Asymptotic equivalence for inhomogeneous jump diffusion processes and white noise

We prove the global asymptotic equivalence between the experiments generated by the discrete (high frequency) or continuous observation of a path of a time inhomogeneous jump-diffusion process and a Gaussian white noise experiment. Here, the considered parameter is the drift function, and we suppose that the observation time $T$ tends to $\infty$. The approximation is given in the sense of the Le Cam $\Delta$-distance, under smoothness conditions on the unknown drift function. These asymptotic equivalences are established by constructing explicit Markov kernels that can be used to reproduce one experiment from the other.Comment: 20 pages; to appear on ESAIM: P\&S. In this version there are some improvements in the exposition following the reports suggestion

Asymptotic equivalence for density estimation and gaussian white noise: An extension

The aim of this paper is to present an extension of the well-known as-ymptotic equivalence between density estimation experiments and a Gaussian white noise model. Our extension consists in enlarging the nonparametric class of the admissible densities. More precisely, we propose a way to allow densities defined on any subinterval of R, and also some discontinuous or unbounded densities are considered (so long as the discontinuity and unboundedness patterns are somehow known a priori). The concept of equivalence that we shall adopt is in the sense of the Le Cam distance between statistical models. The results are constructive: all the asymptotic equivalences are established by constructing explicit Markov kernels.Comment: 11 pages. arXiv admin note: text overlap with arXiv:1503.0453

L_1-distance for additive processes with time-homogeneous L\'evy measures

We give an explicit bound for the $L_1$-distance between two additive processes of local characteristics $(f_j(\cdot),\sigma^2(\cdot),\nu_j)$, $j = 1,2$. The cases $\sigma =0$ and $\sigma > 0$ are both treated. We allow $\nu_1$ and $\nu_2$ to be equivalent time-homogeneous L\'evy measures, possibly with infinite variation. Some examples of possible applications are discussed.Comment: 9 pages; extended introduction and added reference

Asymptotic equivalence for pure jump Lévy processes with unknown Lévy density and Gaussian white noise

50 pages. The definition of the parameter space has changed and some proofs have been expanded and corrected.The aim of this paper is to establish a global asymptotic equivalence between the experiments generated by the discrete (high frequency) or continuous observation of a path of a Lévy process and a Gaussian white noise experiment observed up to a time T, with T tending to ∞. These approximations are given in the sense of the Le Cam distance, under some smoothness conditions on the unknown Lévy density. All the asymptotic equivalences are established by constructing explicit Markov kernels that can be used to reproduce one experiment from the other

A Bayesian nonparametric approach to log-concave density estimation

The estimation of a log-concave density on $\mathbb{R}$ is a canonical problem in the area of shape-constrained nonparametric inference. We present a Bayesian nonparametric approach to this problem based on an exponentiated Dirichlet process mixture prior and show that the posterior distribution converges to the log-concave truth at the (near-) minimax rate in Hellinger distance. Our proof proceeds by establishing a general contraction result based on the log-concave maximum likelihood estimator that prevents the need for further metric entropy calculations. We also present two computationally more feasible approximations and a more practical empirical Bayes approach, which are illustrated numerically via simulations.Comment: 39 pages, 17 figures. Simulation studies were significantly expanded and one more theorem has been adde