Sequential Estimation of the Parameters in a Trigonometric Regression Model with the Gaussian Coloured Noise

This paper considers the estimation problem for a trigonometric regression model with the noise specified by the Ornstein–Uhlenbeck process with unknown parameter. We propose a sequential procedure which ensures a prescribed mean square precision uniformly in the nuisance parameter. The asymptotic behaviour of the procedure duration mean has been studied. This revised version was published online in August 2006 with corrections to the Cover Date.     

Adaptive efficient analysis for big data ergodic diffusion models

Statistical Inference for Stochastic Processes - We consider drift estimation problems for high dimension ergodic diffusion processes in nonparametric setting based on observations at discrete...     

Robust adaptive efficient estimation for semi-Markov nonparametric regression models

Statistical Inference for Stochastic Processes - We consider the nonparametric robust estimation problem for regression models in continuous time with semi-Markov noises. An adaptive model...     

Uniform concentration inequality for ergodic diffusion processes observed at discrete times

In this paper a concentration inequality is proved for the deviation in the ergodic theorem for diffusion processes in the case of discrete time observations. The proof is based on geometric ergodicity of diffusion processes. We consider as an application the nonparametric pointwise estimation problem of the drift coefficient when the process is observed at discrete times.     

Improved Model Selection Method for a Regression Function with Dependent Noise

This paper is devoted to nonparametric estimation, through the -risk, of a regression function based on observations with spherically symmetric errors, which are dependent random variables (except in the normal case). We apply a model selection approach using improved estimates. In a nonasymptotic setting, an upper bound for the risk is obtained (oracle inequality). Moreover asymptotic properties are given, such as upper and lower bounds for the risk, which provide optimal rate of convergence for penalized estimators.     

Improved estimation method for high dimension semimartingale regression models based on discrete data

Statistical Inference for Stochastic Processes - In this paper we study a high dimension (Big Data) regression model in continuous time observed in the discrete time moments with dependent noises...     

Ruin probability in the presence of risky investments

We consider an insurance company in the case when the premium rate is a bounded non-negative random function _ct$c_t$ and the capital of the insurance company is invested in a risky asset whose price follows a geometric Brownian motion with mean return a   and volatility σ>0$\sigma >0$. If β?2a/σ²-1>0$\beta ?2a/{\sigma }^2-1>0$ we find exact the asymptotic upper and lower bounds for the ruin probability Ψ(u)$\Psi (u)$ as the initial endowment u   tends to infinity, i.e. we show that _C*u^-β?Ψ(u)?C^*u^-β$C_{*}{u}^{-\beta }?\Psi (u)?C^{*}{u}^{-\beta }$ for sufficiently large u  . Moreover if _ct=c^*e^γt$c_t=c^{*}{\mathrm{e}}^{\gamma t}$ with γ?0$\gamma ?0$ we find the exact asymptotics of the ruin probability, namely Ψ(u)∼u^-β$\Psi (u)\sim u^{-\beta }$. If β?0$\beta ?0$, we show that Ψ(u)=1$\Psi (u)=1$ for any u?0$u?0$.     

Asymptotic Expansions for the Stochastic Approximation Averaging Procedure in Continuous Time

This paper considers the statistical estimation problem of the root of a nonlinear function under observations of the nonlinear regression model in continuous time. Asymptotic expansions for stochastic approximation averaging procedure are constructed, the gain factor, which minimizes the asymptotic bias is found, asymptotic normality for the procedure is proved. This revised version was published online in June 2006 with corrections to the Cover Date.     

Efficient estimation methods for non-Gaussian regression models in continuous time

Annals of the Institute of Statistical Mathematics - In this paper, we develop an efficient nonparametric estimation theory for continuous time regression models with non-Gaussian Lévy noises...