Investigating the correlation structure of quadrivariate udder infection times through hierarchical Archimedean copulas

The correlation structure imposed on multivariate time to event data is often of a simple nature, such as in the shared frailty model where pairwise correlations between event times in a cluster are all the same. In modeling the infection times of the four udder quarters clustered within the cow, more complex correlation structures are possibly required, and if so, such more complex correlation structures give more insight in the infection process. In this article, we will choose a marginal approach to study more complex correlation structures, therefore leaving the modeling of marginal distributions unaffected by the association parameters. The dependency of failure times will be induced through copula functions. The methods are shown for (mixtures of) the Clayton copula, but can be generalized to mixtures of Archimedean copulas for which the nesting conditions are met (McNeil in J Stat Comput Simul 6:567–581, 2008; Hofert in Comput Stat Data Anal 55:57–70, 2011).     

Conditioning of Marked Point Processes within a Bayesian Framework

Shale units with low permeability create barriers to fluid flow in a sandstone reservoir. A spatial stochastic model for the location of shale units in a reservoir is defined. The model is based on a marked point process formulation, where the marks are parameterized by random functions for the shape of a shale unit. This extends the traditional formulation in the sense that conditioning on the actual observations of the shale units is allowed in an arbitrary number of wells penetrating the reservoir. The marked point process for the shale units includes spatial interaction of units and allows a random number of units to be present. The model is defined in a Bayesian setting with prior pdfs assigned to size–shape parameters of shale units. The observations of shales in wells are associated with a likelihood function. The posterior pdf of the marked point process can only partially be developed analytically; the final solution must be determined by sampling using the Metropolis–Hastings algorithm. An example is presented, demonstrating the consequences of increasing the number of wells in which observations are made.     

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.     

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.

     

Flexible Modeling in the Koziol-Green Model by a Copula Function

In survival analysis, the classical Koziol-Green random censorship model is commonly used to describe informative censoring. Hereby, it is assumed that the distribution of the censoring time is a power of the distribution of the survival time. In this article, we extend this model by assuming a general function between these distributions. We determine this function from a relationship between the observable random variables which is described by a copula family that depends on an unknown parameter θ. For this setting, we develop a semi-parametric estimator for the distribution of the survival time in which we propose a pseudo-likelihood estimator for the copula parameter θ. As results, we show first the consistency and asymptotic normality of the estimator for θ. Afterwards, we prove the weak convergence of the process associated to the semi-parametric distribution estimator. Furthermore, we investigate the finite sample performance of these estimators through a simulation study and finally apply it to a practical data set on survival with malignant melanoma.     

State estimation for aoristic models

Aoristic data can be described by a marked point process in time in which the points cannot be observed directly but are known to lie in observable intervals, the marks. We consider Bayesian state estimation for the latent points when the marks are modeled in terms of an alternating renewal process in equilibrium and the prior is a Markov point process. We derive the posterior distribution, estimate its parameters and present some examples that illustrate the influence of the prior distribution. The model is then used to estimate times of occurrence of interval censored crimes.     

NONPARAMETRIC ESTIMATION OF CONDITIONAL CUMULATIVE HAZARDS FOR MISSING POPULATION MARKS

A new function for the competing risks model, the conditional cumulative hazard function, is introduced, from which the conditional distribution of failure times of individuals failing due to cause  j  can be studied. The standard Nelson–Aalen estimator is not appropriate in this setting, as population membership (mark) information may be missing for some individuals owing to random right-censoring. We propose the use of imputed population marks for the censored individuals through fractional risk sets. Some asymptotic properties, including uniform strong consistency, are established. We study the practical performance of this estimator through simulation studies and apply it to a real data set for illustration.     

Joint modelling of longitudinal biomarker and gap time between recurrent events: copula-based dependence

In this paper, we will extend the joint model of longitudinal biomarker and recurrent event via copula function for accounting the dependence between the two processes. The general idea of joining separate processes by allowing model-specific random effect may come from different families distribution. It is a main advantage of the proposed method that a copula construction does not constrain the choice of marginal distributions of random effects. A maximum likelihood estimation with importance sampling technique as a simple and easy understanding method is employed to model inference. To evaluate and verify the validation of the proposed joint model, a bootstrapping method as a model-based resampling is developed. Our proposed joint model is also applied to pemphigus disease data for assessing the effect of biomarker trajectory on risk of recurrence.     

Nonparametric estimation of copula functions for dependence modelling

Copulas characterize the dependence among components of random vectors. Unlike marginal and joint distributions, which are directly observable, the copula of a random vector is a hidden dependence structure that links the joint distribution with its margins. Choosing a parametric copula model is thus a nontrivial task but it can be facilitated by relying on a nonparametric estimator. Here the authors propose a kernel estimator of the copula that is mean square consistent everywhere on the support. They determine the bias and variance of this estimator. They also study the effects of kernel smoothing on copula estimation. They then propose a smoothing bandwidth selection rule based on the derived bias and variance. After confirming their theoretical findings through simulations, they use their kernel estimator to formulate a goodness-of-fit test for parametric copula models.     

A Method for Bayesian Monotonic Multiple Regression

Abstract. When applicable, an assumed monotonicity property of the regression function w.r.t. covariates has a strong stabilizing effect on the estimates. Because of this, other parametric or structural assumptions may not be needed at all. Although monotonic regression in one dimension is well studied, the question remains whether one can find computationally feasible generalizations to multiple dimensions. Here, we propose a non‐parametric monotonic regression model for one or more covariates and a Bayesian estimation procedure. The monotonic construction is based on marked point processes, where the random point locations and the associated marks (function levels) together form piecewise constant realizations of the regression surfaces. The actual inference is based on model‐averaged results over the realizations. The monotonicity of the construction is enforced by partial ordering constraints, which allows it to asymptotically, with increasing density of support points, approximate the family of all monotonic bounded continuous functions.     

Dynamic Random Effects Models for Times Between Repeated Events

We consider recurrent event data when the duration or gap times between successive event occurrences are of intrinsic interest. Subject heterogeneity not attributed to observed covariates is usually handled by random effects which result in an exchangeable correlation structure for the gap times of a subject. Recently, efforts have been put into relaxing this restriction to allow non-exchangeable correlation. Here we consider dynamic models where random effects can vary stochastically over the gap times. We extend the traditional Gaussian variance components models and evaluate a previously proposed proportional hazards model through a simulation study and some examples. Besides, semiparametric estimation of the proportional hazards models is considered. Both models are easily used. The Gaussian models are easily interpreted in terms of the variance structure. On the other hand, the proportional hazards models would be more appropriate in the context of survival analysis, particularly in the interpretation of the regression parameters. They can be sensitive to the choice of model for random effects but not to the choice of the baseline hazard function.     

Mixed-Poisson Point Process with Partially-Observed Covariates: Ecological Momentary Assessment of Smoking

Ecological Momentary Assessment is an emerging method of data collection in behavioral research that may be used to capture the times of repeated behavioral events on electronic devices, and information on subjects' psychological states through the electronic administration of questionnaires at times selected from a probability-based design as well as the event times. A method for fitting a mixed Poisson point process model is proposed for the impact of partially-observed, time-varying covariates on the timing of repeated behavioral events. A random frailty is included in the point-process intensity to describe variation among subjects in baseline rates of event occurrence. Covariate coefficients are estimated using estimating equations constructed by replacing the integrated intensity in the Poisson score equations with a design-unbiased estimator. An estimator is also proposed for the variance of the random frailties. Our estimators are robust in the sense that no model assumptions are made regarding the distribution of the time-varying covariates or the distribution of the random effects. However, subject effects are estimated under gamma frailties using an approximate hierarchical likelihood. The proposed approach is illustrated using smoking data.     

A bayesian estimator for the dependence function of a bivariate extreme‐value distribution

Any continuous bivariate distribution can be expressed in terms of its margins and a unique copula. In the case of extreme‐value distributions, the copula is characterized by a dependence function while each margin depends on three parameters. The authors propose a Bayesian approach for the simultaneous estimation of the dependence function and the parameters defining the margins. They describe a nonparametric model for the dependence function and a reversible jump Markov chain Monte Carlo algorithm for the computation of the Bayesian estimator. They show through simulations that their estimator has a smaller mean integrated squared error than classical nonparametric estimators, especially in small samples. They illustrate their approach on a hydrological data set.     

Maximum likelihood analysis of semicompeting risks data with semiparametric regression models

The “semicompeting risks” include a terminal event and a non-terminal event. The terminal event may censor the non-terminal event but not vice versa. Because times to the two events are usually correlated, the non-terminal event is subject to dependent/informative censoring by the terminal event. We seek to conduct marginal regressions and joint association analyses for the two event times under semicompeting risks. The proposed method is based on the modeling setup where the semiparametric transformation models are assumed for marginal regressions, and a copula model is assumed for the joint distribution. We propose a nonparametric maximum likelihood approach for inferences, which provides a martingale representation for the score function and an analytical expression for the information matrix. Direct theoretical developments and computational implementation are allowed for the proposed approach. Simulations and a real data application demonstrate the utility of the proposed methodology.     

Functional approach of flexibly modelling generalized longitudinal data and survival time

We propose a flexible functional approach for modelling generalized longitudinal data and survival time using principal components. In the proposed model the longitudinal observations can be continuous or categorical data, such as Gaussian, binomial or Poisson outcomes. We generalize the traditional joint models that treat categorical data as continuous data by using some transformations, such as CD4 counts. The proposed model is data-adaptive, which does not require pre-specified functional forms for longitudinal trajectories and automatically detects characteristic patterns. The longitudinal trajectories observed with measurement error or random error are represented by flexible basis functions through a possibly nonlinear link function, combining dimension reduction techniques resulting from functional principal component (FPC) analysis. The relationship between the longitudinal process and event history is assessed using a Cox regression model. Although the proposed model inherits the flexibility of non-parametric methods, the estimation procedure based on the EM algorithm is still parametric in computation, and thus simple and easy to implement. The computation is simplified by dimension reduction for random coefficients or FPC scores. An iterative selection procedure based on Akaike information criterion (AIC) is proposed to choose the tuning parameters, such as the knots of spline basis and the number of FPCs, so that appropriate degree of smoothness and fluctuation can be addressed. The effectiveness of the proposed approach is illustrated through a simulation study, followed by an application to longitudinal CD4 counts and survival data which were collected in a recent clinical trial to compare the efficiency and safety of two antiretroviral drugs.     

Semiparametric estimation method for accelerated failure time model with dependent censoring

Independent censoring is commonly assumed in survival analysis. However, it may be questionable when censoring is related to event time. We model the event and censoring time marginally through accelerated failure time models, and model their association by a known copula. An iteration algorithm is proposed to estimate the regression parameters. Simulation results show the improvement of the proposed method compared to the naive method under independent censoring. Sensitivity analysis gives the evidences that the proposed method can obtain reasonable estimates even when the forms of copula are misspecified. We illustrate its application by analyzing prostate cancer data.     

Measures of non-exchangeability for bivariate random vectors

We introduce a set of axioms for measures of non-exchangeability for bivariate vectors of continuous and identically distributed random variables and give some examples together with possible applications in statistical models based on the copula function.     

Symmetry of functions and exchangeability of random variables

We present a new approach for measuring the degree of exchangeability of two continuous, identically distributed random variables or, equivalently, the degree of symmetry of their corresponding copula. While the opposite of exchangeability does not exist in probability theory, the contrary of symmetry is quite obvious from an analytical point of view. Therefore, leaving the framework of probability theory, we introduce a natural measure of symmetry for bivariate functions in an arbitrary normed function space. Restricted to the set of copulas this yields a general concept for measures of (non-)exchangeability of random variables. The fact that copulas are never antisymmetric leads to the notion of maximal degree of antisymmetry of copulas. We illustrate our approach by various norms on function spaces, most notably the Sobolev norm for copulas.     

Modeling of semi-competing risks by means of first passage times of a stochastic process

In semi-competing risks one considers a terminal event, such as death of a person, and a non-terminal event, such as disease recurrence. We present a model where the time to the terminal event is the first passage time to a fixed level c in a stochastic process, while the time to the non-terminal event is represented by the first passage time of the same process to a stochastic threshold S, assumed to be independent of the stochastic process. In order to be explicit, we let the stochastic process be a gamma process, but other processes with independent increments may alternatively be used. For semi-competing risks this appears to be a new modeling approach, being an alternative to traditional approaches based on illness-death models and copula models. In this paper we consider a fully parametric approach. The likelihood function is derived and statistical inference in the model is illustrated on both simulated and real data.     

Statistical Modeling of Temporal Dependence in Financial Data via a Copula Function

In financial analysis it is useful to study the dependence between two or more time series as well as the temporal dependence in a univariate time series. This article is concerned with the statistical modeling of the dependence structure in a univariate financial time series using the concept of copula. We treat the series of financial returns as a first order Markov process. The Archimedean two-parameter BB7 copula is adopted to describe the underlying dependence structure between two consecutive returns, while the log-Dagum distribution is employed to model the margins marked by skewness and kurtosis. A simulation study is carried out to evaluate the performance of the maximum likelihood estimates. Furthermore, we apply the model to the daily returns of four stocks and, finally, we illustrate how its fitting to data can be improved when the dependence between consecutive returns is described through a copula function.