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 $(\varepsilon n^\rho)^{-1} =O(1)$ for some $\rho>0$. 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.

翻译：我们考虑从离散观测中用一个小分散参数对多维扩散过程进行参数估计和测试。关于对扩散过程的参数估计,主要目标是估计漂移参数和扩散参数。在本文件中,我们建议两种参数的适应性估计值,并用美元至美元、美元至美元、美元至美元和美元=O(1)美元的平衡条件,即美元至美元至美元至美元至美元至美元至美元。我们采用这些适应性估计值,还采用一致的适应性测试方法,并证明适应性试验的统计在完全假设下是无症状分布的。在模拟研究中,我们检查并比较两种适应性估计和试验统计的无症状行为。此外,我们处理SIR模型,该模型描述一种简单的流行病在生物应用中的传播。