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Search Results: 1 - 10 of 4387 matches for " Christophe Ley "
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Flexible modelling in statistics: past, present and future
Christophe Ley
Statistics , 2014,
Abstract: 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.
Stein's density approach and information inequalities
Christophe Ley,Yvik Swan
Mathematics , 2012,
Abstract: We provide a new perspective on Stein's so-called density approach by introducing a new operator and characterizing class which are valid for a much wider family of probability distributions on the real line. We prove an elementary factorization property of this operator and propose a new Stein identity which we use to derive information inequalities in terms of what we call the \emph{generalized Fisher information distance}. We provide explicit bounds on the constants appearing in these inequalities for several important cases. We conclude with a comparison between our results and known results in the Gaussian case, hereby improving on several known inequalities from the literature.
Local Pinsker inequalities via Stein's discrete density approach
Christophe Ley,Yvik Swan
Mathematics , 2012,
Abstract: Pinsker's inequality states that the relative entropy $d_{\mathrm{KL}}(X, Y)$ between two random variables $X$ and $Y$ dominates the square of the total variation distance $d_{\mathrm{TV}}(X,Y)$ between $X$ and $Y$. In this paper we introduce generalized Fisher information distances $\mathcal{J}(X, Y)$ between discrete distributions $X$ and $Y$ and prove that these also dominate the square of the total variation distance. To this end we introduce a general discrete Stein operator for which we prove a useful covariance identity. We illustrate our approach with several examples. Whenever competitor inequalities are available in the literature, the constants in ours are at least as good, and, in several cases, better.
Discrete Stein characterizations and discrete information distances
Christophe Ley,Yvik Swan
Mathematics , 2011,
Abstract: We construct two different Stein characterizations of discrete distributions and use these to provide a natural connection between Stein characterizations for discrete distributions and discrete information functionals.
A unified approach to Stein characterizations
Christophe Ley,Yvik Swan
Mathematics , 2011,
Abstract: This article deals with Stein characterizations of probability distributions. We provide a general framework for interpreting these in terms of the parameters of the underlying distribution. In order to do so we introduce two concepts (a class of functions and an operator) which generalize those which were developed in the 70's by Charles Stein and Louis Chen for characterizing the Gaussian and the Poisson distributions. Our methodology (i) allows for writing many (if not all) known univariate Stein characterizations, (ii) permits to identify clearly minimal conditions under which these results hold and (iii) provides a straightforward tool for constructing new Stein characterizations. Our parametric interpretation of Stein characterizations also raises a number of questions which we outline at the end of the paper.
Skew-symmetric distributions and Fisher information -- a tale of two densities
Marc Hallin,Christophe Ley
Statistics , 2012, DOI: 10.3150/12-BEJ346
Abstract: Skew-symmetric densities recently received much attention in the literature, giving rise to increasingly general families of univariate and multivariate skewed densities. Most of those families, however, suffer from the inferential drawback of a potentially singular Fisher information in the vicinity of symmetry. All existing results indicate that Gaussian densities (possibly after restriction to some linear subspace) play a special and somewhat intriguing role in that context. We dispel that widespread opinion by providing a full characterization, in a general multivariate context, of the information singularity phenomenon, highlighting its relation to a possible link between symmetric kernels and skewing functions -- a link that can be interpreted as the mismatch of two densities.
Local powers of optimal one- and multi-sample tests for the concentration of Fisher-von Mises-Langevin distributions
Christophe Ley,Thomas Verdebout
Statistics , 2013,
Abstract: One-sample and multi-sample tests on the concentration parameter of Fisher-von Mises-Langevin (FvML) distributions have been well studied in the literature. However, only very little is known about their behavior under local alternatives, which is due to complications inherent to the curved nature of the parameter space. The aim of the present paper therefore consists in filling that gap by having recourse to the Le Cam methodology, which has been adapted from the linear to the spherical setup in Ley \emph{et al.} (2013). We obtain explicit expressions of the powers for the most efficient one- and multi-sample tests; these tests are those considered in Watamori and Jupp (2005). As a nice by-product, we are also able to write down the powers (against local FvML alternatives) of the celebrated Rayleigh (1919) test of uniformity. A Monte Carlo simulation study confirms our theoretical findings and shows the finite-sample behavior of the above-mentioned procedures.
Simple, asymptotically distribution-free, optimal tests for circular reflective symmetry about a known median direction
Christophe Ley,Thomas Verdebout
Statistics , 2013,
Abstract: In this paper, we propose optimal tests for circular reflective symmetry about a fixed median direction. The distributions against which optimality is achieved are the so-called k-sine-skewed distributions of Umbach and Jammalamadaka (2009). We first show that sequences of k-sine-skewed models are locally and asymptotically normal in the vicinity of reflective symmetry. Following the Le Cam methodology, we then construct optimal (in the maximin sense) parametric tests for reflective symmetry, which we render semi-parametric by a studentization argument. These asymptotically distribution-free tests happen to be uniformly optimal (under any reference density) and are moreover of a very simple and intuitive form. They furthermore exhibit nice small sample properties, as we show through a Monte Carlo simulation study. Our new tests also allow us to re-visit the famous red wood ants data set of Jander (1957). We further show that one of the proposed parametric tests can as well serve as a test for uniformity against cardioid alternatives; this test coincides with the famous circular Rayleigh (1919) test for uniformity which is thus proved to be (also) optimal against cardioid alternatives. Moreover, our choice of k-sine-skewed alternatives, which are the circular analogues of the classical linear skew-symmetric distributions, permits us a Fisher singularity analysis \`a la Hallin and Ley (2012) with the result that only the prominent sine-skewed von Mises distribution suffers from these inferential drawbacks. Finally, we conclude the paper by discussing the unspecified location case.
A tractable, parsimonious and highly flexible model for cylindrical data, with applications
Toshihiro Abe,Christophe Ley
Statistics , 2015,
Abstract: 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, 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.
Efficient inference about the tail weight in multivariate Student $t$ distributions
Christophe Ley,Anouk Neven
Statistics , 2013,
Abstract: We propose a new testing procedure about the tail weight parameter of multivariate Student $t$ distributions by having recourse to the Le Cam methodology. Our test is asymptotically as efficient as the classical likelihood ratio test, but outperforms the latter by its flexibility and simplicity: indeed, our approach allows to estimate the location and scatter nuisance parameters by any root-$n$ consistent estimators, hereby avoiding numerically complex maximum likelihood estimation. The finite-sample properties of our test are analyzed in a Monte Carlo simulation study, and we apply our method on a financial data set. We conclude the paper by indicating how to use this framework for efficient point estimation.
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