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We study the problem of the non-parametric estimation for the density $pi$ of the stationary distribution of a stochastic two-dimensional damping Hamiltonian system $(Z_t)_{tin[0,T]}=(X_t,Y_t)_{t in [0,T]}$. From the continuous observation of the sampling path on $[0,T]$, we study the rate of estimation for $pi(x_0,y_0)$ as $T to infty$. We show that kernel based estimators can achieve the rate $T^{-v}$ for some explicit exponent $v in (0,1/2)$. One finding is that the rate of estimation depends on the smoothness of $pi$ and is completely different with the rate appearing in the standard i.i.d. setting or in the case of two-dimensional non degenerate diffusion processes. Especially, this rate depends also on $y_0$. Moreover, we obtain a minimax lower bound on the $L^2$-risk for pointwise estimation, with the same rate $T^{-v}$, up to $log(T)$ terms.
We consider a stochastic individual-based model in continuous time to describe a size-structured population for cell divisions. This model is motivated by the detection of cellular aging in biology. We address here the problem of nonparametric estima
Estimating the mixing density of a mixture distribution remains an interesting problem in statistics literature. Using a stochastic approximation method, Newton and Zhang (1999) introduced a fast recursive algorithm for estimating the mixing density
We consider the semi-parametric estimation of a scale parameter of a one-dimensional Gaussian process with known smoothness. We suggest an estimator based on quadratic variations and on the moment method. We provide asymptotic approximations of the m
In this paper we consider multivariate Hawkes processes with baseline hazard and kernel functions that depend on time. This defines a class of locally stationary processes. We discuss estimation of the time-dependent baseline hazard and kernel functi
This paper deals with the maximum likelihood estimator for the mean-reverting parameter of a first order autoregressive models with exogenous variables, which are stationary Gaussian noises (Colored noise). Using the method of the Laplace transform,