We consider parametric estimation and tests for multi-dimensional diffusion processes with a small dispersion parameter $\varepsilon$ from discrete observations. For parametric estimation of diffusion processes, the main target is to estimate the drift parameter and the diffusion parameter. In this paper, we propose two types of adaptive estimators for both parameters and show their asymptotic properties under $\varepsilon\to0$, $n\to\infty$ and the balance condition that $(\varepsilon n^ρ)^{-1} =O(1)$ for some $ρ>0$. Using these adaptive estimators, we also introduce consistent adaptive testing methods and prove that test statistics for adaptive tests have asymptotic distributions under null hypothesis. In simulation studies, we examine and compare asymptotic behaviors of the two kinds of adaptive estimators and test statistics. Moreover, we treat the SIR model which describes a simple epidemic spread for a biological application.
Parametric estimation for partially hidden diffusion processes sampled at discrete times
对于一个一维扩散过程$X=\{X(t) ; 0\leq t \leq T \}$，我们假设如果$X(t)$低于某个固定且已知的阈值$τ$，则它是隐藏的，否则它是可见的。这意味着一个部分隐藏的扩散过程。本文讨论的问题是在离散抽样方案得到的部分隐扩散过程下，估计漂移系数和扩散系数中的有限维参数。假设采样发生在长度为$h_n$的规则间隔的时间间隔上，使得$n h_n=T$。渐近是当$h_n\to0$、$T\to\infty$和$n h_n^2\to 0$为$n\to\infty$时。证明了漂移系数和扩散系数参数估计的相合性和渐近正态性。
Adaptive nonparametric drift estimation for diffusion processes using Faber-Schauder expansions
本文研究了一维扩散方程的周期漂移的非参数估计问题。出于计算考虑，van der Meulen e. a.（2014）将漂移的先验定义为具有高斯系数的随机截断和随机缩放的Faber-Schauder级数展开。我们研究的行为后获得的前从频率论渐近的观点。如果产生漂移的真实数据是平滑的，则证明了后验是自适应的，其中后验收缩率对于$L_2$ -范数是最优的，直到对数因子。此外，还导出了具有$p\in (2,\infty]$的$L_p$ -范数的收缩率.
