Gabriel Lang

Quadratic variations and estimation of the local Hölder index of a Gaussian process

We study the convergence of a generalisation of the quadratic variations of a Gaussian process. We build a convergent estimator of the local Holder index of the sample paths and prove a central limit theorem.

Generalization of the Partitioning of Shannon Diversity

Traditional measures of diversity, namely the number of species as well as Simpsons and Shannons indices, are particular cases of Tsallis entropy. Entropy decomposition, i.e. decomposing gamma entropy into alpha and beta components, has been previously derived in the literature. We propose a generalization of the additive decomposition of Shannon entropy applied to Tsallis entropy. We obtain a self-contained definition of beta entropy as the information gain brought by the knowledge of each community composition. We propose a correction of the estimation bias allowing to estimate alpha, beta and gamma entropy from the data and eventually convert them into true diversity. We advocate additive decomposition in complement of multiplicative partitioning to allow robust estimation of biodiversity.

Analysis of the Spatial Organization of Molecules with Robust Statistics

One major question in molecular biology is whether the spatial distribution of observed molecules is random or organized in clusters. Indeed, this analysis gives information about molecules’ interactions and physical interplay with their environment. The standard tool for analyzing molecules’ distribution statistically is the Ripley’s K function, which tests spatial randomness through the computation of its critical quantiles. However, quantiles’ computation is very cumbersome, hindering its use. Here, we present an analytical expression of these quantiles, leading to a fast and robust statistical test, and we derive the characteristic clusters’ size from the maxima of the Ripley’s K function. Subsequently, we analyze the spatial organization of endocytic spots at the cell membrane and we report that clathrin spots are randomly distributed while clathrin-independent spots are organized in clusters with a radius of , which suggests distinct physical mechanisms and cellular functions for each pathway.

Asymptotics of weighted empirical processes of linear fields with long-range dependence

We discuss the asymptotic behavior of weighted empirical processes of stationary linear random fields in Z d with long-range dependence. It is shown that an appropriately standardized empirical process converges weakly in the uniform-topology to a degenerated process of the form fZ ,w hereZ is a standard normal random variable and f is the marginal probability density of the underlying random field.  2002 Editions scientifiques et medicales Elsevier SAS MSC: 60F17; 60G60 RESUME. - Nous etudions le comportement asymptotique du processus empirique pondere pour un champ lineaire stationnaire a longue dependance sur Z d . Nous montrons que ce processus convenablement normalise converge faiblement pour la topologie uniforme vers un processus degenere de la forme fZ ,o uZ est une variable normale standard et f est la densite de la marginale du champ considere.  2002 Editions scientifiques et medicales Elsevier SAS

Rates in the Empirical Central Limit Theorem for Stationary Weakly Dependent Random Fields

A weak dependence condition is derived as the natural generalization to random fields on notions developed in Doukhan and Louhichi (1999). Examples of such weakly dependent fields are defined. In the context of a weak dependence coefficient series with arithmetic or geometric decay, we give explicit bounds in Prohorov metric for the convergence in the empirical central limit theorem. For random fields indexed by &Zopfd, in the geometric decay case, rates have the form n−1/(8d+24)L(n), where L(n) is a power of log(n).

Semi-parametric Estimation of the Hölder Exponent of a Stationary Gaussian Process with Minimax Rates

Let (X(t))t∈[0,1] be a centered Gaussian process with stationary increments such that IE[(X{u+t-Xu)2] = C|t|s+r(t). Assume that there exists an extra parameter β > 0 and a polynomial P of degree smaller than s + β such that |r(t)-P(t)| is bounded with respect to |t|s+β. We consider the problem of estimating the parameter s ∈ (0,2) in the asymptotic framework given by n equispaced observations in [0, 1]. Adding possibly stronger regularity conditions to r, we define classes of such processes over which we show that s cannot be estimated at a better rate than nmin(1/2, β). Then, we study increment (or, more generally, discrete variation) estimators. We obtained precise bounds of the bias of the variance which show that the bias mainly depend on the parameter β and the variance on two terms, one depending on the parameter s and one on some regularity properties of r. A central limit theorem is given when the variance term relying on s dominates the bias and the other variance term. Eventually, we exhibit an estimator which achieves the minimax rate over a wide range of classes for which sufficient regularity conditions are assumed on r.

Estimation of long memory in the presence of a smooth nonparametric trend

We consider semiparametric estimation of the long-memory parameter of a stationary process in the presence of an additive nonparametric mean function. We use a semiparametric Whittle-type estimator, applied to the tapered, differenced series. Because the mean function is not necessarily a polynomial of finite order, no amount of differencing will completely remove the mean. We establish a central limit theorem for the estimator of the memory parameter, assuming that a slowly increasing number of low frequencies are trimmed from the estimators objective function. We find in simulations that tapering and trimming, applied either separately or together, are essential for the good performance of the estimator in practice. In our simulation study, we also compare the proposed estimator of the long-memory parameter with a direct estimator obtained from the raw data without differencing or tapering, and finally we study the question of feasible inference for the regression function. We find that the proposed estimator of the long-memory parameter is potentially far less biased than the direct estimator, and consequently that the proposed estimator may lead to more accurate inference on the regression function.

Dependent Wild Bootstrap for the Empirical Process

In this paper, we propose a model-free bootstrap method for the empirical process under absolute regularity. More precisely, consistency of an adapted version of the so-called dependent wild bootstrap, that was introduced by Shao (2010) and is very easy to implement, is proved under minimal conditions on the tuning parameter of the procedure. We apply our results to construct confidence intervals for unknown parameters and to approximate critical values for statistical tests. A simulation study shows that our method is competitive to standard block bootstrap methods in finite samples.

Evaluation for moments of a ratio with application to regression estimation

Many problems involve ratios in probability or in statistical applications. We aim at approximating the moments of such ratios un- der specic assumptions. Using ideas from Collomb (1977) (7), we propose sharper bounds for the moments of randomly weighted sums which also may appear as a ratio of two random variables. Suitable applications are given in more detail here in the elds of functional estimation, in nance and for

Phantom distribution functions for some stationary sequences

The notion of a phantom distribution function (phdf) was introduced by O’Brien (Ann. Probab. 15, 281–292 (1987)). We show that the existence of a phdf is a quite common phenomenon for stationary weakly dependent sequences. It is proved that any α-mixing stationary sequence with continuous marginals admits a continuous phdf. Sufficient conditions are given for stationary sequences exhibiting weak dependence, what allows the use of attractive models beyond mixing. The case of discontinuous marginals is also discussed for α-mixing. Special attention is paid to examples of processes which admit a continuous phantom distribution function while their extremal index is zero. We show that Asmussen (Ann. Appl. Probab. 8, 354–374 1998) and Roberts et al. (Extremes. 9, 213–229 2006) provide natural examples of such processes. We also construct a non-ergodic stationary process of this type.