Abstract

We study the parameter estimation for ergodic diffusion process X_t from perturbed observations $Y_{t_j}=X_{t_j}+{\epsilon }_{t_j},j=1,\dots ,n$ where $t_1,\dots ,t_n$ are observation times and the noise ${\epsilon }_t$ is a strongly mixing stationary noisy process with the density function g. We construct an estimator of the diffusion parameters based on the minimum Hellinger distance between the density of the invariant distribution of diffusion process X_t and the nonparametric deconvolution kernel estimator of this density. This article focuses on the ordinary smooth noise density class with the assumption $A_1{(1+\left|p\right|)}^{-\kappa }\le \left|g^{ft}(p)\right|\le A_2{(1+\left|p\right|)}^{-\kappa }$ (where $\kappa >0$ and $g^{ft}(p)$ is characteristic function of noisy random variable). This assumption is more general than the condition $\underset{p\to \infty }{\lim }{g}^{ft}(p)\cdot p^{\kappa }=C$ (where C is a constant) which is used in a lot of articles. We also discuss the asymptotic normality for both the estimator of deconvolution kernel density and the estimator of diffusion parameters. Finally, we illustrate the properties of the estimator by two examples of diffusion processes.

摘要

我们从扰动观测研究遍历扩散过程X _t 的参数估计${ÿ}_{{Ť}_{Ĵ}}=X_{{Ť}_{Ĵ}}+{\epsilon }_{{Ť}_{Ĵ}}，Ĵ=\mathrm{1个}，\dots ，ñ$ 哪里 ${Ť}_{\mathrm{1个}}，\dots ，{Ť}_{ñ}$ 是观察时间和噪音 ${\epsilon }_{Ť}$是一个具有密度函数g的强混合静态噪声过程。我们基于扩散过程X _t 不变分布的密度与该密度的非参数反卷积核估计器之间的最小Hellinger距离，构造了扩散参数的估计器。本文重点关注具有假设的普通平滑噪声密度类${\mathrm{一种}}_{\mathrm{1个}}{（\mathrm{1个}+\left|p\right|）}^{-\kappa }\le \left|G^{FŤ}（p）\right|\le {\mathrm{一种}}_2{（\mathrm{1个}+\left|p\right|）}^{-\kappa }$ （哪里 $\kappa >0$ 和 $G^{FŤ}（p）$是噪声随机变量的特征函数）。这个假设比条件更普遍$\underset{p\to \infty }{林}{G}^{FŤ}（p）\cdot p^{\kappa }=C$（其中C是常数），这在很多文章中都使用过。我们还讨论了反卷积核密度的估计量和扩散参数的估计量的渐近正态性。最后，我们通过两个扩散过程的例子来说明估计器的性质。