论文标题
在各向异性持有人类上的固定分布的固定分布的估计率
Rate of estimation for the stationary distribution of jump-processes over anisotropic Holder classes
论文作者
论文摘要
我们研究了与跳跃（XT）的多元随机微分方程的固定分布密度的非参数估计的问题，当尺寸D大于3时，从对[0，T]上的采样路径的连续观察到[0，T]时，我们表明，在型号持有器平滑度约束基于基础的基于核心的基于基于核心的估算率速度快速估算率。特别是，它们的速度与Dalalyan和Reiss [9]所发现的那样快，以估算不变密度，而在各向同性持有人的平滑度约束下没有跳跃。此外，它们比Strauch [29]在各向异性支架平滑度约束下对连续随机微分方程的不变密度估计的速度快。此外，我们在L2风险上获得了最小的下限以进行点式估计，其速率相同，直至log（t）项。这意味着，在一系列扩散的情况下，它们不变密度属于我们正在考虑的各向异性持有人类类别，就不可能找到比我们提出的估计速率快的估计速率。
We study the problem of the non-parametric estimation for the density of the stationary distribution of the multivariate stochastic differential equation with jumps (Xt) , when the dimension d is bigger than 3. From the continuous observation of the sampling path on [0, T ] we show that, under anisotropic Holder smoothness constraints, kernel based estimators can achieve fast convergence rates. In particular , they are as fast as the ones found by Dalalyan and Reiss [9] for the estimation of the invariant density in the case without jumps under isotropic Holder smoothness constraints. Moreover, they are faster than the ones found by Strauch [29] for the invariant density estimation of continuous stochastic differential equations, under anisotropic Holder smoothness constraints. Furthermore, we obtain a minimax lower bound on the L2-risk for pointwise estimation, with the same rate up to a log(T) term. It implies that, on a class of diffusions whose invariant density belongs to the anisotropic Holder class we are considering, it is impossible to find an estimator with a rate of estimation faster than the one we propose.
