The copula directional dependence by stochastic volatility models

This paper proposes a copula directional dependence by using a bivariate Gaussian copula beta regression with Stochastic Volatility (SV) models for marginal distributions. With the asymmetric copula generated by the composition of two Plackett copulas, we show that our SV copula directional dependence by the Gaussian copula beta regression model is superior to the Kim and Hwang (2016) copula directional dependence by an asymmetric GARCH model in terms of the percent relative efficiency of bias and mean squared error. To validate our proposed method with the real data, we use Brent Crude Daily Price (BRENT), West Texas Intermediate Daily Price (WTI), the Standard & Poor’s 500 (SP) and US 10-Year Treasury Constant Maturity Rate (TCM) so that our copula SV directional dependence is overall superior to the Kim and Hwang (2016) copula directional dependence by an asymmetric GARCH model in terms of precision by the percent relative efficiency of mean squared error. In terms of forecasting using the real financial data, we also show that the Bayesian SV model of the uniform transformed data by a copula conditional distribution yields an improvement on the volatility models such as GARCH and SV.     

Mixtures of Gaussian copula factor analyzers for clustering high dimensional data

Mixtures of factor analyzers is a useful model-based clustering method which can avoid the curse of dimensionality in high-dimensional clustering. However, this approach is sensitive to both diverse non-normalities of marginal variables and outliers, which are commonly observed in multivariate experiments. We propose mixtures of Gaussian copula factor analyzers (MGCFA) for clustering high-dimensional clustering. This model has two advantages; (1) it allows different marginal distributions to facilitate fitting flexibility of the mixture model, (2) it can avoid the curse of dimensionality by embedding the factor-analytic structure in the component-correlation matrices of the mixture distribution.An EM algorithm is developed for the fitting of MGCFA. The proposed method is free of the curse of dimensionality and allows any parametric marginal distribution which fits best to the data. It is applied to both synthetic data and a microarray gene expression data for clustering and shows its better performance over several existing methods.     

Directional dependence via Gaussian copula beta regression model with asymmetric GARCH marginals

This article proposes a new directional dependence by using the Gaussian copula beta regression model. In particular, we consider an asymmetric Generalized AutoRegressive Conditional Heteroscedasticity (GARCH) model for the marginal distribution of standardized residuals to make data exhibiting conditionally heteroscedasticity to white noise process. With the simulated data generated by an asymmetric bivariate copula, we verify our proposed directional dependence method. For the multivariate direction dependence by using the Gaussian copula beta regression model, we employ a three-dimensional archemedian copula to generate trivariate data and then show the directional dependence for one random variable given two other random variables. With West Texas Intermediate Daily Price (WTI) and the Standard & Poor’s 500 (S&P 500), our proposed directional dependence by the Gaussian copula beta regression model reveals that the directional dependence from WTI to S&P 500 is greater than that from S&P 500 to WTI. To validate our empirical result, the Granger causality test is conducted, confirming the same result produced by our method.     

基于copula回归模型的损失预测

在非寿险损失预测的广义线性模型中,通常假设损失次数与损失强度相互独立,事实上二者之间往往存在一定的相依关系,可通过copula函数来刻画.在损失已经发生的条件下,假设损失次数服从零截断泊松分布,损失强度服从伽玛分布,可以建立损失次数与损失强度相互依赖的copula回归模型.把损失强度的分布扩展到逆高斯分布,并将此模型应用于一组车险保单数据进行实证研究.结果表明:该模型不但在损失预测方面优于独立假设下的广义线性模型,而且也优于损失强度服从伽马分布假设下的copula回归模型.     

On the Correlation Structure of Gaussian Copula Models for Geostatistical Count Data

We describe a class of random field models for geostatistical count data based on Gaussian copulas. Unlike hierarchical Poisson models often used to describe this type of data, Gaussian copula models allow a more direct modelling of the marginal distributions and association structure of the count data. We study in detail the correlation structure of these random fields when the family of marginal distributions is either negative binomial or zero‐inflated Poisson; these represent two types of overdispersion often encountered in geostatistical count data. We also contrast the correlation structure of one of these Gaussian copula models with that of a hierarchical Poisson model having the same family of marginal distributions, and show that the former is more flexible than the latter in terms of range of feasible correlation, sensitivity to the mean function and modelling of isotropy. An exploratory analysis of a dataset of Japanese beetle larvae counts illustrate some of the findings. All of these investigations show that Gaussian copula models are useful alternatives to hierarchical Poisson models, specially for geostatistical count data that display substantial correlation and small overdispersion.     

Bivariate log Birnbaum–Saunders distribution

Univariate Birnbaum–Saunders distribution has received a considerable amount of attention in recent years. Rieck and Nedelman (A log-linear model for the Birnbaum–Saunders distribution. Technometrics, 1991;33:51–60) introduced a log Birnbaum–Saunders distribution. The main aim of this paper is to introduce bivariate log Birnbaum–Saunders distribution. The proposed model is symmetric and it has five parameters. It can be obtained using Gaussian copula. Different properties can be obtained using copula structure. It is observed that the maximum likelihood estimators (MLEs) cannot be obtained explicitly. Two-dimensional profile likelihood approach may be adopted to compute the MLEs. We propose some alternative estimators also, which can be obtained quite conveniently. The analysis of one data set is performed for illustrative purposes. Finally, it is observed that this model can be used as a bivariate log-linear model, and its multivariate generalization is also quite straight forward.     

Eliciting Dirichlet and Gaussian copula prior distributions for multinomial models

In this paper, we propose novel methods of quantifying expert opinion about prior distributions for multinomial models. Two different multivariate priors are elicited using median and quartile assessments of the multinomial probabilities. First, we start by eliciting a univariate beta distribution for the probability of each category. Then we elicit the hyperparameters of the Dirichlet distribution, as a tractable conjugate prior, from those of the univariate betas through various forms of reconciliation using least-squares techniques. However, a multivariate copula function will give a more flexible correlation structure between multinomial parameters if it is used as their multivariate prior distribution. So, second, we use beta marginal distributions to construct a Gaussian copula as a multivariate normal distribution function that binds these marginals and expresses the dependence structure between them. The proposed method elicits a positive-definite correlation matrix of this Gaussian copula. The two proposed methods are designed to be used through interactive graphical software written in Java.     

A taxonomy of mixing and outcome distributions based on conjugacy and bridging

Abstract

The generalized linear mixed model (GLMM) is commonly used for the analysis of hierarchical non Gaussian data. It combines an exponential family model formulation with normally distributed random effects. A drawback is the difficulty of deriving convenient marginal mean functions with straightforward parametric interpretations. Several solutions have been proposed, including the marginalized multilevel model (directly formulating the marginal mean, together with a hierarchical association structure) and the bridging approach (choosing the random-effects distribution such that marginal and hierarchical mean functions share functional forms). Another approach, useful in both a Bayesian and a maximum-likelihood setting, is to choose a random-effects distribution that is conjugate to the outcome distribution. In this paper, we contrast the bridging and conjugate approaches. For binary outcomes, using characteristic functions and cumulant generating functions, it is shown that the bridge distribution is unique. Self-bridging is introduced as the situation in which the outcome and random-effects distributions are the same. It is shown that only the Gaussian and degenerate distributions have well-defined cumulant generating functions for which self-bridging holds.     

Simultaneous inference in structured additive conditional copula regression models: a unifying Bayesian approach

While most regression models focus on explaining distributional aspects of one single response variable alone, interest in modern statistical applications has recently shifted towards simultaneously studying multiple response variables as well as their dependence structure. A particularly useful tool for pursuing such an analysis are copula-based regression models since they enable the separation of the marginal response distributions and the dependence structure summarised in a specific copula model. However, so far copula-based regression models have mostly been relying on two-step approaches where the marginal distributions are determined first whereas the copula structure is studied in a second step after plugging in the estimated marginal distributions. Moreover, the parameters of the copula are mostly treated as a constant not related to covariates and most regression specifications for the marginals are restricted to purely linear predictors. We therefore propose simultaneous Bayesian inference for both the marginal distributions and the copula using computationally efficient Markov chain Monte Carlo simulation techniques. In addition, we replace the commonly used linear predictor by a generic structured additive predictor comprising for example nonlinear effects of continuous covariates, spatial effects or random effects and furthermore allow to make the copula parameters covariate-dependent. To facilitate Bayesian inference, we construct proposal densities for a Metropolis–Hastings algorithm relying on quadratic approximations to the full conditionals of regression coefficients avoiding manual tuning. The performance of the resulting Bayesian estimates is evaluated in simulations comparing our approach with penalised likelihood inference, studying the choice of a specific copula model based on the deviance information criterion, and comparing a simultaneous approach with a two-step procedure. Furthermore, the flexibility of Bayesian conditional copula regression models is illustrated in two applications on childhood undernutrition and macroecology.     

Semiparametric Estimation in Copulas with the Same Marginals

We consider semiparametric multivariate data models based on copula representation of the common distribution function. A copula is characterized by a parameter of association and marginal distribution functions. This parameter and the marginal distributions are unknown. In this article, we study the estimator of the parameter of association in copulas with the marginal distribution functions assumed as nuisance parameters restricted by the assumption that the components are identically distributed. Results of this work could be used to construct special kinds of tests of homogeneity for random vectors having dependent components.     

Bayesian nonparametric estimation of a copula

A copula can fully characterize the dependence of multiple variables. The purpose of this paper is to provide a Bayesian nonparametric approach to the estimation of a copula, and we do this by mixing over a class of parametric copulas. In particular, we show that any bivariate copula density can be arbitrarily accurately approximated by an infinite mixture of Gaussian copula density functions. The model can be estimated by Markov Chain Monte Carlo methods and the model is demonstrated on both simulated and real data sets.     

Copula‐based semiparametric analysis for time series data with detection limits

The analysis of time series data with detection limits is challenging due to the high‐dimensional integral involved in the likelihood. Existing methods are either computationally demanding or rely on restrictive parametric distributional assumptions. We propose a semiparametric approach, where the temporal dependence is captured by parametric copula, while the marginal distribution is estimated non‐parametrically. Utilizing the properties of copulas, we develop a new copula‐based sequential sampling algorithm, which provides a convenient way to calculate the censored likelihood. Even without full parametric distributional assumptions, the proposed method still allows us to efficiently compute the conditional quantiles of the censored response at a future time point, and thus construct both point and interval predictions. We establish the asymptotic properties of the proposed pseudo maximum likelihood estimator, and demonstrate through simulation and the analysis of a water quality data that the proposed method is more flexible and leads to more accurate predictions than Gaussian‐based methods for non‐normal data. The Canadian Journal of Statistics 47: 438–454; 2019 © 2019 Statistical Society of Canada     

Modeling binary familial data using Gaussian copula

Modeling binary familial data has been a challenging task due to the dependence among family members and the constraints imposed on the joint probability distribution of the binary responses. This paper investigates some useful familial dependence structures and proposes analyzing binary familial data using Gaussian copula model. Advantages of this approach are discussed as well as some computational details. An numerical example is also presented with an aim to show the capability of Gaussian copula model in more sophisticated data analysis.     

A bivariate model of claim frequencies and severities

Bivariate claim data come from a population that consists of insureds who may claim either one, both or none of the two types of benefits covered by a policy. In the present paper, we develop a statistical procedure to fit bivariate distributions of claims in presence of covariates. This allows for a more accurate study of insureds' choice and size in the frequency and severity of the two types of claims. A generalised logistic model is employed to examine the frequency probabilities, whilst the three parameter Burr distribution is suggested to model the underlying severity distributions. The bivariate copula model is exploited in such a way that it allows us to adjust for a range of frequency dependence structures; a method for assessing the adequacy of the fitted severity model is outlined. A health claims dataset illustrates the methods; we describe the use of orthogonal polynomials for characterising the relationship between age and the frequency and severity models.     

Goodness-of-fit tests for binomial AR(1) processes

The binomial AR(1) model describes a nonlinear process with a first-order autoregressive (AR(1)) structure and a binomial marginal distribution. To develop goodness-of-fit tests for the binomial AR(1) model, we investigate the observed marginal distribution of the binomial AR(1) process, and we tackle its autocorrelation structure. Motivated by the family of power-divergence statistics for handling discrete multivariate data, we derive the asymptotic distribution of certain categorized power-divergence statistics for the case of a binomial AR(1) process. Then we consider Bartlett's formula, which is widely used in time series analysis to provide estimates of the asymptotic covariance between sample autocorrelations, but which is not applicable when the underlying process is nonlinear. Hence, we derive a novel Bartlett-type formula for the asymptotic distribution of the sample autocorrelations of a binomial AR(1) process, which is then applied to develop tests concerning the autocorrelation structure. Simulation studies are carried out to evaluate the size and power of the proposed tests under diverse alternative process models. Several real examples are used to illustrate our methods and findings.     

Bayesian bivariate survival analysis using the power variance function copula

Copula models have become increasingly popular for modelling the dependence structure in multivariate survival data. The two-parameter Archimedean family of Power Variance Function (PVF) copulas includes the Clayton, Positive Stable (Gumbel) and Inverse Gaussian copulas as special or limiting cases, thus offers a unified approach to fitting these important copulas. Two-stage frequentist procedures for estimating the marginal distributions and the PVF copula have been suggested by Andersen (Lifetime Data Anal 11:333–350, 2005), Massonnet et al. (J Stat Plann Inference 139(11):3865–3877, 2009) and Prenen et al. (J R Stat Soc Ser B 79(2):483–505, 2017) which first estimate the marginal distributions and conditional on these in a second step to estimate the PVF copula parameters. Here we explore an one-stage Bayesian approach that simultaneously estimates the marginal and the PVF copula parameters. For the marginal distributions, we consider both parametric as well as semiparametric models. We propose a new method to simulate uniform pairs with PVF dependence structure based on conditional sampling for copulas and on numerical approximation to solve a target equation. In a simulation study, small sample properties of the Bayesian estimators are explored. We illustrate the usefulness of the methodology using data on times to appendectomy for adult twins in the Australian NH&MRC Twin registry. Parameters of the marginal distributions and the PVF copula are simultaneously estimated in a parametric as well as a semiparametric approach where the marginal distributions are modelled using Weibull and piecewise exponential distributions, respectively.     

Copula-Based Models for the Power of Independence Tests

ABSTRACT

We consider independence tests and the methods to evaluate their efficiency. First, we observe that many of the most used independence tests are functions of the empirical copula, which is a sufficient statistic. Hence, the power of these tests, such as the tests based on Spearman's ρ, on Kendall's τ, and on Gini's γ, depend solely on the theoretical copula, and not on the marginal distributions. Then, we consider monotone dependence tests and we propose a parametric model to define the power function. Such a model is based on a path of copulas, from the copula of discordance to the copula of concordance, and can be characterized by the copula of the underlying joint distribution. Moreover, we introduce a consistent estimator of the path of copulas. Finally, we provide some examples of applications, and in particular, a bootstrap-plug-in estimator of the power curve, all useful for power comparison.     

Monitoring correlated processes with binomial marginals

Few approaches for monitoring autocorrelated attribute data have been proposed in the literature. If the marginal process distribution is binomial, then the binomial AR(1) model as a realistic and well-interpretable process model may be adequate. Based on known and newly derived statistical properties of this model, we shall develop approaches to monitor a binomial AR(1) process, and investigate their performance in a simulation study. A case study demonstrates the applicability of the binomial AR(1) model and of the proposed control charts to problems from statistical process control.     

Multiple event times in the presence of informative censoring: modeling and analysis by copulas

Motivated by a breast cancer research program, this paper is concerned with the joint survivor function of multiple event times when their observations are subject to informative censoring caused by a terminating event. We formulate the correlation of the multiple event times together with the time to the terminating event by an Archimedean copula to account for the informative censoring. Adapting the widely used two-stage procedure under a copula model, we propose an easy-to-implement pseudo-likelihood based procedure for estimating the model parameters. The approach yields a new estimator for the marginal distribution of a single event time with semicompeting-risks data. We conduct both asymptotics and simulation studies to examine the proposed approach in consistency, efficiency, and robustness. Data from the breast cancer program are employed to illustrate this research.

     

CONDITIONAL EXPECTATION FORMULAE FOR COPULAS

Not only are copula functions joint distribution functions in their own right, they also provide a link between multivariate distributions and their lower‐dimensional marginal distributions. Copulas have a structure that allows us to characterize all possible multivariate distributions, and therefore they have the potential to be a very useful statistical tool. Although copulas can be traced back to 1959, there is still much scope for new results, as most of the early work was theoretical rather than practical. We focus on simple practical tools based on conditional expectation, because such tools are not widely available. When dealing with data sets in which the dependence throughout the sample is variable, we suggest that copula‐based regression curves may be more accurate predictors of specific outcomes than linear models. We derive simple conditional expectation formulae in terms of copulas and apply them to a combination of simulated and real data.