论文标题
稳定定律的三角近似值的最大似然估计
Trigonometrically approximated maximum likelihood estimation for stable law
论文作者
论文摘要
提出了$α$稳定法律的三角近似值的最大似然估计。估算器求解了近似似然方程，该方程是通过在三角函数跨越的空间上投影真实的得分函数来获得的。预计得分仅由特征函数及其衍生物的真实和虚构部分表示，因此我们可以明确构建目标估计方程。我们研究所提出的估计量的渐近特性，并显示一致性和渐近正态性。此外，随着三角函数的数量的增加，估计值将其收敛到确切的最大似然估计量，因为它们具有相同的渐近定律。仿真研究表明，我们的估计器的表现优于其他力矩类型的估计器，其标准偏差几乎达到了Cramér-Rao下限。我们将方法应用于$α$ - 稳定的Ornstein- uhlenbeck过程的估计问题。获得的结果证明了渐近混合正态性的理论。
A trigonometrically approximated maximum likelihood estimation for $α$-stable laws is proposed. The estimator solves the approximated likelihood equation, which is obtained by projecting a true score function on the space spanned by trigonometric functions. The projected score is expressed only by real and imaginary parts of the characteristic function and their derivatives, so that we can explicitly construct the targeting estimating equation. We study the asymptotic properties of the proposed estimator and show consistency and asymptotic normality. Furthermore, as the number of trigonometric functions increases, the estimator converges to the exact maximum likelihood estimator, in the sense that they have the same asymptotic law. Simulation studies show that our estimator outperforms other moment-type estimators, and its standard deviation almost achieves the Cramér--Rao lower bound. We apply our method to the estimation problem for $α$-stable Ornstein--Uhlenbeck processes in a high-frequency setting. The obtained result demonstrates the theory of asymptotic mixed normality.
