182 research outputs found
Moment bounds and central limit theorems for Gaussian subordinated arrays
A general moment bound for sums of products of Gaussian vector's functions
extending the moment bound in Taqqu (1977, Lemma 4.5) is established. A general
central limit theorem for triangular arrays of nonlinear functionals of
multidimensional non-stationary Gaussian sequences is proved. This theorem
extends the previous results of Breuer and Major (1981), Arcones (1994) and
others. A Berry-Esseen-type bound in the above-mentioned central limit theorem
is derived following Nourdin, Peccati and Podolskij (2011). Two applications of
the above results are discussed. The first one refers to the asymptotic
behavior of a roughness statistic for continuous-time Gaussian processes and
the second one is a central limit theorem satisfied by long memory locally
stationary process
Non-parametric estimation of time varying AR(1)--processes with local stationarity and periodicity
Extending the ideas of [7], this paper aims at providing a kernel based
non-parametric estimation of a new class of time varying AR(1) processes (Xt),
with local stationarity and periodic features (with a known period T), inducing
the definition Xt = at(t/nT)X t--1 + t for t N and with a t+T
at. Central limit theorems are established for kernel estima-tors
as(u) reaching classical minimax rates and only requiring low order moment
conditions of the white noise (t)t up to the second order
Detecting changes in the fluctuations of a Gaussian process and an application to heartbeat time series
The aim of this paper is first the detection of multiple abrupt changes of
the long-range dependence (respectively self-similarity, local fractality)
parameters from a sample of a Gaussian stationary times series (respectively
time series, continuous-time process having stationary increments). The
estimator of the change instants (the number is supposed to be known)
is proved to satisfied a limit theorem with an explicit convergence rate.
Moreover, a central limit theorem is established for an estimator of each
long-range dependence (respectively self-similarity, local fractality)
parameter. Finally, a goodness-of-fit test is also built in each time domain
without change and proved to asymptotically follow a Khi-square distribution.
Such statistics are applied to heart rate data of marathon's runners and lead
to interesting conclusions
Detecting abrupt changes of the long-range dependence or the self-similarity of a Gaussian process
In this paper, an estimator of instants ( is known) of abrupt changes
of the parameter of long-range dependence or self-similarity is proved to
satisfy a limit theorem with an explicit convergence rate for a sample of a
Gaussian process. In each estimated zone where the parameter is supposed not to
change, a central limit theorem is established for the parameter's (of
long-range dependence, self-similarity) estimator and a goodness-of-fit test is
also built. {\it To cite this article: J.M. Bardet, I. Kammoun, C. R. Acad.
Sci. Paris, Ser. I 340 (2007).
Semiparametric stationarity and fractional unit roots tests based on data-driven multidimensional increment ratio statistics
In this paper, we show that the central limit theorem (CLT) satisfied by the
data-driven Multidimensional Increment Ratio (MIR) estimator of the memory
parameter d established in Bardet and Dola (2012) for d (--0.5, 0.5) can
be extended to a semiparametric class of Gaussian fractionally integrated
processes with memory parameter d (--0.5, 1.25). Since the asymptotic
variance of this CLT can be estimated, by data-driven MIR tests for the two
cases of stationarity and non-stationarity, so two tests are constructed
distinguishing the hypothesis d \textless{} 0.5 and d 0.5, as well as a
fractional unit roots test distinguishing the case d = 1 from the case d
\textless{} 1. Simulations done on numerous kinds of short-memory, long-memory
and non-stationary processes, show both the high accuracy and robustness of
this MIR estimator compared to those of usual semiparametric estimators. They
also attest of the reasonable efficiency of MIR tests compared to other usual
stationarity tests or fractional unit roots tests. Keywords: Gaussian
fractionally integrated processes; semiparametric estimators of the memory
parameter; test of long-memory; stationarity test; fractional unit roots test.Comment: arXiv admin note: substantial text overlap with arXiv:1207.245
Identification of the multiscale fractional Brownian motion with biomechanical applications
In certain applications, for instance biomechanics, turbulence, finance, or
Internet traffic, it seems suitable to model the data by a generalization of a
fractional Brownian motion for which the Hurst parameter is depending on
the frequency as a piece-wise constant function. These processes are called
multiscale fractional Brownian motions. In this contribution, we provide a
statistical study of the multiscale fractional Brownian motions. We develop a
method based on wavelet analysis. By using this method, we find initially the
frequency changes, then we estimate the different parameters and afterwards we
test the goodness-of-fit. Lastly, we give the numerical algorithm.
Biomechanical data are then studied with these new tools
A new non-parametric detector of univariate outliers for distributions with unbounded support
The purpose of this paper is to construct a new non-parametric detector of
univariate outliers and to study its asymptotic properties. This detector is
based on a Hill's type statistic. It satisfies a unique asymptotic behavior for
a large set of probability distributions with positive unbounded support (for
instance: for the absolute value of Gaussian, Gamma, Weibull, Student or
regular variations distributions). We have illustrated our results by numerical
simulations which show the accuracy of this detector with respect to other
usual univariate outlier detectors (Tukey, MAD or Local Outlier Factor
detectors). The detection of outliers in a database providing the prices of
used cars is also proposed as an application to real-life database
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