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 + $\xi$t for t $\in$ N and with a t+T $\not\equiv$ 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 ($\xi$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 $m$ change instants (the number $m$ 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 $m$ instants ($m$ 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 $\in$ (--0.5, 0.5) can be extended to a semiparametric class of Gaussian fractionally integrated processes with memory parameter d $\in$ (--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 $\ge$ 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 $H$ 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