non-normal Distributions: An Elementary Ratio Technique

This article presents an elementary informal technique for deriving the convergence of known distributions to limiting normal or non-normal distributions. The presentation should be of interest to teachers and students of first year graduate level courses in probability and statistics.     

Strong Consistency of the Maximum Likelihood Estimator for Finite Mixtures of Location–Scale Distributions When Penalty is Imposed on the Ratios of the Scale Parameters

Abstract.  In finite mixtures of location–scale distributions, if there is no constraint or penalty on the parameters, then the maximum likelihood estimator does not exist because the likelihood is unbounded. To avoid this problem, we consider a penalized likelihood, where the penalty is a function of the minimum of the ratios of the scale parameters and the sample size. It is shown that the penalized maximum likelihood estimator is strongly consistent. We also analyse the consistency of a penalized maximum likelihood estimator where the penalty is imposed on the scale parameters themselves.     

Testing for a Change of the Innovation Distribution in Nonparametric Autoregression: The Sequential Empirical Process Approach

We consider a nonparametric autoregression model under conditional heteroscedasticity with the aim to test whether the innovation distribution changes in time. To this end, we develop an asymptotic expansion for the sequential empirical process of nonparametrically estimated innovations (residuals). We suggest a Kolmogorov–Smirnov statistic based on the difference of the estimated innovation distributions built from the first ?ns?and the last n ? ?ns? residuals, respectively (0 ≤ s ≤ 1). Weak convergence of the underlying stochastic process to a Gaussian process is proved under the null hypothesis of no change point. The result implies that the test is asymptotically distribution‐free. Consistency against fixed alternatives is shown. The small sample performance of the proposed test is investigated in a simulation study and the test is applied to a data example.     

Analysing the interevent time distribution to identify seismicity phases: a Bayesian nonparametric approach to the multiple-changepoint problem

In the study of earthquakes, several aspects of the underlying physical process, such as the time non-stationarity of the process, are not yet well understood, because we lack clear indications about its evolution in time. Taking as our point of departure the theory that the seismic process evolves in phases with different activity patterns, we have attempted to identify these phases through the variations in the interevent time probability distribution within the framework of the multiple-changepoint problem. In a nonparametric Bayesian setting, the distribution under examination has been considered a random realization from a mixture of Dirichlet processes, the parameter of which is proportional to a generalized gamma distribution. In this way we could avoid making precise assumptions about the functional form of the distribution. The number and location in time of the phases are unknown and are estimated at the same time as the interevent time distributions. We have analysed the sequence of main shocks that occurred in Irpinia, a particularly active area in southern Italy: the method consistently identifies changepoints at times when strong stress releases were recorded. The estimation problem can be solved by stochastic simulation methods based on Markov chains, the implementation of which is improved, in this case, by the good analytical properties of the Dirichlet process.     

Mixtures of concentrated multivariate sine distributions with applications to bioinformatics

Motivated by examples in protein bioinformatics, we study a mixture model of multivariate angular distributions. The distribution treated here (multivariate sine distribution) is a multivariate extension of the well-known von Mises distribution on the circle. The density of the sine distribution has an intractable normalizing constant and here we propose to replace it in the concentrated case by a simple approximation. We study the EM algorithm for this distribution and apply it to a practical example from protein bioinformatics.     

Stochastic Volatility Models Based on OU-Gamma Time Change: Theory and Estimation

We consider stochastic volatility models that are defined by an Ornstein–Uhlenbeck (OU)-Gamma time change. These models are most suitable for modeling financial time series and follow the general framework of the popular non-Gaussian OU models of Barndorff-Nielsen and Shephard. One current problem of these otherwise attractive nontrivial models is, in general, the unavailability of a tractable likelihood-based statistical analysis for the returns of financial assets, which requires the ability to sample from a nontrivial joint distribution. We show that an OU process driven by an infinite activity Gamma process, which is an OU-Gamma process, exhibits unique features, which allows one to explicitly describe and exactly sample from relevant joint distributions. This is a consequence of the OU structure and the calculus of Gamma and Dirichlet processes. We develop a particle marginal Metropolis–Hastings algorithm for this type of continuous-time stochastic volatility models and check its performance using simulated data. For illustration we finally fit the model to S&P500 index data.     

Semiparametric Bayesian hierarchical models for heterogeneous population in nonlinear mixed effect model: application to gastric emptying studies

Gastric emptying studies are frequently used in medical research, both human and animal, when evaluating the effectiveness and determining the unintended side-effects of new and existing medications, diets, and procedures or interventions. It is essential that gastric emptying data be appropriately summarized before making comparisons between study groups of interest and to allow study the comparisons. Since gastric emptying data have a nonlinear emptying curve and are longitudinal data, nonlinear mixed effect (NLME) models can accommodate both the variation among measurements within individuals and the individual-to-individual variation. However, the NLME model requires strong assumptions that are often not satisfied in real applications that involve a relatively small number of subjects, have heterogeneous measurement errors, or have large variation among subjects. Therefore, we propose three semiparametric Bayesian NLMEs constructed with Dirichlet process priors, which automatically cluster sub-populations and estimate heterogeneous measurement errors. To compare three semiparametric models with the parametric model we propose a penalized posterior Bayes factor. We compare the performance of our semiparametric hierarchical Bayesian approaches with that of the parametric Bayesian hierarchical approach. Simulation results suggest that our semiparametric approaches are more robust and flexible. Our gastric emptying studies from equine medicine are used to demonstrate the advantage of our approaches.