Copula-Type Estimators for Flexible Multivariate Density Modeling Using Mixtures

Copulas are popular as models for multivariate dependence because they allow the marginal densities and the joint dependence to be modeled separately. However, they usually require that the transformation from uniform marginals to the marginals of the joint dependence structure is known. This can only be done for a restricted set of copulas, for example, a normal copula. Our article introduces copula-type estimators for flexible multivariate density estimation which also allow the marginal densities to be modeled separately from the joint dependence, as in copula modeling, but overcomes the lack of flexibility of most popular copula estimators. An iterative scheme is proposed for estimating copula-type estimators and its usefulness is demonstrated through simulation and real examples. The joint dependence is modeled by mixture of normals and mixture of normal factor analyzer models, and mixture of t and mixture of t-factor analyzer models. We develop efficient variational Bayes algorithms for fitting these in which model selection is performed automatically. Based on these mixture models, we construct four classes of copula-type densities which are far more flexible than current popular copula densities, and outperform them in a simulated dataset and several real datasets. Supplementary material for this article is available online.     

Efficient algorithms for basket default swap pricing with multivariate Archimedean copulas

We introduce a new importance sampling method for pricing basket default swaps employing exchangeable Archimedean copulas and nested Gumbel copulas. We establish more realistic dependence structures than existing copula models for credit risks in the underlying portfolio, and propose an appropriate density for importance sampling by analyzing multivariate Archimedean copulas. To justify efficiency and accuracy of the proposed algorithms, we present numerical examples and compare them with the crude Monte Carlo simulation, and finally show that our proposed estimators produce considerably smaller variances.     

Thresholding methods to estimate copula density

This paper deals with the problem of multivariate copula density estimation. Using wavelet methods we provide two shrinkage procedures based on thresholding rules for which knowledge of the regularity of the copula density to be estimated is not necessary. These methods, said to be adaptive, have proved to be very effective when adopting the minimax and the maxiset approaches. Moreover we show that these procedures can be discriminated in the maxiset sense. We provide an estimation algorithm and evaluate its properties using simulation. Finally, we propose a real life application for financial data.     

On the strong approximation of bootstrapped empirical copula processes with applications

The purpose of the present paper is to provide a strong invariance principle for the generalized bootstrapped empirical copula processwith the rate of the approximation for multivariate empirical processes. As a by-product, we obtain a uniform-in-bandwidth consistency result for kernel-type estimators of copula derivatives, which is of its own interest. We introduce also the delta-sequence estimators of the copula derivatives. The applications discussed here are change-point detection in multivariate copula models, nonparametric tests of stochastic vectorial independence and the law of iterated logarithm for the generalized bootstrapped empirical copula process. Finally, a general notion of bootstrapped empirical copula process constructed by exchangeably weighting the sample is presented.     

Mixture Modeling for Longitudinal Data

In this article, we propose an unbiased estimating equation approach for a two-component mixture model with correlated response data. We adapt the mixture-of-experts model and a generalized linear model for component distribution and mixing proportion, respectively. The new approach only requires marginal distributions of both component densities and latent variables. We use serial correlations from subjects’ subgroup memberships, which improves estimation efficiency and classification accuracy, and show that estimation consistency does not depend on the choice of the working correlation matrix. The proposed estimating equation is solved by an expectation-estimating-equation (EEE) algorithm. In the E-step of the EEE algorithm, we propose a joint imputation based on the conditional linear property for the multivariate Bernoulli distribution. In addition, we establish asymptotic properties for the proposed estimators and the convergence property using the EEE algorithm. Our method is compared to an existing competitive mixture model approach in both simulation studies and an election data application. Supplementary materials for this article are available online.     

Modeling dependence based on mixture copulas and its application in risk management

This paper is concerned with the statistical modeling of the dependence structure of multivariate financial data using the copula, and the application of copula functions in VaR valuation. After the introduction of the pure copula method and the maximum and minimum mixture copula method, authors present a new algorithm based on the more generalized mixture copula functions and the dependence measure, and apply the method to the portfolio of Shanghai stock composite index and Shenzhen stock component index. Comparing with the results from various methods, one can find that the mixture copula method is better than the pure Gaussian copula method and the maximum and minimum mixture copula method on different VaR level.     

A hybrid Pareto model for asymmetric fat-tailed data: the univariate case

Density estimators that can adapt to asymmetric heavy tails are required in many applications such as finance and insurance. Extreme value theory (EVT) has developed principled methods based on asymptotic results to estimate the tails of most distributions. However, the finite sample approximation might introduce a severe bias in many cases. Moreover, the full range of the distribution is often needed, not only the tail area. On the other hand, non-parametric methods, while being powerful where data are abundant, fail to extrapolate properly in the tail area. We put forward a non-parametric density estimator that brings together the strengths of non-parametric density estimation and of EVT. A hybrid Pareto distribution that can be used in a mixture model is proposed to extend the generalized Pareto (GP) to the whole real axis. Experiments on simulated data show the following. On one hand, the mixture of hybrid Paretos converges faster in terms of log-likelihood and provides good estimates of the tail of the distributions when compared with other density estimators including the GP distribution. On the other hand, the mixture of hybrid Paretos offers an alternate way to estimate the tail index which is comparable to the one estimated with the standard GP methodology. The mixture of hybrids is also evaluated on the Danish fire insurance data set.      

Structural test in regression on functional variables

Many papers deal with structural testing procedures in multivariate regression. More recently, various estimators have been proposed for regression models involving functional explanatory variables. Thanks to these new estimators, we propose a theoretical framework for structural testing procedures adapted to functional regression. The procedures introduced in this paper are innovative and make the link between former works on functional regression and others on structural testing procedures in multivariate regression. We prove asymptotic properties of the level and the power of our procedures under general assumptions that cover a large scope of possible applications: tests for no effect, linearity, dimension reduction, …     

An EM-Like Algorithm for Semi- and Nonparametric Estimation in Multivariate Mixtures

We propose an algorithm for nonparametric estimation for finite mixtures of multivariate random vectors that strongly resembles a true EM algorithm. The vectors are assumed to have independent coordinates conditional upon knowing from which mixture component they come, but otherwise their density functions are completely unspecified. Sometimes, the density functions may be partially specified by Euclidean parameters, a case we call semiparametric. Our algorithm is much more flexible and easily applicable than existing algorithms in the literature; it can be extended to any number of mixture components and any number of vector coordinates of the multivariate observations. Thus it may be applied even in situations where the model is not identifiable, so care is called for when using it in situations for which identifiability is difficult to establish conclusively. Our algorithm yields much smaller mean integrated squared errors than an alternative algorithm in a simulation study. In another example using a real dataset, it provides new insights that extend previous analyses. Finally, we present two different variations of our algorithm, one stochastic and one deterministic, and find anecdotal evidence that there is not a great deal of difference between the performance of these two variants. The computer code and data used in this article are available online.     

Real-Time Density and Mode Estimation With Application to Time-Dynamic Mode Tracking

We introduce a nonparametric time-dynamic kernel type density estimate for the situation where an underlying multivariate distribution evolves with time. Based on this time-dynamic density estimate, we propose nonparametric estimates for the time-dynamic mode of the underlying distribution. Our estimators involve boundary kernels for the time dimension so that the estimator is always centered at current time, and multivariate kernels for the spatial dimension of the time-evolving distribution. Under certain mild conditions, the asymptotic behavior of density and mode estimators, especially their uniform convergence in both time and space, is derived. A time-dynamic algorithm for mode tracking is proposed, including automatic bandwidth choices, and is implemented via a mean update algorithm. Simulation studies and real data illustrations demonstrate that the proposed methods work well in practice.     

On the Bayesian Mixture Model and Identifiability

This article is concerned with Bayesian mixture models and identifiability issues. There are two sources of unidentifiability: the well-known likelihood invariance under label switching and the perhaps less well-known parameter identifiability problem. When using latent allocation variables determined by the mixture model, these sources of unidentifiability create arbitrary labeling that renders estimation of the model very difficult. We endeavor to tackle these problems by proposing a prior distribution on the allocations, which provides an explicit interpretation for the labeling by removing gaps with high probability. We propose a Markov chain Monte Carlo (MCMC) estimation method and present supporting illustrations.     

Stein's idea and minimax admissible estimation of a multivariate normal mean

We consider estimation of a multivariate normal mean vector under sum of squared error loss.We propose a new class of minimax admissible estimator which are generalized Bayes with respect to a prior distribution which is a mixture of a point prior at the origin and a continuous hierarchical type prior. We also study conditions under which these generalized Bayes minimax estimators improve on the James–Stein estimator and on the positive-part James–Stein estimator.     

Rank-based inference tools for copula regression,with property and casualty insurance applications

Rank-based procedures are commonly used for inference in copula models for continuous responses whose behavior does not depend on covariates. This paper describes how these procedures can be adapted to the broader framework in which (possibly non-linear) regression models for the marginal responses are linked by a copula that does not depend on covariates. The validity of many of these techniques can be derived from the asymptotic equivalence between the classical empirical copula process and its analog based on suitable residuals from the marginal models. Moment-based parameter estimation and copula goodness-of-fit tests are shown to remain valid under weak conditions on the marginal error term distributions, even when the residual-based empirical copula process fails to converge weakly. The performance of these procedures is evaluated through simulation in the context of two general insurance applications: micro-level multivariate insurance claims, and dependent loss triangles.     

Comparison of three semiparametric methods for estimating dependence parameters in copula models

Three semiparametric methods for estimating dependence parameters in copula models are compared, namely maximum pseudo-likelihood estimation and the two method-of-moment approaches based on the inversion of Spearman’s rho and Kendall’s tau. For each of these three asymptotically normal estimators, an estimator of their asymptotic (co)variance is stated in three different situations, namely the bivariate one-parameter case, the multivariate one-parameter case and the multivariate multiparameter case. An extensive Monte Carlo study is carried out to compare the finite-sample performance of the three estimators under consideration in these three situations. In the one-parameter case, it involves up to six bivariate and four-variate copula families, and up to five levels of dependence. In the multiparameter case, attention is restricted to trivariate and four-variate normal and t copulas. The maximum pseudo-likelihood estimator appears as the best choice in terms of mean square error in all situations except for small and weakly dependent samples. It is followed by the method-of-moment estimator based on Kendall’s tau, which overall appears to be significantly better than its analogue based on Spearman’s rho. The simulation results are complemented by asymptotic relative efficiency calculations. The numerical computation of Spearman’s rho, Kendall’s tau and their derivatives in the case of copula families for which explicit expressions are not available is also investigated.     

Estimation of nonstrict Archimedean copulas and its application to quantum networks

Bivariate nonstrict Archimedean copulas form a subclass of Archimedean copulas and are able to model the dependence structure of random variables that do not take on low quantiles simultaneously; i.e. their domain includes a set, the so‐called zero set, with positive Lebesgue measure but zero probability mass. Standard methods to fit a parametric Archimedean copula, e.g. classical maximum likelihood estimation, are either getting computationally more involved or even fail when dealing with this subclass. We propose an alternative method for estimating the parameter of a nonstrict Archimedean copula that is based on the zero set and the functional form of its boundary curve. This estimator is fast to compute and can be applied to absolutely continuous copulas but also allows singular components. In a simulation study, we compare its performance to that of the standard estimators. Finally, the estimator is applied when modeling the dependence structure of quantities describing the quality of transmission in a quantum network, and it is shown how this model can be used effectively to detect potential intruders in this network. Copyright © 2014 John Wiley & Sons, Ltd.     

Asymptotic properties of pseudo maximum likelihood estimators and test in semi-parametric copula models with multiple change points

The main purpose of the present paper is to establish the asymptotic properties of pseudo maximum likelihood estimators of the parameters of a multiple change-point model in the multivariate copula models when marginal distributions are unspecified but the copula function is parametrized. A pseudo likelihood ratio-type statistic is proposed for testing a sequence of observations for no change in the copula parameter against possible changes. Finally, a weighted bootstrap procedure that aims at evaluating the limiting distributions is examined.     

Variable Selection for Clustering by Separability Based on Ridgelines

A new variable selection algorithm is developed for clustering based on mode association. In conventional mixture-model-based clustering, each mixture component is treated as one cluster and the separation between clusters is usually measured by the ratio of between- and within-component dispersion. In this article, we allow one cluster to contain several components depending on whether they merge into one mode. The extent of separation between clusters is quantified using critical points on the ridgeline between two modes, which reflects the exact geometry of the density function. The computational foundation consists of the recently developed Modal expectation–maximization (MEM) algorithm which solves the modes of a Gaussian mixture density, and the Ridgeline expectation–maximization (REM) algorithm which solves the ridgeline passing through the critical points of the mixed density of two unimode clusters. Forward selection is used to find a subset of variables that maximizes an aggregated index of pairwise cluster separability. Theoretical analysis of the procedure is provided. We experiment with both simulated and real datasets and compare with several state-of-the-art variable selection algorithms. Supplemental materials including an R-package, datasets, and appendices for proofs are available online.     

TVaR-based capital allocation with copulas

Because of regulation projects from control organisations such as the European solvency II reform and recent economic events, insurance companies need to consolidate their capital reserve with coherent amounts allocated to the whole company and to each line of business. The present study considers an insurance portfolio consisting of several lines of risk which are linked by a copula and aims to evaluate not only the capital allocation for the overall portfolio but also the contribution of each risk over their aggregation. We use the tail value at risk (TVaR) as risk measure. The handy form of the FGM copula permits an exact expression for the TVaR of the sum of the risks and for the TVaR-based allocations when claim amounts are exponentially distributed and distributed as a mixture of exponentials. We first examine the bivariate model and then the multivariate case. We also show how to approximate the TVaR of the aggregate risk and the contribution of each risk when using any copula.     

Applying copula models to individual claim loss reserving methods

The estimation of loss reserves for incurred but not reported (IBNR) claims presents an important task for insurance companies to predict their liabilities. Recently, individual claim loss models have attracted a great deal of interest in the actuarial literature, which overcome some shortcomings of aggregated claim loss models. The dependence of the event times with the delays is a crucial issue for estimating the claim loss reserving. In this article, we propose to use semi-competing risks copula and semi-survival copula models to fit the dependence structure of the event times with delays in the individual claim loss model. A nonstandard two-step procedure is applied to our setting in which the associate parameter and one margin are estimated based on an ad hoc estimator of the other margin. The asymptotic properties of the estimators are established as well. A simulation study is carried out to evaluate the performance of the proposed methods.     

Superefficient estimation of the marginals by exploiting knowledge on the copula

We consider the problem of estimating the marginals in the case where there is knowledge on the copula. If the copula is smooth, it is known that it is possible to improve on the empirical distribution functions: optimal estimators still have a rate of convergence n^−1/2, but a smaller asymptotic variance. In this paper we show that for non-smooth copulas it is sometimes possible to construct superefficient estimators of the marginals: we construct both a copula and, exploiting the information our copula provides, estimators of the marginals with the rate of convergence logn/n.