Search

We introduce one-sided versions of Huber's contamination model, in which corrupted samples tend to take larger values than uncorrupted ones. Two intertwined problems are addressed: estimation of the mean of uncorrupted samples (minimum effect) and selection of corrupted samples (outliers). Regarding the minimum effect estimation, we derive the minimax risks and introduce adaptive estimators to the unknown number of contaminations. Interestingly, the optimal convergence rate highly differs from that in classical Huber's contamination model. Also, our analysis uncovers the effect of particular structural assumptions on the distribution of the contaminated samples. As for the problem of selecting the outliers, we formulate the problem in a multiple testing framework for which the location/scaling of the null hypotheses are unknown. We rigorously prove how estimating the null hypothesis is possible while maintaining a theoretical guarantee on the amount of the falsely selected outliers, both through false discovery rate (FDR) or post hoc bounds. As a by-product, we address a long-standing open issue on FDR control under equi-correlation, which reinforces the interest of removing dependency when making multiple testing., Comment: 70 pages; 7 figures

In this paper, we are interested in the following problem: find a curve $f$ minimizing the quantity $\mathbb{E}[\min_{t\in[0,1]}\|X-f(t)\|^{2}]$, where $X$ is a random variable, under a length constraint. This question is known in the probability and statistical learning context as length-constrained principal curves optimization, as introduced in (IEEE Trans. Pattern Anal. Mach. Intell. 22 (2000) 281–297), and it also corresponds to a version of the “average-distance problem” studied in the calculus of variation and shape optimization community (Ann. Sc. Norm. Super. Pisa Cl. Sci. II (4) (2003) 631–678; Progr. Nonlinear Differential Equations Appl. 51 (2002) 41–65). ¶ We investigate the theoretical properties satisfied by a principal curve $f:[0,1]\to\mathbb{R}^{d}$ with length at most $L$, associated to a probability distribution with second-order moment. We suppose that the probability distribution is not supported on the image of a curve with length $L$. Studying open as well as closed optimal curves, we show that they have finite curvature. We also derive an Euler–Lagrange equation. This equation is then used to show that a length-constrained principal curve in two dimensions has no multiple point. Finally, some examples of optimal curves are presented.

We study in detail the bias and variance of the entropy estimator proposed by Kozachenko and Leonenko (1987) for a large class of densities on R d . We then use the work of Bickel and Breiman (1983) to prove a central limit theorem in dimensions 1 and 2 . In higher dimensions, we provide a development of the bias in terms of powers of N − 2 / d . This allows us to use a Richardson extrapolation to build, in any dimension, a root- n consistent entropy estimator satisfying a central limit theorem which allows for explicit (asymptotic) confidence intervals. To our knowledge, all the previous general root- n consistency results were concerning dimension 1 .

We address the issue of the proximity of interacting diffusion models on large graphs with a uniform degree property and a corresponding mean field model, i.e. a model on the complete graph with a suitably renormalized interaction parameter. Examples include Erd\H{o}s-R\'enyi graphs with edge probability $p_n$, $n$ is the number of vertices, such that $\lim_{n \to \infty}p_n n= \infty$. The purpose of this note it twofold: (1) to establish this proximity on finite time horizon, by exploiting the fact that both systems are accurately described by a Fokker-Planck PDE (or, equivalently, by a nonlinear diffusion process) in the $n=\infty$ limit; (2) to remark that in reality this result is unsatisfactory when it comes to applying it to systems with $N$ large but finite, for example the values of $N$ that can be reached in simulations or that correspond to the typical number of interacting units in a biological system., Comment: 14 pages, 1 figure. A few corrections, streamlined some explanations and improved Corollary 1.2

We study the convergence of the false discovery proportion (FDP) of the Benjamini–Hochberg procedure in the Gaussian equi-correlated model, when the correlation ρ m converges to zero as the hypothesis number m grows to infinity. In this model, the FDP converges to the false discovery rate (FDR) at rate { min ( m , 1 / ρ m ) } 1 / 2 , which is different from the standard convergence rate m 1 / 2 holding under independence.

International audience; We generalise the construction of multivariate Hawkes processes to a possibly infinite network of counting processes on a directed graph G. The process is constructed as the solution to a system of Poisson driven stochastic differential equations, for which we prove pathwise existence and uniqueness under some reasonable conditions. We next investigate how to approximate a standard N -dimensional Hawkes process by a simple inhomogeneous Poisson process in the mean-field framework where each pair of individuals interact in the same way, in the limit N → ∞. In the so-called linear case for the interaction, we further investigate the large time behaviour of the process. We study in particular the stability of the central limit theorem when exchanging the limits N, T → ∞ and exhibit different possible behaviours. We finally consider the case G = Z d with nearest neighbour interactions. In the linear case, we prove some (large time) laws of large numbers and exhibit different behaviours, reminiscent of the infinite setting. Finally we study the propagation of a single impulsion started at a given point of Z d at time 0. We compute the probability of extinction of such an impulsion and, in some particular cases, we can accurately describe how it propagates to the whole space. Mathematics Subject Classification (2010): 60F05, 60G55, 60G57.

We consider a population of N N individuals, of which we observe the number of actions until time t t . For each couple of individuals (i,j) i j , j j may or not influence i i , which we model by i.i.d. Bernoulli(p) p -random variables, for some unknown parameter p∈(0,1] p 0 1 . Each individual acts autonomously at some unknown rate μ>0 μ 0 and acts by mimetism at some rate proportional to the sum of some function φ φ of the ages of the actions of the individuals which influence him. The function φ φ is unknown but assumed, roughly, to be decreasing and with fast decay. The goal of this paper is to estimate p p , which is the main characteristic of the graph of interactions, in the asymptotic N→∞ N ∞ , t→∞ t ∞ . The main issue is that the mean field limit (as N→∞ N ∞ ) of this model is unidentifiable, in that it only depends on the parameters μ μ and pφ p φ . Fortunately, this mean field limit is not valid for large times. We distinguish the subcritical case, where, roughly, the mean number mt m t of actions per individual increases linearly and the supercritical case, where mt m t increases exponentially. Although the nuisance parameter φ φ is non-parametric, we are able, in both cases, to estimate p p without estimating φ φ in a nonparametric way, with a precision of order N−1/2+N1/2m−1t N 1 2 N 1 2 m t 1 , up to some arbitrarily small loss. We explain, using a Gaussian toy model, the reason why this rate of convergence might be (almost) optimal.

We consider one dimensional probability distributions μ having a continuous and positive probability density function. We find the asymptotic of the size and the mass of the Voronoi cells and we prove that the local distortion associated with stationary or optimal quantizers is asymptotically uniform. Numerical simulations and computations illustrate the theoretical results and lead to the design of some good-fit test for the stationary equilibria.

We study the problem of the efficient estimation of the jumps for stochastic processes. We assume that the stochastic jump process $(X_t)_{t\in[0,1]}$ is observed discretely, with a sampling step of size $1/n$. In the spirit of Hajek's convolution theorem, we show some lower bounds for the estimation error of the sequence of the jumps $(\Delta X_{T_k})_k$. As an intermediate result, we prove a LAMN property, with rate $\sqrt{n}$, when the marks of the underlying jump component are deterministic. We deduce then a convolution theorem, with an explicit asymptotic minimal variance, in the case where the marks of the jump component are random. To prove that this lower bound is optimal, we show that a threshold estimator of the sequence of jumps $(\Delta X_{T_k})_k$ based on the discrete observations, reaches the minimal variance of the previous convolution theorem., Comment: Published in at http://dx.doi.org/10.3150/13-BEJ515 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

International audience; We introduce a new stochastic model for the variations of asset prices at the tick-by-tick level in dimension 1 (for a single asset) and 2 (for a pair of assets). The construction is based on marked point pro- cesses and relies on linear self and mutually exciting stochastic inten- sities as introduced by Hawkes. We associate a counting process with the positive and negative jumps of an asset price. By coupling suitably the stochastic intensities of upward and downward changes of prices for several assets simultaneously, we can reproduce microstructure noise (i.e. strong microscopic mean reversion at the level of seconds to a few minutes) and the Epps effect (i.e. the decorrelation of the increments in microscopic scales) while preserving a standard Brownian diffusion behaviour on large scales. More effectively, we obtain analytical closed-form formulae for the mean signature plot and the correlation of two price increments that enable to track across scales the effect of the mean-reversion up to the diffusive limit of the model. We show that the theoretical results are consistent with empirical fits on futures Euro-Bund and Euro-Bobl in several situations.

A Special Issue on the Occasion of the 2013 International Year of Statistics; International audience; Abstract In the context of statistics for random processes, we prove a law of large numbers and a functional central limit theorem for multivariate Hawkes processes observed over a time interval [ 0 , T ] when T ? ? . We further exhibit the asymptotic behaviour of the covariation of the increments of the components of a multivariate Hawkes process, when the observations are imposed by a discrete scheme with mesh ? over [ 0 , T ] up to some further time shift ? . The behaviour of this functional depends on the relative size of ? and ? with respect to T and enables to give a full account of the second-order structure. As an application, we develop our results in the context of financial statistics. We introduced in Bacry et al. (2013) [7] a microscopic stochastic model for the variations of a multivariate financial asset, based on Hawkes processes and that is confined to live on a tick grid. We derive and characterise the exact macroscopic diffusion limit of this model and show in particular its ability to reproduce the important empirical stylised fact such as the Epps effect and the lead?lag effect. Moreover, our approach enables to track these effects across scales in rigorous mathematical terms.

26 pages; International audience; This paper proposes a general approach to obtain asymptotic lower bounds for the estimation of random functionals. The main result is an abstract convolution theorem in a non parametric setting, based on an associated LAMN property. This result is then applied to the estimation of the integrated volatility, or related quantities, of a diffusion process, when the diffusion coefficient depends on an independent Brownian motion.

This paper introduces a new framework to study the asymptotical behavior of the empirical distribution function (e.d.f.) of Gaussian vector components, whose correlation matrix $\Gamma^{(m)}$ is dimension-dependent. Hence, by contrast with the existing literature, the vector is not assumed to be stationary. Rather, we make a ''vanishing second order" assumption ensuring that the covariance matrix $\Gamma^{(m)}$ is not too far from the identity matrix, while the behavior of the e.d.f. is affected by $\Gamma^{(m)}$ only through the sequence $\gamma_m=m^{-2} \sum_{i\neq j} \Gamma_{i,j}^{(m)}$, as $m$ grows to infinity. This result recovers some of the previous results for stationary long-range dependencies while it also applies to various, high-dimensional, non-stationary frameworks, for which the most correlated variables are not necessarily next to each other. Finally, we present an application of this work to the multiple testing problem, which was the initial statistical motivation for developing such a methodology.

http://ecp.ejpecp.org/; International audience; We consider the following statistical problem: based on an i.i.d. sample of size n of integer valued random variables with common law mu, is it possible to test whether or not the support of mu is finite as n goes to infinity? This question is in particular connected to a simple case of Tsirelson's equation, for which it is natural to distinguish between two main configurations, the first one leading only to laws with finite support, and the second one including laws with infinite support. We show that it is in fact not possible to discriminate between the two situations, even using a very weak notion of statistical test.

International audience; We consider linear inverse problems in a nonparametric statistical framework. Both the signal and the operator are unknown and subject to error measurements. We establish minimax rates of convergence under squared error loss when the operator admits a blockwise singular value decomposition (blockwise SVD) and the smoothness of the signal is measured in a Sobolev sense. We construct a nonlinear procedure adapting simultaneously to the unknown smoothness of both the signal and the operator and achieving the optimal rate of convergence to within logarithmic terms. When the noise level in the operator is dominant, by taking full advantage of the blockwise SVD property, we demonstrate that the block SVD procedure outperforms classical methods based on Galerkin projection or nonlinear wavelet thresholding. We subsequently apply our abstract framework to the specific case of blind deconvolution on the torus and on the sphere.

International audience; We consider statistical hypothesis testing simultaneously over a fairly general, possibly uncountably infinite, set of null hypotheses, under the assumption that a suitable single test (and corresponding $p$-value) is known for each individual hypothesis. We extend to this setting the notion of false discovery rate (FDR) as a measure of type I error. Our main result studies specific procedures based on the observation of the $p$-value process. Control of the FDR at a nominal level is ensured either under arbitrary dependence of $p$-values, or under the assumption that the finite dimensional distributions of the $p$-value process have positive correlations of a specific type (weak PRDS). Both cases generalize existing results established in the finite setting. Its interest is demonstrated in several non-parametric examples: testing the mean/signal in a Gaussian white noise model, testing the intensity of a Poisson process and testing the c.d.f. of i.i.d. random variables.

International audience; We consider the problem of adaptive estimation of the regression function in a framework where we replace ergodicity assumptions (such as independence or mixing) by another structural assumption on the model. Namely, we propose adaptive upper bounds for kernel estimators with data-driven bandwidth (Lepski's selection rule) in a regression model where the noise is an increment of martingale. It includes, as very particular cases, the usual i.i.d. regression and auto-regressive models. The cornerstone tool for this study is a new result for self-normalized martingales, called "stability", which is of independent interest. In a first part, we only use the martingale increment structure of the noise. We give an adaptive upper bound using a random rate, that involves the occupation time near the estimation point. Thanks to this approach, the theoretical study of the statistical procedure is disconnected from usual ergodicity properties like mixing. Then, in a second part, we make a link with the usual minimax theory of deterministic rates. Under a beta-mixing assumption on the covariates process, we prove that the random rate considered in the first part is equivalent, with large probability, to a deterministic rate which is the usual minimax adaptive one

For a probability measure $P$ on $\R^d$ and $n\!\! \in \! \N$ consider $e_n = \inf \displaystyle \int \min_{a \in \alpha} V(\| x-a \| )dP(x)$ where the infimum is taken over all subsets $\alpha$ of $\R^d$ with $\mbox{card} (\alpha) \leq n$ and $V$ is a nondecreasing function. Under certain conditions on $V$, we derive the precise $n$-asymptotics of $e_n$ for nonsingular and for (singular) self-similar distributions $P$ and we find the asymptotic performance of optimal quantizers using weighted empirical measures.