Predictive assessment of copula models

Copulas are powerful explanatory tools for studying dependence patterns in multivariate data. While the primary use of copula models is in multivariate dependence modelling, they also offer predictive value for regression analysis. This article investigates the utility of copula models for model‐based predictions from two angles. We assess whether, where, and by how much various copula models differ in their predictions of a conditional mean and conditional quantiles. From a model selection perspective, we then evaluate the predictive discrepancy between copula models using in‐sample and out‐of‐sample predictions both in bivariate and higher‐dimensional settings. Our findings suggest that some copula models are more difficult to distinguish in terms of their overall predictive power than others, and depending on the quantity of interest, the differences in predictions can be detected only in some targeted regions. The situations where copula‐based regression approaches would be advantageous over traditional ones are discussed using simulated and real data. The Canadian Journal of Statistics 47: 8–26; 2019 © 2018 Statistical Society of Canada     

Semiparametric estimation in copula models

The author recalls the limiting behaviour of the empirical copula process and applies it to prove some asymptotic properties of a minimum distance estimator for a Euclidean parameter in a copula model. The estimator in question is semiparametric in that no knowledge of the marginal distributions is necessary. The author also proposes another semiparametric estimator which he calls “rank approximate Z‐estimator” and whose asymptotic normality he derives. He further presents Monte Carlo simulation results for the comparison of various estimators in four well‐known bivariate copula models.     

Ridge estimation in generalized linear models and proportional hazards regressions

Conventionally, a ridge parameter is estimated as a function of regression parameters based on ordinary least squares. In this article, we proposed an iterative procedure instead of the one-step or conventional ridge method. Additionally, we construct an indicator that measures the potential degree of improvement in mean squared error when ridge estimates are employed. Simulations show that our methods are appropriate for a wide class of non linear models including generalized linear models and proportional hazards (PHs) regressions. The method is applied to a PH regression with highly collinear covariates in a cancer recurrence study.     

A GLM approach to step-stress accelerated life testing with interval censoring

In this paper, we present a statistical inference procedure for the step-stress accelerated life testing (SSALT) model with Weibull failure time distribution and interval censoring via the formulation of generalized linear model (GLM). The likelihood function of an interval censored SSALT is in general too complicated to obtain analytical results. However, by transforming the failure time to an exponential distribution and using a binomial random variable for failure counts occurred in inspection intervals, a GLM formulation with a complementary log-log link function can be constructed. The estimations of the regression coefficients used for the Weibull scale parameter are obtained through the iterative weighted least square (IWLS) method, and the shape parameter is updated by a direct maximum likelihood (ML) estimation. The confidence intervals for these parameters are estimated through bootstrapping. The application of the proposed GLM approach is demonstrated by an industrial example.     

Sample size determination for clustered right-censored data using copula models

Determination of an adequate sample size is critical to the design of research ventures. For clustered right-censored data, Manatunga and Chen [Sample size estimation for survival outcomes in cluster-randomized studies with small cluster sizes. Biometrics. 2000;56(2):616–621] proposed a sample size calculation based on considering the bivariate marginal distribution as Clayton copula model. In addition to the Clayton copula, other important family of copulas, such as Gumbel and Frank copulas are also well established in multivariate survival analysis. However, sample size calculation based on these assumptions has not been fully investigated yet. To broaden the scope of Manatunga and Chen [Sample size estimation for survival outcomes in cluster-randomized studies with small cluster sizes. Biometrics. 2000;56(2):616–621]'s research and achieve a more flexible sample size calculation for clustered right-censored data, we extended the work by assuming the marginal distribution as bivariate Gumbel and Frank copulas. We evaluate the performance of the proposed method and investigate the impacts of the accrual times, follow-up times and the within-clustered correlation effect of the study. The proposed method is applied to two real-world studies, and the R code is made available to users.     

Vine copula constructions of higher-dimensional dependent reliability systems

Copulas have proved to be very successful tools for the flexible modeling of dependence. Bivariate copulas have been deeply researched in recent years, while building higher-dimensional copulas is still recognized to be a difficult task. In this paper, we study the higher-dimensional dependent reliability systems using a type of decomposition called “vine,” by which a multivariate distribution can be decomposed into a cascade of bivariate copulas. Some equations of system reliability for parallel, series, and k-out-of-n systems are obtained and then decomposed based on C-vine and D-vine copulas. Finally, a shutdown system is considered to illustrate the results obtained in the paper.     

Doubly inflated Poisson model using Gaussian copula

Multivariate data are present in many research areas. Its analysis is challenging when assumptions of normality are violated and the data are discrete. The Poisson discrete data can be thought of as very common discrete type, but the inflated and the doubly inflated correspondence are gaining popularity (Sengupta, Chaganty, and Sabo 2015; Lee, Jung, and Jin 2009; Agarwal, Gelfand, and Citron-Pousty 2002).

Our aim is to build a statistical model that can be tractable and used to estimate the model parameters for the multivariate doubly inflated Poisson. To keep the correlation structure, we incorporate ideas from the copula distributions. A multivariate doubly inflated Poisson distribution using Gaussian copula is introduced. Data simulation and parameter estimation algorithms are also provided. Residual checks are carried out to assess any substantial biases. The model dimensionality has been increased to test the performance of the provided estimation method. All results show high-efficiency and promising outcomes in the modeling of discrete data and particularly the doubly inflated Poisson count type data, under a novel modified algorithm.     

S-estimator in partially linear regression models

In this paper, a robust estimator is proposed for partially linear regression models. We first estimate the nonparametric component using the penalized regression spline, then we construct an estimator of parametric component by using robust S-estimator. We propose an iterative algorithm to solve the proposed optimization problem, and introduce a robust generalized cross-validation to select the penalized parameter. Simulation studies and a real data analysis illustrate that the our proposed method is robust against outliers in the dataset or errors with heavy tails.     

Testing independence based on Bernstein empirical copula and copula density

In this paper we provide three nonparametric tests of independence between continuous random variables based on the Bernstein copula distribution function and the Bernstein copula density function. The first test is constructed based on a Cramér-von Mises divergence-type functional based on the empirical Bernstein copula process. The two other tests are based on the Bernstein copula density and use Cramér-von Mises and Kullback–Leibler divergence-type functionals, respectively. Furthermore, we study the asymptotic null distribution of each of these test statistics. Finally, we consider a Monte Carlo experiment to investigate the performance of our tests. In particular we examine their size and power which we compare with those of the classical nonparametric tests that are based on the empirical distribution function.     

Estimation of association parameters in copula models for bivariate left-truncated and right-censored data

We investigate the problem of estimating the association between two related survival variables when they follow a copula model and bivariate left-truncated and right-censored data are available. By expressing truncation probability as the functional of marginal survival functions, we propose a two-stage estimation procedure for estimating the parameters of Archimedean copulas. The asymptotic properties of the proposed estimators are established. Simulation studies are conducted to investigate the finite sample properties of the proposed estimators. The proposed method is applied to a bivariate RNA data.     

Penalized contrast estimation in functional linear models with circular data

Our aim is to estimate the unknown slope function in the functional linear model when the response Y is real and the random function X is a second-order stationary and periodic process. We obtain our estimator by minimizing a standard (and very simple) mean-square contrast on linear finite dimensional spaces spanned by trigonometric bases. Our approach provides a penalization procedure which allows to automatically select the adequate dimension, in a non-asymptotic point of view. In fact, we can show that our penalized estimator reaches the optimal (minimax) rate of convergence in the sense of the prediction error. We complete the theoretical results by a simulation study and a real example that illustrates how the procedure works in practice.     

Progressively Type-II censored conditionally N-ordered statistics from a unified elliptically contoured copula

Progressively Type-II censored conditionally N-ordered statistics (PCCOS-N) arising from iid random vectors X_i = (X¹_i, X²_i, …, X_i^p), i = 1, 2…, n, were investigated by Bairamov (2006 Bairamov, I. (2006). Progressive Type II censored order statistics for multivariate observations. J. Mult. Anal. 97:797–809.[Crossref], [Web of Science ®] , [Google Scholar]), with respect to the magnitudes of N(X_i), i = 1, 2, …, n, where N( · ) is a p-variate measurable function defined on the support set of X₁ satisfying certain regularity conditions and N(X_i) denotes the lifetime of the random vector X_i, i = 1, …, n. Under the PCCOS-N sampling scheme, n independent units are placed on a life-test and after the ith failure, R_i (i = 1, …, m) of the surviving units are removed at random from the remaining observations. In this article, we consider PCCOS-N arising from a vector with identical as well as non identical dependent components, jointly distributed according to a unified elliptically contoured copula (PCCOSDUECC-N). Results established here contain the previous results as particular cases. Illustrative examples and simulation studies show that PCCOSDUECC-N enables us to analyze the lifetime of several systems, including repairable systems and systems with standby components, more efficiently than PCCOS-N.     

Parameter estimation in nonlinear regression models 1

Fisher information contained in record values, inter-record times and their concomitants from a sample of fixed size is derived in general and explicit expressions are deduced for some specific known bivariate classes of distributions. A comparison between fixed sampling and inverse sampling schemes with equal number of records and concomitants is also carried out. We also consider parameter estimation based on bivariate records and a small simulation study is done.     

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.     

Piecewise Cox models with right-censored data

We study a general class of piecewise Cox models. We discuss the computation of the semi-parametric maximum likelihood estimates (SMLE) of the parameters, with right-censored data, and a simplified algorithm for the maximum partial likelihood estimates (MPLE). Our simulation study suggests that the relative efficiency of the PMLE of the parameter to the SMLE ranges from 96% to 99.9%, but the relative efficiency of the existing estimators of the baseline survival function to the SMLE ranges from 3% to 24%. Thus, the SMLE is much better than the existing estimators.     

Power and sample size calculation for paired right-censored data based on survival copula models

Sample size determination is essential during the planning phases of clinical trials. To calculate the required sample size for paired right-censored data, the structure of the within-paired correlations needs to be pre-specified. In this article, we consider using popular parametric copula models, including the Clayton, Gumbel, or Frank families, to model the distribution of joint survival times. Under each copula model, we derive a sample size formula based on the testing framework for rank-based tests and non-rank-based tests (i.e., logrank test and Kaplan–Meier statistic, respectively). We also investigate how the power or the sample size was affected by the choice of testing methods and copula model under different alternative hypotheses. In addition to this, we examine the impacts of paired-correlations, accrual times, follow-up times, and the loss to follow-up rates on sample size estimation. Finally, two real-world studies are used to illustrate our method and R code is available to the user.     

Sure independence screening for real medical Poisson data

The statistical modeling of big data bases constitutes one of the most challenging issues, especially nowadays. The issue is even more critical in case of a complicated correlation structure. Variable selection plays a vital role in statistical analysis of large data bases and many methods have been proposed so far to deal with the aforementioned problem. One of such methods is the Sure Independence Screening which has been introduced to reduce dimensionality to a relatively smaller scale. This method, though simple, produces remarkable results even under both ultra high dimensionality and big scale in terms of sample size problems. In this paper we dealt with the analysis of a big real medical data set assuming a Poisson regression model. We support the analysis by conducting simulated experiments taking into consideration the correlation structure of the design matrix.     

Bayesian model selection for D‐vine pair‐copula constructions

In recent years analyses of dependence structures using copulas have become more popular than the standard correlation analysis. Starting from Aas et al. ( 2009 ) regular vine pair‐copula constructions (PCCs) are considered the most flexible class of multivariate copulas. PCCs are involved objects but (conditional) independence present in data can simplify and reduce them significantly. In this paper the authors detect (conditional) independence in a particular vine PCC model based on bivariate t copulas by deriving and implementing a reversible jump Markov chain Monte Carlo algorithm. However, the methodology is general and can be extended to any regular vine PCC and to all known bivariate copula families. The proposed approach considers model selection and estimation problems for PCCs simultaneously. The effectiveness of the developed algorithm is shown in simulations and its usefulness is illustrated in two real data applications. The Canadian Journal of Statistics 39: 239–258; 2011 © 2011 Statistical Society of Canada     

Parametric models for response-biased sampling

Suppose that subjects in a population follow the model f   ( y ^* x ^*; ) where y ^* denotes a response, x ^* denotes a vector of covariates and is the parameter to be estimated. We consider response-biased sampling, in which a subject is observed with a probability which is a function of its response. Such response-biased sampling frequently occurs in econometrics, epidemiology and survey sampling. The semiparametric maximum likelihood estimate of is derived, along with its asymptotic normality, efficiency and variance estimates. The estimate proposed can be used as a maximum partial likelihood estimate in stratified response-selective sampling. Some computation algorithms are also provided.     

On the estimation of normal copula discrete regression models using the continuous extension and simulated likelihood

The continuous extension of a discrete random variable is amongst the computational methods used for estimation of multivariate normal copula-based models with discrete margins. Its advantage is that the likelihood can be derived conveniently under the theory for copula models with continuous margins, but there has not been a clear analysis of the adequacy of this method. We investigate the asymptotic and small-sample efficiency of two variants of the method for estimating the multivariate normal copula with univariate binary, Poisson, and negative binomial regressions, and show that they lead to biased estimates for the latent correlations, and the univariate marginal parameters that are not regression coefficients. We implement a maximum simulated likelihood method, which is based on evaluating the multidimensional integrals of the likelihood with randomized quasi-Monte Carlo methods. Asymptotic and small-sample efficiency calculations show that our method is nearly as efficient as maximum likelihood for fully specified multivariate normal copula-based models. An illustrative example is given to show the use of our simulated likelihood method.