In this paper, we investigate the parameter estimation for threshold Ornstein$\mathit{-}$Uhlenbeck processes. Least squares method is used to obtain continuous-type and discrete-type estimators for the drift parameters based on continuous and discrete observations, respectively. The strong consistency and asymptotic normality of the proposed least squares estimators are studied. We also propose a modified quadratic variation estimator based on the long-time observations for the diffusion parameters and prove its consistency. Our simulation results suggest that the performance of our proposed estimators for the drift parameters may show improvements compared to generalized moment estimators. Additionally, the proposed modified quadratic variation estimator exhibits potential advantages over the usual quadratic variation estimator with relatively small sample sizes. In particular, our method can be applied to the multi-regime cases ($m>2$), while the generalized moment method only deals with the two regime cases ($m=2$). The U.S. treasury rate data is used to illustrate the theoretical results.

翻译：本文研究了阈值奥恩斯坦-乌伦贝克过程的参数估计问题。基于连续观测和离散观测，分别采用最小二乘法获得漂移参数的连续型估计量和离散型估计量，并探讨了所提最小二乘估计量的强相合性与渐近正态性。同时，我们基于长期观测数据提出了一种改进的二次变分估计量用于扩散参数估计，并证明了其相合性。模拟结果表明，与广义矩估计量相比，本文提出的漂移参数估计量性能有所提升。此外，在样本量较小时，改进的二次变分估计量相较于常规二次变分估计量展现出潜在优势。特别地，本方法可应用于多体制情形（m>2），而广义矩方法仅能处理两体制情形（m=2）。最后，采用美国国债利率数据对理论结果进行了实证验证。