Large sample properties of nonparametric copula estimators under bivariate censoring

A new nonparametric estimator is proposed for the copula function of a bivariate survival function for data subject to random right-censoring. We consider two censoring models: univariate and copula censoring. We show strong consistency and we obtain an i.i.d. representation for the copula estimator. In a simulation study we compare the new estimator to the one of Gribkova and Lopez [Nonparametric copula estimation under bivariate censoring; doi:10.1111/sjos.12144].     

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     

Pair‐copula constructions for non‐Gaussian DAG models

We propose a new type of multivariate statistical model that permits non‐Gaussian distributions as well as the inclusion of conditional independence assumptions specified by a directed acyclic graph. These models feature a specific factorisation of the likelihood that is based on pair‐copula constructions and hence involves only univariate distributions and bivariate copulas, of which some may be conditional. We demonstrate maximum‐likelihood estimation of the parameters of such models and compare them to various competing models from the literature. A simulation study investigates the effects of model misspecification and highlights the need for non‐Gaussian conditional independence models. The proposed methods are finally applied to modeling financial return data. The Canadian Journal of Statistics 40: 86–109; 2012 © 2012 Statistical Society of Canada     

Do stock returns have an Archimedean copula?

The flexible class of Archimedean copulas plays an important role in multivariate statistics. While there is a large number of goodness-of-fit tests for copulas and parametric families of copulas, the question if a given data set belongs to an arbitrary Archimedean copula or not has not yet received much attention in the literature. This paper suggests a new, straightforward method to test whether a copula is an Archimedean copula without the need to specify its parametric family. We conduct Monte Carlo simulations to assess the power of the test. The approach is applied to (bivariate) joint distributions of stock asset returns. We find that, in general, stock returns may have Archimedean copulas.     

Comparison of estimation methods for the Weibull distribution

Weibull distributions have received wide ranging applications in many areas including reliability, hydrology and communication systems. Many estimation methods have been proposed for Weibull distributions. But there has not been a comprehensive comparison of these estimation methods. Most studies have focused on comparing the maximum likelihood estimation (MLE) with one of the other approaches. In this paper, we first propose an L-moment estimator for the Weibull distribution. Then, a comprehensive comparison is made of the following methods: the method of maximum likelihood estimation (MLE), the method of logarithmic moments, the percentile method, the method of moments and the method of L-moments.     

Some theory and practical uses of trimmed L-moments

Trimmed L-moments, defined by Elamir and Seheult [2003. Trimmed L-moments. Comput. Statist. Data Anal. 43, 299–314], summarize the shape of probability distributions or data samples in a way that remains viable for heavy-tailed distributions, even those for which the mean may not exist. We derive some further theoretical results concerning trimmed L-moments: a relation with the expansion of the quantile function as a weighted sum of Jacobi polynomials; the bounds that must be satisfied by trimmed L-moments; recurrences between trimmed L-moments with different degrees of trimming; and the asymptotic distributions of sample estimators of trimmed L-moments. We also give examples of how trimmed L-moments can be used, analogously to L-moments, in the analysis of heavy-tailed data. Examples include identification of distributions using a trimmed L-moment ratio diagram, shape parameter estimation for the generalized Pareto distribution, and fitting generalized Pareto distributions to a heavy-tailed data sample of computer network traffic.     

Gluing Copulas

We present a new way of constructing n-copulas, by scaling and gluing finitely many n-copulas. Gluing for bivariate copulas produces a copula that coincides with the independence copula on some grid of horizontal and vertical sections. Examples illustrate how gluing can be applied to build complicated copulas from simple ones. Finally, we investigate the analytical as well as statistical properties of the copulas obtained by gluing, in particular, the behavior of Spearman's ρ and Kendall's τ.     

Optimal designs for copula models

Copula modelling has in the past decade become a standard tool in many areas of applied statistics. However, a largely neglected aspect concerns the design of related experiments. Particularly the issue of whether the estimation of copula parameters can be enhanced by optimizing experimental conditions and how robust all the parameter estimates for the model are with respect to the type of copula employed. In this paper an equivalence theorem for (bivariate) copula models is provided that allows formulation of efficient design algorithms and quick checks of whether designs are optimal or at least efficient. Some examples illustrate that in practical situations considerable gains in design efficiency can be achieved. A natural comparison between different copula models with respect to design efficiency is provided as well.     

Modified inference function for margins for the bivariate clayton copula-based SUN Tobit Model

This paper extends the analysis of the bivariate Seemingly Unrelated Regression (SUN) Tobit model by modeling its nonlinear dependence structure through the Clayton copula. The ability to capture/model the lower tail dependence of the SUN Tobit model where some data are censored (generally, left-censored at zero) is an useful feature of the Clayton copula. We propose a modified version of the (classical) Inference Function for Margins (IFS) method by Joe and XP [H. Joe and J.J. XP, The estimation method of inference functions for margins for multivariate models, Tech. Rep. 166, Department of Statistics, University of British Columbia, 1996], which we refer to as Modified Inference Function for Margins (MIFF) method, to obtain the (point) estimates of the marginal and Clayton copula parameters. More specifically, we employ the (frequenting) data augmentation technique at the second stage of the IFS method (the first stage of the MIFF method is equivalent to the first stage of the IFS method) to generate the censored observations and then estimate the Clayton copula parameter. This process (data augmentation and copula parameter estimation) is repeated until convergence. Such modification at the second stage of the usual estimation method is justified in order to obtain continuous marginal distributions, which ensures the uniqueness of the resulting Clayton copula, as stated by Solar's [A. Solar, Fonctions de répartition à n dimensions et leurs marges, Publ. de l'Institut de Statistique de l'Université de Paris 8 (1959), pp. 229–231] theorem; and also to provide an unbiased estimate of the association parameter (the IFS method provides a biased estimate of the Clayton copula parameter in the presence of censored observations in both margins). Since the usual asymptotic approach, that is the computation of the asymptotic covariance matrix of the parameter estimates, is troublesome in this case, we also propose the use of resampling procedures (bootstrap methods, such as standard normal and percentile, by Efron and Tibshirani [B. Efron and R.J. Tibshirani, An Introduction to the Bootstrap, Chapman & Hall, New York, 1993] to obtain confidence intervals for the model parameters.     

A class of distributions with the quadratic mean residual quantile function

Abstract

The present paper introduces a new family of distributions with quadratic mean residual quantile function. Various distributional properties as well as reliability characteristics are discussed. Some characterizations of the class of distributions are presented. The estimation of parameters of the model using method of L-moments is studied. The practical application of the class of models is illustrated with a real life data set.     

A goodness-of-fit test for multivariate multiparameter copulas based on multiplier central limit theorems

Recent large scale simulations indicate that a powerful goodness-of-fit test for copulas can be obtained from the process comparing the empirical copula with a parametric estimate of the copula derived under the null hypothesis. A first way to compute approximate p-values for statistics derived from this process consists of using the parametric bootstrap procedure recently thoroughly revisited by Genest and Rémillard. Because it heavily relies on random number generation and estimation, the resulting goodness-of-fit test has a very high computational cost that can be regarded as an obstacle to its application as the sample size increases. An alternative approach proposed by the authors consists of using a multiplier procedure. The study of the finite-sample performance of the multiplier version of the goodness-of-fit test for bivariate one-parameter copulas showed that it provides a valid alternative to the parametric bootstrap-based test while being orders of magnitude faster. The aim of this work is to extend the multiplier approach to multivariate multiparameter copulas and study the finite-sample performance of the resulting test. Particular emphasis is put on elliptical copulas such as the normal and the t as these are flexible models in a multivariate setting. The implementation of the procedure for the latter copulas proves challenging and requires the extension of the Plackett formula for the t distribution to arbitrary dimension. Extensive Monte Carlo experiments, which could be carried out only because of the good computational properties of the multiplier approach, confirm in the multivariate multiparameter context the satisfactory behavior of the goodness-of-fit test.     

A Nonparametric Shewhart-Type Signed-Rank Control Chart Based on Runs

Measures of distributional symmetry based on quantiles, L-moments, and trimmed L-moments are briefly reviewed, and (asymptotic) sampling properties of commonly used estimators considered. Standard errors are estimated using both analytical and computer-intensive methods. Simulation is used to assess results when sampling from some known distributions; bootstrapping is used on sample data to estimate standard errors, construct confidence intervals, and test a hypothesis of distributional symmetry. Symmetry measures based on 2- or 3-trimmed L-moments have some advantages over other measures in terms of their existence. Their estimators are generally well behaved, even in relatively small samples.     

Copulas checker-type approximations: Application to quantiles estimation of sums of dependent random variables

Abstract

Several approximations of copulas have been proposed in the literature. By using empirical versions of checker-type copulas approximations, we propose non parametric estimators of the copula. Under some conditions, the proposed estimators are copulas and their main advantage is that they can be sampled from easily. One possible application is the estimation of quantiles of sums of dependent random variables from a small sample of the multivariate law and a full knowledge of the marginal laws. We show that estimations may be improved by including in an easy way in the approximated copula some additional information on the law of a sub-vector for example. Our approach is illustrated by numerical examples.     

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     

Approximation Multivariate Distribution with Pair Copula Using the Orthonormal Polynomial and Legendre Multiwavelets Basis Functions

We concentrate on constructing higher dimensional distributions using a fast growing graphical model called Vine/ pair-copula model which has been introduced and developed by Joe, Cooke, Bedford, Kurowica, Daneshkhah, and others. They first construct a n-dimensional copula density by stacking together n(n ? 1)/2 bivariate copula density, and they then approximate arbitrarily well these bivariate copulas and the corresponding multivariate distribution using a semi-parametric method. One constructive approach involves the use of minimum information copulas that can be specified to any required degree of precision based on the available data (or possibly based on the experts’ judgments). By using this method, one is able to use a fixed finite dimensional family of copulas to be employed in terms of a vine construction, with the promise of a uniform level of approximation.

The basic idea behind this method is to use a two-dimensional ordinary polynomial series to approximate any log-density of a bivariate copula function by truncating the series at an appropriate point. We make this approximation method more accurate and computationally faster by using the orthonormal polynomial and Legendre multiwavelets (LMW) series as the basis functions. We show the derived approximations are more precise and computationally faster with better properties than the one proposed previous method in the literature. We then apply our method to modeling a dataset of Norwegian financial data that was previously analyzed in the series of articles, and finally compare our results by them. At the end, we present a method to simulate from the approximated models, and validate our approximation using the simulation results to recover the same dependency structure of the original data.     

Non‐parametric Copula Estimation Under Bivariate Censoring

In this paper, we consider non‐parametric copula inference under bivariate censoring. Based on an estimator of the joint cumulative distribution function, we define a discrete and two smooth estimators of the copula. The construction that we propose is valid for a large range of estimators of the distribution function and therefore for a large range of bivariate censoring frameworks. Under some conditions on the tails of the distributions, the weak convergence of the corresponding copula processes is obtained in l^∞([0,1]²). We derive the uniform convergence rates of the copula density estimators deduced from our smooth copula estimators. Investigation of the practical behaviour of these estimators is performed through a simulation study and two real data applications, corresponding to different censoring settings. We use our non‐parametric estimators to define a goodness‐of‐fit procedure for parametric copula models. A new bootstrap scheme is proposed to compute the critical values.     

Constructing copula functions with weighted geometric means

The weighted arithmetic mean of two copulas is a copula. In some cases, geometric and harmonic means also provide copulas. There are copulas specially appropriate to be combined by using weighted geometric means. With this method of construction we combine Farlie–Gumbel–Morgentern and Ali–Mikhail–Haq copulas to obtain families of copulas which can be expressed in terms of double power series. The Gumbel–Barnett copula is also considered and a new copula is proposed, which arises as the first order approximation of the weighted geometric mean of two copulas. Invariance of two multivariate distributions (Cuadras–Augé and Johnson–Kotz) by weighted geometric and arithmetic means is also studied.     

Rectangular Patchwork for Bivariate Copulas and Tail Dependence

We present a method for constructing bivariate copulas by changing the values that a given copula assumes on some subrectangles of the unit square. Some applications of this method are discussed, especially in relation to the construction of copulas with different tail dependencies.     

Multivariate Return Decomposition: Theory and Implications

In this paper, we propose a model based on multivariate decomposition of multiplicative – absolute values and signs – components of asset returns. In the m-variate case, the marginals for the m absolute values and the binary marginals for the m directions are linked through a 2m-dimensional copula. The approach is detailed in the case of a bivariate decomposition. We outline the construction of the likelihood function and the computation of different conditional measures. The finite-sample properties of the maximum likelihood estimator are assessed by simulation. An application to predicting bond returns illustrates the usefulness of the proposed method.     

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.