Volume 7, Issue 1

The kurtosis of the distribution of financial returns characterized by high volatility persistence and thick tails is notoriously difficult to estimate precisely. We propose a simple but effective procedure of estimating the kurtosis coefficient (and variance) based on volatility filtering that uses a simple GARCH model. In addition to an estimate, the proposed algorithm issues a signal of whether the kurtosis (or variance) is finite or infinite. We also show how to construct confidence intervals around the proposed estimates. Simulations indicate that the proposed estimates are much less median biased than the usual method-of-moments estimates, their confidence intervals having much more precise coverage probabilities. The procedure alsoworks well when the underlying volatility process is not the one the filtering technique is based on. We illustrate how the algorithm works using several actual series of returns.

The relationship between the European stock market and the crude oil depends on the significance of the different industries in the European economy. The literature points to a structural change after the 2008 crisis without getting into details of which sectors lead this regime switch. The co-movement between oil prices and stock market is known to exhibit (1) non-linearity, (2) asymmetric tail dependence and (3) variation over time. I combine a copula approach with Switching Markov models to capture this complex linkage while the CoVaR measure translates the consequences of the tail dependence into potential losses. The results indicate a change in the lower tail dependence from negative to positive association between oil and Eurostoxx, meaning a shift in the exposure of our stock portfolio to commodity risk. There is a structural change in dependence after the 2008 financial crisis led by energy-intensive sector, e.g. basic materials and consumer goods. The economic cycle and its implications for profit margin and oil demand might explain this switch. Healthcare sector responds to oil shocks in an opposite way than Eurostoxx, displaying useful features to reduce the exposure of the stock portfolio to oil spillovers.

The Copula Multivariate GARCH (CMGARCH) model is based on a dynamic copula function with time-varying parameters. It is particularly suited for modelling dynamic dependence of non-elliptically distributed financial returns series. The model allows for capturing more flexible dependence patterns than a multivariate GARCH model and also generalizes static copula dependence models. Nonetheless, the model is subject to a number of parameter constraints that ensure positivity of variances and covariance stationarity of the modeled stochastic processes. As such, the resulting distribution of parameters of interest is highly irregular, characterized by skewness, asymmetry, and truncation, hindering the applicability and accuracy of asymptotic inference. In this paper, we propose Bayesian analysis of the CMGARCH model based on Constrained Hamiltonian Monte Carlo (CHMC), which has been shown in other contexts to yield efficient inference on complicated constrained dependence structures. In the CMGARCH context, we contrast CHMC with traditional random-walk sampling used in the previous literature and highlight the benefits of CHMC for applied researchers. We estimate the posterior mean, median and Bayesian confidence intervals for the coefficients of tail dependence. The analysis is performed in an application to a recent portfolio of S&P500 financial asset returns.

We present basic properties and discuss potential insurance applications of a new class of probability distributions on positive integers with power law tails. The distributions in this class are zero-inflated discrete counterparts of the Pareto distribution. In particular, we obtain the probability of ruin in the compound binomial risk model where the claims are zero-inflated discrete Pareto distributed and correlated by mixture.

Standard methods of fitting finite mixture models take into account the majority of observations in the center of the distribution. This paper considers the case where the decision maker wants to make sure that the tail of the fitted distribution is at least as heavy as the tail of the empirical distribution. For instance, in nuclear engineering, where probability of exceedance (POE) needs to be estimated, it is important to fit correctly tails of the distributions. The goal of this paper is to supplement the standard methodology and to assure an appropriate heaviness of the fitted tails. We consider a new Conditional Value-at-Risk (CVaR) distance between distributions, that is a convex function with respect to weights of the mixture. We have conducted a case study demonstrating e˚ciency of the approach. Weights of mixture are found by minimizing CVaR distance between the mixture and the empirical distribution. We have suggested convex constraints on weights, assuring that the tail of the mixture is as heavy as the tail of empirical distribution.

We consider a dependent lifetime model for systemic risk, whose basic idea was for the first time presented by Freund. This model allows to model cascading effects of defaults for arbitrarily many economic agents. We study in particular the pertaining bivariate copula function. This copula does not have a closed form and does not belong to the class of Archimedean copulas, either.We derive some monotonicity properties of it and show how to use this copula for modelling the cascade effect implicitly contained in observed CDS spreads.

For a long time, the Dirichlet process has been the gold standard discrete random measure in Bayesian nonparametrics. The Pitman-Yor process provides a simple and mathematically tractable generalization, allowing for a very flexible control of the clustering behaviour. Two commonly used representations of the Pitman-Yor process are the stick-breaking process and the Chinese restaurant process. The former is a constructive representation of the process which turns out very handy for practical implementation, while the latter describes the partition distribution induced. Obtaining one from the other is usually done indirectly with use of measure theory. In contrast, we propose here an elementary proof of Pitman-Yor’s Chinese Restaurant process from its stick-breaking representation.

Úbeda-Flores showed that the range of multivariate Spearman’s footrule for copulas of dimension d ≥ 2 is contained in the interval [−1/ d , 1], that the upper bound is attained exclusively by the upper Fréchet-Hoeffding bound, and that the lower bound is sharp in the case where d = 2. The present paper provides characterizations of the copulas attaining the lower bound of multivariate Spearman’s footrule in terms of the copula measure but also via the copula’s diagonal section.

Exponential inequalities are main tools in machine learning theory. To prove exponential inequalities for non i.i.d random variables allows to extend many learning techniques to these variables. Indeed, much work has been done both on inequalities and learning theory for time series, in the past 15 years. However, for the non independent case, almost all the results concern stationary time series. This excludes many important applications: for example any series with a periodic behaviour is nonstationary. In this paper, we extend the basic tools of [19] to nonstationary Markov chains. As an application, we provide a Bernsteintype inequality, and we deduce risk bounds for the prediction of periodic autoregressive processes with an unknown period.

In this paper we discuss a natural extension of infinite discrete partition-of-unity copulas which were recently introduced in the literature to continuous partition of copulas with possible applications in risk management and other fields. We present a general simple algorithm to generate such copulas on the basis of the empirical copula from high-dimensional data sets. In particular, our constructions also allow for an implementation of positive tail dependence which sometimes is a desirable property of copula modelling, in particular for internal models under Solvency II.

Two simulation algorithms for hierarchical Archimedean copulas in the case when intra-group generators are not necessarily completely monotone are presented. Both generalize existing algorithms for the completely monotone case. The underlying stochastic models for both algorithms arise as a particular instance of a more general probability space studied recently in Ressel, P. (2018): A multivariate version of Williamson’s theorem, ℓ 1 -symmetric survival functions, and generalized Archimedean copulas. Depend. Model. 6 , 356–368. On this probability space the inter-group dependence need not be Archimedean, however, we highlight two particular circumstances that guarantee that a hierarchical Archimedean copula is obtained.

A latent class model is proposed to examine couples’ breadwinning typologies and explain the wage differentials according to the socio-demographic characteristics of the society with data collected through surveys. We derive an ordinal variable indicating the couple’s income provision-role type and suppose the existence of an underlying discrete latent variable to model the effect of covariates. We use a two-step maximum likelihood inference conducted to account for concomitant variables, informative sampling scheme and missing responses. The weighted log-likelihood is maximised through the Expectation-Maximization algorithm and information criteria are used to develop the model selection. Predictions are made on the basis of the maximum posterior probabilities. Disposing of data collected in Japan over thirty years we compare couples’ breadwinning patterns across time. We provide some evidence of the gender wage-gap and we show that it can be attributed to the fact that, especially in Japan, duties and responsibilities for the child care are supported exclusively by women.

For functions of several variables there exist many notions of monotonicity, three of them being characteristic for resp. distribution, survival and co-survival functions. In each case the “degree” of monotonicity is just the basic one of a whole scale. Copulas are special distribution functions, and stable tail dependence functions are special co-survival functions. It will turn out that for both classes the basic degree of monotonicity is the only one possible, apart from the (trivial) independence functions. As a consequence a “nesting” of such functions depends on particular circumstances. For nested Archimedean copulas the rather restrictive conditions known so far are considerably weakened.

We show that any distribution function on ℝ d with nonnegative, nonzero and integrable marginal distributions can be characterized by a norm on ℝ d +1 , called F -norm. We characterize the set of F -norms and prove that pointwise convergence of a sequence of F -norms to an F -norm is equivalent to convergence of the pertaining distribution functions in the Wasserstein metric. On the statistical side, an F -norm can easily be estimated by an empirical F -norm, whose consistency and weak convergence we establish. The concept of F -norms can be extended to arbitrary random vectors under suitable integrability conditions fulfilled by, for instance, normal distributions. The set of F -norms is endowed with a semigroup operation which, in this context, corresponds to ordinary convolution of the underlying distributions. Limiting results such as the central limit theorem can then be formulated in terms of pointwise convergence of products of F -norms. We conclude by showing how, using the geometry of F -norms, we may characterize nonnegative integrable distributions in ℝ d by simple compact sets in ℝ d +1 . We then relate convergence of those distributions in the Wasserstein metric to convergence of these characteristic sets with respect to Hausdorff distances.

Genest and Segers (2010) gave conditions under which the empirical copula process associated with a random sample from a bivariate continuous distribution has a smaller asymptotic covariance than the standard empirical process based on a random sample from the underlying copula. An extension of this result to the multivariate case is provided.

We study nonparametric estimators of conditional Kendall’s tau, a measure of concordance between two random variables given some covariates. We prove non-asymptotic pointwise and uniform bounds, that hold with high probabilities. We provide “direct proofs” of the consistency and the asymptotic law of conditional Kendall’s tau. A simulation study evaluates the numerical performance of such nonparametric estimators. An application to the dependence between energy consumption and temperature conditionally to calendar days is finally provided.

We study the dynamics of the family of copulas { C t } t ≥0 of a pair of stochastic processes given by stochastic differential equations (SDE). We associate to it a parabolic partial differential equation (PDE). Having embedded the set of bivariate copulas in a dual of a Sobolev Hilbert space H 1 (ℝ 2 ) * we calculate the derivative with respect to t and the *weak topology i.e. the tangent vector field to the image of the curve t → C t . Furthermore we show that the family { C t } t ≥0 is an orbit of a strongly continuous semigroup of transformations and provide the infinitesimal generator of this semigroup.

The linear correlation coefficient of Bravais-Pearson is considered a powerful indicator when the dependency relationship is linear and the error variate is normally distributed. Unfortunately in finance and in survival analysis the dependency relationship may not be linear. In such case, the use of rank-based measures of dependence, like Kendall’s tau or Spearman rho are recommended. In this direction, under length-biased sampling, measures of the degree of dependence between the survival time and the covariates appear to have not received much intention in the literature. Our goal in this paper, is to provide an alternative indicator of dependence measure, based on the concept of information gain, using the parametric copulas. In particular, the extension of the Kent’s [18] dependence measure to length-biased survival data is proposed. The performance of the proposed method is demonstrated through simulations studies.

In this paper, we propose a data driven bandwidth selection of the recursive Gumbel kernel estimators of a probability density function based on a stochastic approximation algorithm. The choice of the bandwidth selection approaches is investigated by a second generation plug-in method. Convergence properties of the proposed recursive Gumbel kernel estimators are established. The uniform strong consistency of the proposed recursive Gumbel kernel estimators is derived. The new recursive Gumbel kernel estimators are compared to the non-recursive Gumbel kernel estimator and the performance of the two estimators are illustrated via simulations as well as a real application.

We introduce a location-scale model for conditional heavy-tailed distributions when the covariate is deterministic. First, nonparametric estimators of the location and scale functions are introduced. Second, an estimator of the conditional extreme-value index is derived. The asymptotic properties of the estimators are established under mild assumptions and their finite sample properties are illustrated both on simulated and real data.