Estimating the association parameter for copula models under dependent censoring

Summary. Many biomedical studies involve the analysis of multiple events. The dependence between the times to these end points is often of scientific interest. We investigate a situation when one end point is subject to censoring by the other. The model assumptions of Day and co-workers and Fine and co-workers are extended to more general structures where the level of association may vary with time. Two types of estimating function are proposed. Asymptotic properties of the proposed estimators are derived. Their finite sample performance is studied via simulations. The inference procedures are applied to two real data sets for illustration.     

A copula‐graphic estimator for the conditional survival function under dependent censoring

Rivest Wells (2001) showed that in situations where the dependence between a lifetime and a censoring variable can be modeled by a given Archimedean copula, the copula‐graphic estimator of Zheng Klein (1995) has an explicit form. The authors extend this work to the fixed design regression case. They show that the copula‐graphic estimator then has an asymptotic representation and a Gaussian limit. They also assess the influence of a misspecified copula function on the performance of the estimator. Their developments are illustrated with data on the survival of the Atlantic halibut.     

Accelerated life regression modelling of dependent bivariate time‐to‐event data

To analyze bivariate time‐to‐event data from matched or naturally paired study designs, researchers frequently use a random effect called frailty to model the dependence between within‐pair response measurements. The authors propose a computational framework for fitting dependent bivariate time‐to‐event data that combines frailty distributions and accelerated life regression models. In this framework users can choose from several parametric options for frailties, as well as the conditional distributions for within‐pair responses. The authors illustrate the flexibility that their framework represents using paired data from a study of laser photocoagulation therapy for retinopathy in diabetic patients.     

Nonparametric estimation of the lower tail dependence λL in bivariate copulas

The lower tail dependence λ_L is a measure that characterizes the tendency of extreme co-movements in the lower tails of a bivariate distribution. It is invariant with respect to strictly increasing transformations of the marginal distribution and is therefore a function of the copula of the bivariate distribution. λ_L plays an important role in modelling aggregate financial risk with copulas. This paper introduces three non-parametric estimators for λ_L. They are weakly consistent under mild regularity conditions on the copula and under the assumption that the number k = k(n) of observations in the lower tail, used for estimation, is asymptotically k ≈ √n. The finite sample properties of the estimators are investigated using a Monte Carlo simulation in special cases. It turns out that these estimators are biased, where amount and sign of the bias depend on the underlying copula, on the sample size n, on k, and on the true value of λ_L.     

Pseudo‐likelihood ratio tests for semiparametric multivariate copula model selection

The authors propose pseudo‐likelihood ratio tests for selecting semiparametric multivariate copula models in which the marginal distributions are unspecified, but the copula function is parameterized and can be misspecified. For the comparison of two models, the tests differ depending on whether the two copulas are generalized nonnested or generalized nested. For more than two models, the procedure is built on the reality check test of White (2000). Unlike White (2000), however, the test statistic is automatically standardized for generalized nonnested models (with the benchmark) and ignores generalized nested models asymptotically. The authors illustrate their approach with American insurance claim data.     

Semiparametric Global Cross-ratio Models for Bivariate Censored Data

Abstract.  Multivariate correlated failure time data arise in many medical and scientific settings. In the analysis of such data, it is important to use models where the parameters have simple interpretations. In this paper, we formulate a model for bivariate survival data based on the Plackett distribution. The model is an alternative to the Gamma frailty model proposed by Clayton and Oakes. The parameter in this distribution has a very appealing odds ratio interpretation for dependence between the two failure times; in addition, it allows for negative dependence. We develop novel semiparametric estimation and inference procedures for the model. The asymptotic results of the estimator are developed. The performance of the proposed techniques in finite samples is examined using simulation studies; in addition, the proposed methods are applied to data from an observational study in cancer.     

Two-Stage Estimation in Copula Models Used in Family Studies

In this paper register based family studies provide the motivation for studying a two-stage estimation procedure in copula models for multivariate failure time data. The asymptotic properties of the estimators in both parametric and semi-parametric models are derived, generalising the approach by Shih and Louis (Biometrics vol. 51, pp. 1384–1399, 1995b) and Glidden (Lifetime Data Analysis vol. 6, pp. 141–156, 2000). Because register based family studies often involve very large cohorts a method for analysing a sampled cohort is also derived together with the asymptotic properties of the estimators. The proposed methods are studied in simulations and the estimators are found to be highly efficient. Finally, the methods are applied to a study of mortality in twins.     

Dynamic Copula-Based Markov Time Series

This article examines a test procedure for checking the constancy of serial dependence via copulas for Markov time series data. It also provides a copula-based modeling approach for the dynamic serial dependence. Various parametric families of copulas offering different dependent structures are investigated. A score test is proposed for checking the constancy of a copula parameter. The score test is constructed and its asymptotic null distribution established under a two-stage estimation procedure. The test does not require specification of the probability distribution for the copula parameter. To capture the dynamics of dependence structure over time, autoregressive moving average and exponential type models are proposed. Illustrations are given based on simulated data and historic coffee prices data.     

Flexible Bivariate INAR(1) Processes Using Copulas

Multivariate count time series data occur in many different disciplines. The class of INteger-valued AutoRegressive (INAR) processes has the great advantage to consider explicitly both the discreteness and autocorrelation characterizing this type of data. Moreover, extensions of the simple INAR(1) model to the multi-dimensional space make it possible to model more than one series simultaneously. However, existing models do not offer great flexibility for dependence modelling, allowing only for positive correlation. In this work, we consider a bivariate INAR(1) (BINAR(1)) process where cross-correlation is introduced through the use of copulas for the specification of the joint distribution of the innovations. We mainly emphasize on the parametric case that arises under the assumption of Poisson marginals. Other marginal distributions are also considered. A short application on a bivariate financial count series illustrates the model.     

Sequential order statistics from dependent random variables

Abstract

This paper provides an extension for “sequential order statistics” (SOS) introduced by Kamps. It is called “developed sequential order statistics” (DSOS) and is useful for describing lifetimes of engineering systems when component lifetimes are dependent. Explicit expressions for the joint density function, the marginal distributions and the means of DSOS are derived. Under the well known “conditional proportional hazard rate” (CPHR) model and the Gumbel families of copulas for dependency among component lifetimes, some findings are reported. For example, it is proved that the joint density functions of DSOS and SOS have the same structure. Various illustrative examples are also given.