Equivalence theory for density estimation, Poisson processes and Gaussian white noise with drift

The Le Cam distance between density estimation, Poisson processes and Gaussian white noise

It is well-known that density estimation on the unit interval is asymptotically equivalent to a Gaussian white noise experiment, provided the densities have Hölder smoothness larger than $1/2$ and are uniformly bounded away from zero. We derive matching lower and constructive upper bounds for the Le Cam deficiencies between these experiments, with explicit dependence on both the sample size and the size of the densities in the parameter space. As a consequence, we derive sharp conditions on how small the densities can be for asymptotic equivalence to hold. The related case of Poisson intensity…

使用深度神经网络估计多维Lévy驱动的随机微分方程的漂移项

Drift Estimation for a Multi-Dimensional Lévy-Driven Stochastic Differential Equation Using Deep Neural Networks

本文用稀疏ReLU神经网络估计多维Lévy驱动随机微分方程的漂移项，在弱集中条件下获得与扩散模型同阶的统计界，并证明估计器在极小极大意义下最优。

Efficient nonparametric inference for discretely observed compound Poisson processes

A compound Poisson process whose parameters are all unknown is observed at finitely many equispaced times. Nonparametric estimators of the jump and Lévy distributions are proposed and functional central limit theorems using the uniform norm are proved for both under mild conditions. The limiting Gaussian processes are identified and efficiency of the estimators is established. Kernel estimators for the mass function, the intensity and the drift are also proposed, their asymptotic properties including efficiency are analysed, and joint asymptotic normality is shown. Inference tools such as conf…

Asymptotic equivalence of spectral density estimation and gaussian white noise

We consider the statistical experiment given by a sample of a stationary Gaussian process with an unknown smooth spectral density f. Asymptotic equivalence, in the sense of Le Cam's deficiency Delta-distance, to two Gaussian experiments with simpler structure is established. The first one is given by independent zero mean Gaussians with variance approximately the value of f in points of a uniform grid (nonparametric Gaussian scale regression). This approximation is closely related to well-known asymptotic independence results for the periodogram and corresponding inference methods. The second …

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 $Δ$-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 exp…

Generating Sparse Stochastic Processes Using Matched Splines

We provide an algorithm to generate trajectories of sparse stochastic processes that are solutions of linear ordinary differential equations driven by Lévy white noises. A recent paper showed that these processes are limits in law of generalized compound-Poisson processes. Based on this result, we derive an off-the-grid algorithm that generates arbitrarily close approximations of the target process. Our method relies on a B-spline representation of generalized compound-Poisson processes. We illustrate numerically the validity of our approach.