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Register a new userStein estimation for the drift of Gaussian processes using the Malliavin calculus
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Added by
Nicolas Privault
Publication date
2008
fields
Mathematical Statistics
and research's language is
English
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We consider the nonparametric functional estimation of the drift of a Gaussian process via minimax and Bayes estimators. In this context, we construct superefficient estimators of Stein type for such drifts using the Malliavin integration by parts formula and superharmonic functionals on Gaussian space. Our results are illustrated by numerical simulations and extend the construction of James--Stein type estimators for Gaussian processes by Berger and Wolpert [J. Multivariate Anal. 13 (1983) 401--424].
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We consider stationary processes with long memory which are non-Gaussian and represented as Hermite polynomials of a Gaussian process. We focus on the corresponding wavelet coefficients and study the asymptotic behavior of the sum of their squares since this sum is often used for estimating the long-memory parameter. We show that the limit is not Gaussian but can be expressed using the non-Gaussian Rosenblatt process defined as a Wiener It^o integral of order 2. This happens even if the original process is defined through a Hermite polynomial of order higher than 2.
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In this paper we consider an ergodic diffusion process with jumps whose drift coefficient depends on an unknown parameter $theta$. We suppose that the process is discretely observed at the instants (t n i)i=0,...,n with $Delta$n = sup i=0,...,n--1 (t n i+1 -- t n i) $rightarrow$ 0. We introduce an estimator of $theta$, based on a contrast function, which is efficient without requiring any conditions on the rate at which $Delta$n $rightarrow$ 0, and where we allow the observed process to have non summable jumps. This extends earlier results where the condition n$Delta$ 3 n $rightarrow$ 0 was needed (see [10],[24]) and where the process was supposed to have summable jumps. Moreover, in the case of a finite jump activity, we propose explicit approximations of the contrast function, such that the efficient estimation of $theta$ is feasible under the condition that n$Delta$ k n $rightarrow$ 0 where k > 0 can be arbitrarily large. This extends the results obtained by Kessler [15] in the case of continuous processes. L{e}vy-driven SDE, efficient drift estimation, high frequency data, ergodic properties, thresholding methods.
This paper aims at providing statistical guarantees for a kernel based estimation of time varying parameters driving the dynamic of local stationary processes. We extend the results of Dahlhaus et al. (2018) considering the local stationary version of the infinite memory processes of Doukhan and Wintenberger (2008). The estimators are computed as localized M-estimators of any contrast satisfying appropriate contraction conditions. We prove the uniform consistency and pointwise asymptotic normality of such kernel based estimators. We apply our result to usual contrasts such as least-square, least absolute value, or quasi-maximum likelihood contrasts. Various local-stationary processes as ARMA, AR(infty), GARCH, ARCH(infty), ARMA-GARCH, LARCH(infty),..., and integer valued processes are also considered. Numerical experiments demonstrate the efficiency of the estimators on both simulated and real data sets.
We provide the strong approximation of empirical copula processes by a Gaussian process. In addition we establish a strong approximation of the smoothed empirical copula processes and a law of iterated logarithm.
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