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Donald Andrew Dawson (Don Dawson) was born in 1937. He received a bachelor’s degree in 1958 and a master’s degree in 1959 from McGill University and a Ph.D. in 1963 from M.I.T. under the supervision of Henry P. McKean, Jr. Following an appointment at McGill University as professor for 7 years, he joined Carleton University in 1970 where he remained for the rest of his career. Among his many contributions to the theory of stochastic processes, his work leading to the creation of the Dawson–Watanabe superprocess and the analysis of its remarkable properties in describing the evolution in space and time of populations, stand out as milestones of modern probability theory. His numerous papers span the whole gamut of contemporary hot areas, notably the study of stochastic evolution equations, measure-valued processes, McKean–Vlasov limits, hierarchical structures, super-Brownian motion, as well as branching, catalytic and historical processes. He has over 200 refereed publications and 8 monographs, with an impressive number of citations, more than 7000. He is elected Fellow of the Royal Society and of the Royal Society of Canada, as well as Gold medalist of the Statistical Society of Canada and elected Fellow of the Institute of Mathematical Statistics. We realized this interview to celebrate the outstanding contribution of Don Dawson to 50 years of Stochastics at Carleton University.

We consider several time series and for each of them, we fit an appropriate dynamic parametric model. This produces serially independent error terms for each time series. The dependence between these error terms is then modeled by a regime-switching copula. The EM algorithm is used for estimating the parameters and a sequential goodness-of- fit procedure based on Cramer-von Mises statistics is proposed to select the appropriate number of regimes. Numerical experiments are performed to assess the validity of the proposed methodology. As an example of application, we evaluate a European put-on-max option on the returns of two assets. In order to facilitate the use of our methodology, we have built a R package HMMcopula available on CRAN.

In this article we show that under weak assumptions, the change-point tests designed for independent random vectors can also be used with pseudo-observations for testing change-point in the joint distribution of non-observable random vectors, the associated copula, or the margins, without modifying the limiting distributions. In particular, change-point tests can be applied to the residuals of stochastic volatility models or conditional distribution functions applied to the observations, which are prime examples of pseudo-observations. Since the limiting distribution of test statistics depends on the unknown joint distribution function or its associated unknown copula when the dimension is greater than one, we also show that iid multipliers and traditional bootstrap can be used with pseudo-observations to approximate P -values for the test statistics. Numerical experiments are performed in order to compare the different statistics and bootstrapping methods. Examples of applications to change-point problems are given. The R package changepointTests (Nasri and Remillard, 2021) includes all the methodologies proposed in this article.

Recently, two different copula-based approaches have been proposed to estimate the conditional quantile function of a variable Y with respect to a vector of covariates X : the first estimator is related to quantile regression weighted by the conditional copula density, while the second estimator is based on the inverse of the conditional distribution function written in terms of margins and the copula. Using empirical processes, we show that even if the two estimators look quite different, their estimation errors have the same limiting distribution. Also, we propose a bootstrap procedure for the limiting process in order to construct uniform confidence bands around the conditional quantile function.

The goal of this paper is to introduce new families of multivariate copulas, extending the chi-square copulas, the Fisher copula, and squared copulas. The new families are constructed from existing copulas by first transforming their margins to standard Gaussian distributions, then transforming these variables into non-central chi-square variables with one degree of freedom, and finally by considering the copula associated with these new variables. It is shown that by varying the non-centrality parameters, one can model non-monotonic dependence, and when one or many non-centrality parameters are outside a given hyper-rectangle, then the copula is almost the same as the one when these parameters are infinite. For these new families, the tail behavior, the monotonicity of dependence measures such as Kendall’s tau and Spearman’s rho are investigated, and estimation is discussed. The R package NCSCopula ( Nasri, 2019 ) can be used to estimate the parameters for several copula families.

In this paper, we consider non-stationary response variables and covariates, where the marginal distributions and the associated copula may be time-dependent. We propose estimators for the unknown parameters and we establish the limiting distribution of the estimators of the copula and the conditional copula, together with a parametric bootstrap method for constructing confidence bands around the estimator and for testing the adequacy of the model. We also consider three examples of functionals of the copula-based model under non-stationarity: conditional quantiles, conditional mean, and conditional expected shortfall. The asymptotic distribution of the estimation errors is shown to be Gaussian, and bootstrapping methods are proposed to estimate their asymptotic variances. The finite sample performance of our estimators is investigated through Monte Carlo experiments, and we show three examples of implementation of the proposed methodology.

In this paper, we propose an intuitive way to couple several dynamic time series models by inducing dependence between the so-called generalized errors of each model. This extends previous work for modelling dependance between innovations of stochastic volatility models. We consider time-independent and time-dependent copula models and we study the asymptotic behavior of some empirical processes constructed from pseudo-observations, as well as the behavior of pseudo-maximum likelihood estimators of the associated copula parameters. The results show that even if the generalized errors depend on unknown parameters, the limiting behaviour of many processes of interest do not depend on the estimation errors. One can easily perform tests of change-point on the full distribution, and the margins or the copula, as if the generalized errors were observed. For some interesting parametric models of time-dependent copulas, the same behavior is observed: one can work with the pseudo-observations, as if we were observing the generalized errors. This interesting property makes it possible to construct consistent tests of specification for the dependence models, without having to consider the dynamic time series models. An example of application with financial data is given.