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Christophe Ley

Publications and source records attributed to Christophe Ley.

At least 37 records · Page 2Linked to original sources
arXiv2020

The Wasserstein Impact Measure (WIM): a generally applicable, practical tool for quantifying prior impact in Bayesian statistics

The prior distribution is a crucial building block in Bayesian analysis, and its choice will impact the subsequent inference. It is therefore important to have a convenient way to quantify this impact, as such a measure of prior impact will help us to choose between two or more priors in a given situation. A recently proposed approach consists in determining the Wasserstein distance between posteriors resulting from two distinct priors, revealing how close or distant they are. In particular, if one prior is the uniform/flat prior, this distance leads to a genuine measure of prior impact for the other prior. While highly appealing and successful from a theoretical viewpoint, this proposal suffers from practical limitations: it requires prior distributions to be nested, posterior distributions should not be of a too complex form, in most considered settings the exact distance was not computed but sharp upper and lower bounds were proposed, and the proposal so far is restricted to scalar parameter settings. In this paper, we overcome all these limitations by introducing a practical version of this theoretical approach, namely the Wasserstein Impact Measure (WIM). In three simulated scenarios, we will compare the WIM to the theoretical Wasserstein approach, as well as to two competitor prior impact measures from the literature. We finally illustrate the versatility of the WIM by applying it on two datasets.

stat.ME↗
arXiv2019

Evaluating one-shot tournament predictions

We introduce the Tournament Rank Probability Score (TRPS) as a measure to evaluate and compare pre-tournament predictions, where predictions of the full tournament results are required to be available before the tournament begins. The TRPS handles partial ranking of teams, gives credit to predictions that are only slightly wrong, and can be modified with weights to stress the importance of particular features of the tournament prediction. Thus, the Tournament Rank Prediction Score is more flexible than the commonly preferred log loss score for such tasks. In addition, we show how predictions from historic tournaments can be optimally combined into ensemble predictions in order to maximize the TRPS for a new tournament.

stat.AP↗
arXiv2019

Optimal tests for elliptical symmetry: specified and unspecified location

Although the assumption of elliptical symmetry is quite common in multivariate analysis and widespread in a number of applications, the problem of testing the null hypothesis of ellipticity so far has not been addressed in a fully satisfactory way. Most of the literature in the area indeed addresses the null hypothesis of elliptical symmetry with specified location and actually addresses location rather than non-elliptical alternatives. In this paper, we are proposing new classes of testing procedures, both for specified and unspecified location. The backbone of our construction is Le Cam's asymptotic theory of statistical experiments, and optimality is to be understood locally and asymptotically within the family of generalized skew-elliptical distributions. The tests we are proposing are meeting all the desired properties of a``good'' test of elliptical symmetry: they have a simple asymptotic distribution under the entire null hypothesis of elliptical symmetry with unspecified radial density and shape parameter; they are affine-invariant, computationally fast, intuitively understandable, and not too demanding in terms of moments. While achieving optimality against generalized skew-elliptical alternatives, they remain quite powerful under a much broader class of non-elliptical distributions and significantly outperform the available competitors.

stat.ME↗
arXiv2019

Simple variance bounds with applications to Bayesian posteriors and intractable distributions

Using coupling techniques based on Stein's method for probability approximation, we revisit classical variance bounding inequalities of Chernoff, Cacoullos, Chen and Klaassen. Taking advantage of modern coupling techniques allows us to establish novel variance bounds in settings where the underlying density function is unknown or intractable. Applications include bounds for asymptotically Gaussian random variables using zero-biased couplings, bounds for random variables which are New Better (Worse) than Used in Expectation, and analysis of the posterior in Bayesian statistics.

math.PR↗
arXiv2019

Sine-skewed toroidal distributions and their application in protein bioinformatics

In the bioinformatics field, there has been a growing interest in modelling dihedral angles of amino acids by viewing them as data on the torus. This has motivated, over the past years, new proposals of distributions on the bivariate torus. The main drawback of most of these models is that the related densities are (pointwise) symmetric, despite the fact that the data usually present asymmetric patterns. This motivates the need to find a new way of constructing asymmetric toroidal distributions starting from a symmetric distribution. We tackle this problem in this paper by introducing the sine-skewed toroidal distributions. The general properties of the new models are derived. Based on the initial symmetric model, explicit expressions for the shape parameters are obtained, a simple algorithm for generating random numbers is provided, and asymptotic results for the maximum likelihood estimators are established. An important feature of our construction is that no normalizing constant needs to be calculated, leading to more flexible distributions without increasing the complexity of the models. The benefit of employing these new sine-skewed distributions is shown on the basis of protein data, where, in general, the new models outperform their symmetric antecedents.

stat.ME↗
arXiv2019

Hybrid Machine Learning Forecasts for the FIFA Women's World Cup 2019

In this work, we combine two different ranking methods together with several other predictors in a joint random forest approach for the scores of soccer matches. The first ranking method is based on the bookmaker consensus, the second ranking method estimates adequate ability parameters that reflect the current strength of the teams best. The proposed combined approach is then applied to the data from the two previous FIFA Women's World Cups 2011 and 2015. Finally, based on the resulting estimates, the FIFA Women's World Cup 2019 is simulated repeatedly and winning probabilities are obtained for all teams. The model clearly favors the defending champion USA before the host France.

stat.ML↗
arXiv2018

Ranking soccer teams on basis of their current strength: a comparison of maximum likelihood approaches

We present ten different strength-based statistical models that we use to model soccer match outcomes with the aim of producing a new ranking. The models are of four main types: Thurstone-Mosteller, Bradley-Terry, Independent Poisson and Bivariate Poisson, and their common aspect is that the parameters are estimated via weighted maximum likelihood, the weights being a match importance factor and a time depreciation factor giving less weight to matches that are played a long time ago. Since our goal is to build a ranking reflecting the teams' current strengths, we compare the 10 models on basis of their predictive performance via the Rank Probability Score at the level of both domestic leagues and national teams. We find that the best models are the Bivariate and Independent Poisson models. We then illustrate the versatility and usefulness of our new rankings by means of three examples where the existing rankings fail to provide enough information or lead to peculiar results.

stat.AP↗
arXiv2018

Prediction of the FIFA World Cup 2018 - A random forest approach with an emphasis on estimated team ability parameters

In this work, we compare three different modeling approaches for the scores of soccer matches with regard to their predictive performances based on all matches from the four previous FIFA World Cups 2002 - 2014: Poisson regression models, random forests and ranking methods. While the former two are based on the teams' covariate information, the latter method estimates adequate ability parameters that reflect the current strength of the teams best. Within this comparison the best-performing prediction methods on the training data turn out to be the ranking methods and the random forests. However, we show that by combining the random forest with the team ability parameters from the ranking methods as an additional covariate we can improve the predictive power substantially. Finally, this combination of methods is chosen as the final model and based on its estimates, the FIFA World Cup 2018 is simulated repeatedly and winning probabilities are obtained for all teams. The model slightly favors Spain before the defending champion Germany. Additionally, we provide survival probabilities for all teams and at all tournament stages as well as the most probable tournament outcome.

stat.AP↗
arXiv2017

Natural (non-)informative priors for skew-symmetric distributions

In this paper, we present an innovative method for constructing proper priors for the skewness (shape) parameter in the skew-symmetric family of distributions. The proposed method is based on assigning a prior distribution on the perturbation effect of the shape parameter, which is quantified in terms of the Total Variation distance. We discuss strategies to translate prior beliefs about the asymmetry of the data into an informative prior distribution of this class. We show via a Monte Carlo simulation study that our noninformative priors induce posterior distributions with good frequentist properties, similar to those of the Jeffreys prior. Our informative priors yield better results than their competitors from the literature. We also propose a scale- and location-invariant prior structure for models with unknown location and scale parameters and provide sufficient conditions for the propriety of the corresponding posterior distribution. Illustrative examples are presented using simulated and real data.

stat.ME↗
arXiv2017

Optimal tests for circular reflective symmetry about an unknown central direction

Parametric and semiparametric tests of circular reflective symmetry about an unknown central direction are developed that are locally and asymptotically optimal in the Le Cam sense against asymmetric $k$-sine-skewed alternatives. The results from Monte Carlo studies comparing the rejection rates of tests with those of previously proposed tests lead to recommendations regarding the use of the various tests with small- to medium-sized samples. Analyses of data on the directions of cracks in cemented femoral components and the times of gun crimes in Pittsburgh illustrate the proposed methodology and its bootstrap extension.

stat.ME↗
arXiv2016

An Interpolating Family of Size Distributions

We introduce a new five-parameter family of size distributions on the semi-finite interval $[x_0, \infty), x_0 \geqslant 0$, with two attractive features. First, it interpolates between power laws, such as the Pareto distribution, and power laws with exponential cut-off, such as the Weibull distribution. The proposed family is thus very flexible and spans over a broad range of well-known size distributions which are special cases of our family. Second, it has important tractability advantages over the popular five-parameter Generalized Beta distribution. We derive the hazard function, survival function, modes and quantiles, propose a random number generation procedure and discuss maximum likelihood estimation issues. Finally, we illustrate the wide applicability and fitting capacities of our new model on basis of three real data sets from very diverse domains, namely actuarial science, environmental science and survival analysis.

math.ST↗
arXiv2016

Bounds for the asymptotic normality of the maximum likelihood estimator using the Delta method

The asymptotic normality of the Maximum Likelihood Estimator (MLE) is a cornerstone of statistical theory. In the present paper, we provide sharp explicit upper bounds on Zolotarev-type distances between the exact, unknown distribution of the MLE and its limiting normal distribution. Our approach to this fundamental issue is based on a sound combination of the Delta method, Stein's method, Taylor expansions and conditional expectations, for the classical situations where the MLE can be expressed as a function of a sum of independent and identically distributed terms. This encompasses in particular the broad exponential family of distributions.

math.ST↗
arXiv2016

Stein's method for comparison of univariate distributions

We propose a new general version of Stein's method for univariate distributions. In particular we propose a canonical definition of the Stein operator of a probability distribution {which is based on a linear difference or differential-type operator}. The resulting Stein identity highlights the unifying theme behind the literature on Stein's method (both for continuous and discrete distributions). Viewing the Stein operator as an operator acting on pairs of functions, we provide an extensive toolkit for distributional comparisons. Several abstract approximation theorems are provided. Our approach is illustrated for comparison of several pairs of distributions : normal vs normal, sums of independent Rademacher vs normal, normal vs Student, and maximum of random variables vs exponential, Frechet and Gumbel.

math.PR↗
arXiv2015

A tractable, parsimonious and flexible model for cylindrical data, with applications

In this paper, we propose cylindrical distributions obtained by combining the sine-skewed von Mises distribution (circular part) with the Weibull distribution (linear part). This new model, the WeiSSVM, enjoys numerous advantages: simple normalizing constant and hence very tractable density, parameter-parsimony and interpretability, good circular-linear dependence structure, easy random number generation thanks to known marginal/conditional distributions, flexibility illustrated via excellent fitting abilities, and a straightforward extension to the case of directional-linear data. Inferential issues, such as independence testing, circular-linear respectively linear-circular regression, can easily be tackled with our model, which we apply on two real data sets. We conclude the paper by discussing future applications of our model.

stat.ME↗
arXiv2015

Distances between nested densities and a measure of the impact of the prior in Bayesian statistics

In this paper we propose tight upper and lower bounds for the Wasserstein distance between any two {univariate continuous distributions} with probability densities $p_1$ and $p_2$ having nested supports. These explicit bounds are expressed in terms of the derivative of the likelihood ratio $p_1/p_2$ as well as the Stein kernel $τ_1$ of $p_1$. The method of proof relies on a new variant of Stein's method which manipulates Stein operators. We give several applications of these bounds. Our main application is in Bayesian statistics : we derive explicit data-driven bounds on the Wasserstein distance between the posterior distribution based on a given prior and the no-prior posterior based uniquely on the sampling distribution. This is the first finite sample result confirming the well-known fact that with well-identified parameters and large sample sizes, reasonable choices of prior distributions will have only minor effects on posterior inferences if the data are benign.

math.PR↗
arXiv2015

Asymptotic properties of QML estimators for VARMA models with time-dependent coefficients: Part I

This paper is about vector autoregressive-moving average (VARMA) models with time-dependent coefficients to represent non-stationary time series. Contrarily to other papers in the univariate case, the coefficients depend on time but not on the length of the series $n$. Under appropriate assumptions, it is shown that a Gaussian quasi-maximum likelihood estimator is almost surely consistent and asymptotically normal. The theoretical results are illustrated by means of two examples of bivariate processes. It is shown that the assumptions underlying the theoretical results apply. In the second example the innovations are also marginally heteroscedastic with a correlation ranging from -0.8 to 0.8. In the two examples, the asymptotic information matrix is obtained in the Gaussian case. Finally, the finite-sample behaviour is checked via a Monte Carlo simulation study for $n$ going from 25 to 400. The results confirm the validity of the asymptotic properties even for short series and reveal that the asymptotic information matrix deduced from the theory is correct.

math.ST↗
arXiv2014

Flexible modelling in statistics: past, present and future

In times where more and more data become available and where the data exhibit rather complex structures (significant departure from symmetry, heavy or light tails), flexible modelling has become an essential task for statisticians as well as researchers and practitioners from domains such as economics, finance or environmental sciences. This is reflected by the wealth of existing proposals for flexible distributions; well-known examples are Azzalini's skew-normal, Tukey's $g$-and-$h$, mixture and two-piece distributions, to cite but these. My aim in the present paper is to provide an introduction to this research field, intended to be useful both for novices and professionals of the domain. After a description of the research stream itself, I will narrate the gripping history of flexible modelling, starring emblematic heroes from the past such as Edgeworth and Pearson, then depict three of the most used flexible families of distributions, and finally provide an outlook on future flexible modelling research by posing challenging open questions.

stat.ME↗