We suppose that a Lévy process is observed at discrete time points. A rather general construction of minimum-distance estimators is shown to give consistent estimators of the Lévy-Khinchine characteristics as the number of observations tends to infinity, keeping the observation distance fixed. For a specific C 2 -criterion this estimator is rate-optimal. The connection with deconvolution and inverse problems is explained. A key step in the proof is a uniform control on the deviations of the empirical characteristic function on the whole real line.
We consider a class of observation-driven Poisson count processes where the
current value of the accompanying intensity process depends on previous values
of both processes. We show under a contractive condition that the bivariate
process has a unique stationary distribution and that a stationary version of
the count process is absolutely regular. Moreover, since the intensities can be
written as measurable functionals of the count variables, we conclude that the
bivariate process is ergodic. As an important application of these results, we
show how a test method previously used in the case of independent Poisson data
can be used in the case of Poisson count processes.Comment: Published in at http://dx.doi.org/10.3150/10-BEJ313 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
In the present paper we consider nonlinear wavelet estimators of the spectral density f of a zero mean, not necessarily Gaussian, stochastic process, which is stationary in the wide sense. It is known in the case of Gaussian regression that these estimators outpefiorm traditional linear methods if the degree of smoothness of the regression function varies considerably over the interval of interest. Such methods are based on a nonlinear treatment of empirical coefficients that arise from an orthonormal series expansion according to a wavelet basis.The main goal of this paper is to transfer these methods to spectral density estimation. This is done by showing the asymptotic normality of certain empirical coefficients based on the tapered periodogram. Using these results we can show the risk equivalence to the Gaussian case for monotone estimators based on such empirical coefficients. The resulting estimator of f keeps all interesting properties such as high spatial adaptivity that are already known for wavelet estimators in the case of Gaussian regression.It turns out that appropriately tuned versions of this estimator attain the optimal uniform rate of convergence of their Lz risk in a wide variety of Besov smoothness classes, including classes where linear estimators (kernel, spline) are not able to attain this rate. Some simulations indicate the usefulness of the new method in cases of high spatial inhomogeneity.
We develop a test for stationarity of a time series against the alternative of a time-varying covariance structure. Using localized versions of the periodogram, we obtain empirical versions of a reasonable notion of a time-varying spectral density. Coef®cients with respect to a Haar wavelet series expansion of such a time-varying periodogram are an indicator of whether there is some deviation from covariance stationarity. We propose a test based on the limit distribution of these empirical coef®cients.
International audienceUsing the idea of weak dependence introduced in Doukhan and Louhichi 1999, we present a new Bernstein type inequality for times series. This result extends on Kallabis and Neumann 2004 but it also holds for non causal type sequences and subgeometric decay rates of the dependence coefficients. Examples of applications and a large variety of models beyond mixing times series are also listed
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