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Christophe Ley

Publications and source records attributed to Christophe Ley.

At least 55 records · Page 3Linked to original sources
arXiv2014

Skew-symmetric distributions and Fisher information: The double sin of the skew-normal

Hallin and Ley [Bernoulli 18 (2012) 747-763] investigate and fully characterize the Fisher singularity phenomenon in univariate and multivariate families of skew-symmetric distributions. This paper proposes a refined analysis of the (univariate) problem, showing that singularity can be more or less severe, inducing $n^{1/4}$ ("simple singularity"), $n^{1/6}$ ("double singularity"), or $n^{1/8}$ ("triple singularity") consistency rates for the skewness parameter. We show, however, that simple singularity (yielding $n^{1/4}$ consistency rates), if any singularity at all, is the rule, in the sense that double and triple singularities are possible for generalized skew-normal families only. We also show that higher-order singularities, leading to worse-than-$n^{1/8}$ rates, cannot occur. Depending on the degree of the singularity, our analysis also suggests a simple reparametrization that offers an alternative to the so-called centred parametrization proposed, in the particular case of skew-normal and skew-$t$ families, by Azzalini [Scand. J. Stat. 12 (1985) 171-178], Arellano-Valle and Azzalini [J. Multivariate Anal. 113 (2013) 73-90], and DiCiccio and Monti [Quaderni di Statistica 13 (2011) 1-21], respectively.

math.ST↗
arXiv2014

Efficient inference about the tail weight in multivariate Student $t$ distributions

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.

stat.ME↗
arXiv2014

Maximum likelihood characterization of distributions

A famous characterization theorem due to C.F. Gauss states that the maximum likelihood estimator (MLE) of the parameter in a location family is the sample mean for all samples of all sample sizes if and only if the family is Gaussian. There exist many extensions of this result in diverse directions, most of them focussing on location and scale families. In this paper, we propose a unified treatment of this literature by providing general MLE characterization theorems for one-parameter group families (with particular attention on location and scale parameters). In doing so, we provide tools for determining whether or not a given such family is MLE-characterizable, and, in case it is, we define the fundamental concept of minimal necessary sample size at which a given characterization holds. Many of the cornerstone references on this topic are retrieved and discussed in the light of our findings, and several new characterization theorems are provided. Of particular interest is that one part of our work, namely the introduction of so-called equivalence classes for MLE characterizations, is a modernized version of Daniel Bernoulli's viewpoint on maximum likelihood estimation.

math.ST↗
arXiv2014

High-dimensional tests for spherical location and spiked covariance

Rotationally symmetric distributions on the p-dimensional unit hypersphere, extremely popular in directional statistics, involve a location parameter theta that indicates the direction of the symmetry axis. The most classical way of addressing the spherical location problem H_0:theta=theta_0, with theta_0 a fixed location, is the so-called Watson test, which is based on the sample mean of the observations. This test enjoys many desirable properties, but its implementation requires the sample size n to be large compared to the dimension p. This is a severe limitation, since more and more problems nowadays involve high-dimensional directional data (e.g., in genetics or text mining). In this work, we therefore introduce a modified Watson statistic that can cope with high-dimensionality. We derive its asymptotic null distribution as both n and p go to infinity. This is achieved in a universal asymptotic framework that allows p to go to infinity arbitrarily fast (or slowly) as a function of n. We further show that our results also provide high-dimensional tests for a problem that has recently attracted much attention, namely that of testing that the covariance matrix of a multinormal distribution has a "theta_0-spiked" structure. Finally, a Monte Carlo simulation study corroborates our asymptotic results.

math.ST↗
arXiv2014

The value at the mode in multivariate $t$ distributions: a curiosity or not?

It is a well-known fact that multivariate Student $t$ distributions converge to multivariate Gaussian distributions as the number of degrees of freedom $ν$ tends to infinity, irrespective of the dimension $k\geq1$. In particular, the Student's value at the mode (that is, the normalizing constant obtained by evaluating the density at the center) $c_{ν,k}=\frac{Γ(\frac{ν+k}{2})}{(πν)^{k/2} Γ( \fracν{2})}$ converges towards the Gaussian value at the mode $c_k=\frac{1}{(2π)^{k/2}}$. In this note, we prove a curious fact: $c_{ν,k}$ tends monotonically to $c_k$ for each $k$, but the monotonicity changes from increasing in dimension $k=1$ to decreasing in dimensions $k\geq3$ whilst being constant in dimension $k=2$. A brief discussion raises the question whether this \emph{a priori} curious finding is a curiosity, \emph{in fine}.

math.ST↗
arXiv2014

Depth-based Runs Tests for Bivariate Central Symmetry

McWilliams (1990) introduced a nonparametric procedure based on runs for the problem of testing univariate symmetry about the origin (equivalently, about an arbitrary specified center). His procedure first reorders the observations according to their absolute values, then rejects the null when the number of runs in the resulting series of signs is too small. This test is universally consistent and enjoys nice robustness properties, but is unfortunately limited to the univariate setup. In this paper, we extend McWilliams' procedure into tests of bivariate central symmetry. The proposed tests first reorder the observations according to their statistical depth in a symmetrized version of the sample, then reject the null when an original concept of simplicial runs is too small. Our tests are affine-invariant and have good robustness properties. In particular, they do not require any finite moment assumption. We derive their limiting null distribution, which establishes their asymptotic distribution-freeness. We study their finite-sample properties through Monte Carlo experiments, and conclude with some final comments.

stat.ME↗
arXiv2013

Characterizations of GIG laws: a survey complemented with two new results

Several characterizations of the Generalized Inverse Gaussian (GIG) distribution have been proposed in the literature, especially over the past two decades. These characterization theorems are surveyed, and two new characterizations are established, one based on maximum likelihood estimation and the other is a Stein characterization.

math.PR↗
arXiv2013

Efficiency combined with simplicity: new testing procedures for Generalized Inverse Gaussian models

The standard efficient testing procedures in the Generalized Inverse Gaussian (GIG) family (also known as Halphen Type A family) are likelihood ratio tests, hence rely on Maximum Likelihood (ML) estimation of the three parameters of the GIG. The particular form of GIG densities, involving modified Bessel functions, prevents in general from a closed-form expression for ML estimators, which are obtained at the expense of complex numerical approximation methods. On the contrary, Method of Moments (MM) estimators allow for concise expressions, but tests based on these estimators suffer from a lack of efficiency compared to likelihood ratio tests. This is why, in recent years, trade-offs between ML and MM estimators have been proposed, resulting in simpler yet not completely efficient estimators and tests. In the present paper, we do not propose such a trade-off but rather an optimal combination of both methods, our tests inheriting efficiency from an ML-like construction and simplicity from the MM estimators of the nuisance parameters. This goal shall be reached by attacking the problem from a new angle, namely via the Le Cam methodology. Besides providing simple efficient testing methods, the theoretical background of this methodology further allows us to write out explicitly power expressions for our tests. A Monte Carlo simulation study shows that, also at small sample sizes, our simpler procedures do at least as good as the complex likelihood ratio tests. We conclude the paper by applying our findings on two real-data sets.

stat.ME↗
arXiv2013

Local Pinsker inequalities via Stein's discrete density approach

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.

math.PR↗
arXiv2013

Parametric Stein operators and variance bounds

Stein operators are differential operators which arise within the so-called Stein's method for stochastic approximation. We propose a new mechanism for constructing such operators for arbitrary (continuous or discrete) parametric distributions with continuous dependence on the parameter. We provide explicit general expressions for location, scale and skewness families. We also provide a general expression for discrete distributions. For specific choices of target distributions (including the Gaussian, Gamma and Poisson) we compare the operators hereby obtained with those provided by the classical approaches from the literature on Stein's method. We use properties of our operators to provide upper and lower variance bounds (only lower bounds in the discrete case) on functionals $h(X)$ of random variables $X$ following parametric distributions. These bounds are expressed in terms of the first two moments of the derivatives (or differences) of $h$. We provide general variance bounds for location, scale and skewness families and apply our bounds to specific examples (namely the Gaussian, exponential, Gamma and Poisson distributions). The results obtained via our techniques are systematically competitive with, and sometimes improve on, the best bounds available in the literature.

math.PR↗
arXiv2013

Stein's density approach and information inequalities

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.

math.PR↗
arXiv2013

Simple, asymptotically distribution-free, optimal tests for circular reflective symmetry about a known median direction

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 à 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.

stat.ME↗
arXiv2013

Local powers of optimal one- and multi-sample tests for the concentration of Fisher-von Mises-Langevin distributions

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.

math.ST↗
arXiv2012

Efficient ANOVA for directional data

In this paper we tackle the ANOVA problem for directional data (with particular emphasis on geological data) by having recourse to the Le Cam methodology usually reserved for linear multivariate analysis. We construct locally and asymptotically most stringent parametric tests for ANOVA for directional data within the class of rotationally symmetric distributions. We turn these parametric tests into semi-parametric ones by (i) using a studentization argument (which leads to what we call pseudo-FvML tests) and by (ii) resorting to the invariance principle (which leads to efficient rank-based tests). Within each construction the semi-parametric tests inherit optimality under a given distribution (the FvML distribution in the first case, any rotationally symmetric distribution in the second) from their parametric antecedents and also improve on the latter by being valid under the whole class of rotationally symmetric distributions. Asymptotic relative efficiencies are calculated and the finite-sample behavior of the proposed tests is investigated by means of a Monte Carlo simulation. We conclude by applying our findings on a real-data example involving geological data.

math.ST↗
arXiv2012

Skew-symmetric distributions and Fisher information -- a tale of two densities

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.

math.ST↗
arXiv2012

Optimal R-Estimation of a Spherical Location

In this paper, we provide $R$-estimators of the location of a rotationally symmetric distribution on the unit sphere of $\R^k$. In order to do so we first prove the local asymptotic normality property of a sequence of rotationally symmetric models; this is a non standard result due to the curved nature of the unit sphere. We then construct our estimators by adapting the Le Cam one-step methodology to spherical statistics and ranks. We show that they are asymptotically normal under any rotationally symmetric distribution and achieve the efficiency bound under a specific density. Their small sample behavior is studied via a Monte Carlo simulation and our methodology is illustrated on geological data.

stat.AP↗
arXiv2011

A unified approach to Stein characterizations

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.

math.PR↗