151 research outputs found
Convergence to stable laws in the space
We study the convergence of centered and normalized sums of i.i.d. random
elements of the space of c{{\'a}}dl{{\'a}}g functions endowed
with Skorohod's topology, to stable distributions in . 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 from 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 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 of the projection
estimator verifies , 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 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
-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
, where
is a family of probability densities and
is a probability measure on . We consider the problem of
identifying the unknown part of this model, the mixing distribution , from
a finite sample of independent observations from . 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
Handy sufficient conditions for the convergence of the maximum likelihood estimator in observation-driven models
This paper generalizes asymptotic properties obtained in the
observation-driven times series models considered by \cite{dou:kou:mou:2013} in
the sense that the conditional law of each observation is also permitted to
depend on the parameter. The existence of ergodic solutions and the consistency
of the Maximum Likelihood Estimator (MLE) are derived under easy-to-check
conditions. The obtained conditions appear to apply for a wide class of models.
We illustrate our results with specific observation-driven times series,
including the recently introduced NBIN-GARCH and NM-GARCH models, demonstrating
the consistency of the MLE for these two models
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
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