Showing papers in "Statistical Inference for Stochastic Processes in 2008"
TL;DR: In this article, the maximum likelihood estimator for stochastic equations with additive fractional Brownian sheet was studied, and the Girsanov transform was used for the two-parameter fractional brownian motion.
Abstract: We study the maximum likelihood estimator for stochastic equations with additive fractional Brownian sheet. We use the Girsanov transform for the the two-parameter fractional Brownian motion, as well as the Malliavin calculus and Gaussian regularity theory.
18 citations
TL;DR: In this article, different estimators of the parameters of the Poisson process were proposed and their performance under the asymptotics was investigated under a small sample size n. Given the complexity of the equations involved only estimators on the first model can be studied analytically.
Abstract: The telegraph process X(t), t ≥ 0, (Goldstein, Q J Mech Appl Math 4:129–156, 1951) and the geometric telegraph process \(S(t) = s_{0} {\rm exp}\{(\mu -\frac12\sigma^{2})t + \sigma X(t)\}\) with μ a known real constant and σ > 0 a parameter are supposed to be observed at n + 1 equidistant time points ti = iΔn,i = 0,1,..., n. For both models λ, the underlying rate of the Poisson process, is a parameter to be estimated. In the geometric case, also σ > 0 has to be estimated. We propose different estimators of the parameters and we investigate their performance under the asymptotics, i.e. Δn → 0, nΔn = T 0 fixed. The process X(t) in non markovian, non stationary and not ergodic thus we build a contrast function to derive an estimator. Given the complexity of the equations involved only estimators on the first model can be studied analytically. Therefore, we run an extensive Monte Carlo analysis to study the performance of the proposed estimators also for small sample size n.
14 citations
TL;DR: In this article, the stationary density of an invertible linear process can be estimated at the parametric rate by a convolution of residual-based kernel estimators, and the convergence is uniform and a functional central limit theorem holds in the space of continuous functions vanishing at infinity.
Abstract: The stationary density of an invertible linear processes can be estimated at the parametric rate by a convolution of residual-based kernel estimators. We have shown elsewhere that the convergence is uniform and that a functional central limit theorem holds in the space of continuous functions vanishing at infinity. Here we show that analogous results hold in weighted L
1-spaces. We do not require smoothness of the innovation density.
13 citations
TL;DR: This paper introduces a nonparametric approach for the estimation of the covariance function of a stationary stochastic process Xt indexed by t, and uses the estimator designed by Parzen for continuous-time data.
Abstract: In this paper we introduce a nonparametric approach for the estimation of the covariance function of a stationary stochastic process Xt indexed by \(t \in {\mathbb{R}}^+.\) The data consist of a finite number of observations of the process at irregularly spaced time points and the aim is to estimate the covariance at any lag point without parametric assumptions and in such a way that it is a positive definite function. After interpolating the process, we use the estimator designed by Parzen (Technometrics 3:167–190,1961) for continuous-time data. Our estimator is shown to be consistent under smoothness assumptions on the covariance. Its performance is evaluated by simulations.
13 citations
TL;DR: In this article, a wide class of recursive estimation procedures for the general statistical model and study convergence are proposed. But they are not recursive in the sense that each successive estimator is obtained from the previous one by a simple adjustment.
Abstract: We consider estimation procedures which are recursive in the sense that each successive estimator is obtained from the previous one by a simple adjustment. We propose a wide class of recursive estimation procedures for the general statistical model and study convergence.
12 citations
TL;DR: In this article, the authors investigated the performance of a linear wavelet-type deconvolution estimator for weakly dependent data and showed that the rates of convergence which are optimal in the case of i.i.d. data are also (almost) attained for strongly mixing observations, provided the mixing coefficients decay fast enough.
Abstract: In this paper we investigate the performance of a linear wavelet-type deconvolution estimator for weakly dependent data. We show that the rates of convergence which are optimal in the case of i.i.d. data are also (almost) attained for strongly mixing observations, provided the mixing coefficients decay fast enough. The results are applied to a discretely observed continuous-time stochastic volatility model.
12 citations
TL;DR: In this paper, the authors obtained exact large deviation rates for the log-likelihood ratio in testing models with observed Ornstein-Uhlenbeck processes and gave explicit rates of decrease for the error probabilities of Neyman-Pearson, Bayes, and minimax tests.
Abstract: We obtain exact large deviation rates for the log-likelihood ratio in testing models with observed Ornstein–Uhlenbeck processes and get explicit rates of decrease for the error probabilities of Neyman–Pearson, Bayes, and minimax tests. Moreover, we give expressions for the rates of decrease for the error probabilities of Neyman–Pearson tests in models with observed processes solving affine stochastic delay differential equations.
12 citations
TL;DR: In this paper, asymptotic properties of the estimated coefficients of the multi-indexed autoregressive model are investigated and they use some kind of self-normalization and obtain limit normal law independent of unknown parameters.
Abstract: In the paper asymptotic properties of the estimated coefficients of multi-indexed autoregressive model are investigated. Considering Least Squares estimates we use some kind of self-normalization and obtain limit normal law independent of unknown parameters. In the proof of this result we use recent result on the Central Limit Theorem for dependent summands.
8 citations
TL;DR: In this paper, a general functional central limit theorem for weak dependent time series was proved for causal or non-causal linear models, such as ARCH(1), bilinear, and Volterra processes.
Abstract: We prove a general functional central limit theorem for weak dependent time series. A very large variety of models, for instance, causal or non causal linear, ARCH(1), bilinear, Volterra processes, satisfies this theorem. Moreover, it provides numerous application as well for bounding the distance between the empirical mean and the Gaussian measure than for obtaining central limit theorem for sample moments and cumulants.
8 citations
TL;DR: In this paper, the L 1 convergence of the filtering and smoothing densities in state space models is studied, when the transition and emission densities are estimated using non parametric kernel estimates.
Abstract: This paper addresses the problem of reconstructing partially observed stochastic processes. The L1 convergence of the filtering and smoothing densities in state space models is studied, when the transition and emission densities are estimated using non parametric kernel estimates. An application to real data is proposed, in which a wave time series is forecasted given a wind time series.
7 citations
TL;DR: In this article, the authors show that all integrable additive functionals of a one-dimensional Harris recurrent diffusion are asymptotically equivalent in probability to deterministic processes.
Abstract: Let X
t
be a one-dimensional Harris recurrent diffusion, with a drift depending on an unknown parameter θ belonging to some metric compact Θ. We firstly show that all integrable additive functionals of X
t
are asymptotically equivalent in probability to some deterministic process v
t
. Then we use this result to study the behavior of the maximum likelihood estimator for the parameter θ. Under mild regularity assumptions, we find an upper rate of its convergence as a function of v
t
, extending some recent results for ergodic diffusions.
TL;DR: Asymptotic properties of bias-corrected estimators for small diffusion models from the viewpoint of information geometry were investigated in this article, where the authors obtained results analogous to those for independent and identically distributed (iid) models.
Abstract: Information geometrical quantities such as metric tensors and connection coefficients for small diffusion models are obtained Asymptotic properties of bias-corrected estimators for small diffusion models are investigated from the viewpoint of information geometry Several results analogous to those for independent and identically distributed (iid) models are obtained by using the asymptotic normality of the statistics appearing in asymptotic expansions In contrast to the asymptotic theory for iidmodels, the geometrical quantities depend on the magnitude of noise