I got my PhD from the Louvain Catholic University (UCL, Belgium) in July 2008. I then moved to Australia to take up a post-doctoral research position at the University of Melbourne under the supervision of Professor Peter Hall. In October 2009, I was offered an academic position at UNSW Sydney.
Most of my research lies in developing nonparametric and semiparametric methods in various contexts. In particular, I’m interested in nonparametric regression models (mainly kernel smoothing methods), semiparametric regression models (mainly Single-Index Models), nonparametric copula models for dependence modelling and nonparametric methods for functional data analysis.
- PhD in Sciences (Statistics), Université catholique de Louvain, Belgium, 2008
- MSc in Statistics, Université catholique de Louvain, Belgium, 2003
- MSc in Engineering (Applied Mathematics orientation), Université catholique de Louvain, Belgium, 2002
- BSc in Engineering, Faculté Polytechnique de Mons, Belgium, 1999
- Develop efficient and flexible nonparametric models for a variety of statistical analyses
- Extend those models to higher dimensional data, in spite of the “Curse of Dimensionality”
- Provide data-driven automatic rules for smoothing parameter selection in those models
Research in Detail
Most of my research lies in developing nonparametric and semiparametric methods in various contexts. Traditional parametric models assume that the functional form of the statistical objects of interest (distribution of a random variable, regression function, etc.) is exactly known up to a finite number or parameters. This is restrictive and, in case of model misspecification, can lead to erroneous conclusions. In contrast, nonparametric models keep the structural prior assumptions as weak as possible, really `letting the data speak for themselves' (as it is commonly quoted). Usually, this (almost) total flexibility is reached by using statistical models that are infinite-dimensional. This, of course, brings in some new challenges, both in theory and in practice. More specifically, several topics I am currently interested in are the following:
- Nonparametric copula modelling: A copula describes how the marginal distributions of two random variables `interact' to produce the joint bivariate distribution of the corresponding random vector. Today, copulas are used extensively in statistical modelling in all areas, from quantitative finance and insurance to medicine and climatology. Therefore, empirically estimating a copula function from a bivariate sample has become one of the most important problems of modern statistical modelling.
- Nonparametric copula-based conditional density estimation: More than a regression model Y = m(X) + e, the conditional density of the dependent variableY given the regressor X provides complete information about the relationship between Y and X. Nonparametrically estimating a conditional density is challenging. However, an elegant and efficient estimator is based on estimating the copula density.
- Nonparametric high-dimensional density estimation via pair-copula construction: In general, any d-variate joint probability density can be expressed as a product of its d marginals times d(d-1)/2 pair-copula densities, acting on several different conditional probability distributions. This `pair-copula construction' breaks down the high-dimensional density in a product of lower-dimensional objects, that should be easier to estimate. This offers a promising path for nonparametrically estimating any high dimensional probability density, which is a difficult problem.
- Nonparametric binary regression: consider a regression problem with a binary response. Then, the usual regression function is nothing else but the conditional probability of the response taking the value 1 given the value of the predictors. In that situation, the panacea seems to be logistic regression, although this model is built on strong parametric constraints whose validity is rarely checked. I work on developing very flexible nonparametric estimators for this conditional probability function, as well as using those flexible estimators in further procedures where binary (or more generally discrete and/or qualitative) variables are involved.
- Nonparametric functional regression: imagine that the random object that you have to work with is actually a whole function. Then we talk about functional data. Until recently, only parametric models had been proposed in that situation, precisely because of the above mentionned dimensionality problems: if a whole random function has to be considered as such, we have to work in an infinite-dimensional space and we are thus facing a severe version of the Curse of Dimensionality. However, recent results have shown that nonparametric models for functional data are no longer that unrealistic. I work on that topic, both theoretically and practically. For instance, a powerful system of signature digital recognition is under consideration, based on the idea that a signature can be regarded as a random function. Estimating the probability of a forgery by analyzing at once a whole signature-function is of obvious interest and I claim that this problem can be tackled via nonparametric functional regression.
- Related applied studies: So far, most of the applied studies lies within a parametric framework, so that any use of nonparametric methods in applications might be innovative. For instance, I have recently developed a nonparametric model aiming at analysing football (soccer) results, which shows interesting and up to now ignored patterns.
- 2014 – 2016 : Faculty Research Grant, Faculty of Science, UNSW
- 2010 – 2013: Early Career Research Grants, Faculty of Science, UNSW
Current Student Projects (PhD and Honours)
Carlos Aya Moreno, “New wavelet-based density estimation in higher dimensions, with application to image registration”, PhD, 2013 - (in co-supervision with A/Prof Spiro Penev)
Students willing to know more about any of the above listed research topics are welcome to contact me. I can suggest Honours/PhD projects.
TEACHING & OUTREACH
Courses I teach
MATH2089: Statistics (2nd year engineering)
MATH3801-MATH3901: (Higher) Probability and Stochastics Processes
MATH5895: Nonparametric Statistics
Professional affiliations and service positions
Director of Postgraduate Studies (Coursework),School of Mathematics and Statistics, UNSW
Associate Editor of “Statistics and Probability Letters”
Member of the Belgian Statistical Society (SBS-BVS)