Convergence to stable laws in the space DD

We study the convergence of centered and normalized sums of i.i.d. random elements of the space $\mathcal{D}$ of c{{\'a}}dl{{\'a}}g functions endowed with Skorohod's $J\_1$ topology, to stable distributions in $\mathcal D$. Our results are based on the concept of regular variation on metric spaces and on point process convergence. We provide some applications, in particular to the empirical process of the renewal-reward process

Nonparametric estimation of the mixing density using polynomials

We consider the problem of estimating the mixing density $f$ from $n$ i.i.d. observations distributed according to a mixture density with unknown mixing distribution. In contrast with finite mixtures models, here the distribution of the hidden variable is not bounded to a finite set but is spread out over a given interval. We propose an approach to construct an orthogonal series estimator of the mixing density $f$ involving Legendre polynomials. The construction of the orthonormal sequence varies from one mixture model to another. Minimax upper and lower bounds of the mean integrated squared error are provided which apply in various contexts. In the specific case of exponential mixtures, it is shown that the estimator is adaptive over a collection of specific smoothness classes, more precisely, there exists a constant A\textgreater{}0 such that, when the order $m$ of the projection estimator verifies $m\sim A \log(n)$, the estimator achieves the minimax rate over this collection. Other cases are investigated such as Gamma shape mixtures and scale mixtures of compactly supported densities including Beta mixtures. Finally, a consistent estimator of the support of the mixing density $f$ is provided

Detection and localization of change-points in high-dimensional network traffic data

We propose a novel and efficient method, that we shall call TopRank in the following paper, for detecting change-points in high-dimensional data. This issue is of growing concern to the network security community since network anomalies such as Denial of Service (DoS) attacks lead to changes in Internet traffic. Our method consists of a data reduction stage based on record filtering, followed by a nonparametric change-point detection test based on $U$-statistics. Using this approach, we can address massive data streams and perform anomaly detection and localization on the fly. We show how it applies to some real Internet traffic provided by France-T\'el\'ecom (a French Internet service provider) in the framework of the ANR-RNRT OSCAR project. This approach is very attractive since it benefits from a low computational load and is able to detect and localize several types of network anomalies. We also assess the performance of the TopRank algorithm using synthetic data and compare it with alternative approaches based on random aggregation.Comment: Published in at http://dx.doi.org/10.1214/08-AOAS232 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

Time-frequency analysis of locally stationary Hawkes processes

Locally stationary Hawkes processes have been introduced in order to generalise classical Hawkes processes away from stationarity by allowing for a time-varying second-order structure. This class of self-exciting point processes has recently attracted a lot of interest in applications in the life sciences (seismology, genomics, neuro-science,...), but also in the modelling of high-frequency financial data. In this contribution we provide a fully developed nonparametric estimation theory of both local mean density and local Bartlett spectra of a locally stationary Hawkes process. In particular we apply our kernel estimation of the spectrum localised both in time and frequency to two data sets of transaction times revealing pertinent features in the data that had not been made visible by classical non-localised approaches based on models with constant fertility functions over time.Comment: Bernoulli journal, A Para{\^i}tr

Nonparametric estimation of mixing densities for discrete distributions

By a mixture density is meant a density of the form $\pi_{\mu}(\cdot)=\int\pi_{\theta}(\cdot)\times\mu(d\theta)$, where $(\pi_{\theta})_{\theta\in\Theta}$ is a family of probability densities and $\mu$ is a probability measure on $\Theta$. We consider the problem of identifying the unknown part of this model, the mixing distribution $\mu$, from a finite sample of independent observations from $\pi_{\mu}$. Assuming that the mixing distribution has a density function, we wish to estimate this density within appropriate function classes. A general approach is proposed and its scope of application is investigated in the case of discrete distributions. Mixtures of power series distributions are more specifically studied. Standard methods for density estimation, such as kernel estimators, are available in this context, and it has been shown that these methods are rate optimal or almost rate optimal in balls of various smoothness spaces. For instance, these results apply to mixtures of the Poisson distribution parameterized by its mean. Estimators based on orthogonal polynomial sequences have also been proposed and shown to achieve similar rates. The general approach of this paper extends and simplifies such results. For instance, it allows us to prove asymptotic minimax efficiency over certain smoothness classes of the above-mentioned polynomial estimator in the Poisson case. We also study discrete location mixtures, or discrete deconvolution, and mixtures of discrete uniform distributions.Comment: Published at http://dx.doi.org/10.1214/009053605000000381 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Function-indexed empirical processes based on an infinite source Poisson transmission stream

We study the asymptotic behavior of empirical processes generated by measurable bounded functions of an infinite source Poisson transmission process when the session length have infinite variance. In spite of the boundedness of the function, the normalized fluctuations of such an empirical process converge to a non-Gaussian stable process. This phenomenon can be viewed as caused by the long-range dependence in the transmission process. Completing previous results on the empirical mean of similar types of processes, our results on non-linear bounded functions exhibit the influence of the limit transmission rate distribution at high session lengths on the asymptotic behavior of the empirical process. As an illustration, we apply the main result to estimation of the distribution function of the steady state value of the transmission process