We consider the quasi-likelihood analysis for a linear regression model driven by a Student-t L\'evy process with constant scale and arbitrary degrees of freedom. The model is observed at a high frequency over an extending period, under which we can quantify how the sampling frequency affects estimation accuracy. In that setting, joint estimation of trend, scale, and degrees of freedom is a non-trivial problem. The bottleneck is that the Student-t distribution is not closed under convolution, making it difficult to estimate all the parameters fully based on the high-frequency time scale. To efficiently deal with the intricate nature from both theoretical and computational points of view, we propose a two-step quasi-likelihood analysis: first, we make use of the Cauchy quasi-likelihood for estimating the regression-coefficient vector and the scale parameter; then, we construct the sequence of the unit-period cumulative residuals to estimate the remaining degrees of freedom. In particular, using full data in the first step causes a problem stemming from the small-time Cauchy approximation, showing the need for data thinning. Also presented is the implementation in a computer through the yuima package and some numerical examples.

翻译：本文针对由Student-t Lévy过程驱动、具有恒定尺度与任意自由度的线性回归模型进行拟似然分析。该模型在延长时段内以高频方式观测，在此框架下可量化采样频率对估计精度的影响。在此设定中，趋势、尺度及自由度的联合估计构成一项非平凡问题。其瓶颈在于Student-t分布不具备卷积封闭性，导致难以完全基于高频时间尺度对所有参数进行充分估计。为从理论与计算双重视角有效应对该复杂特性，我们提出两步拟似然分析法：首先采用柯西拟似然估计回归系数向量与尺度参数；继而构建单位周期累积残差序列以估计剩余自由度。特别需要指出的是，在第一步中使用全量数据会因短时柯西近似引发问题，由此揭示数据稀疏化的必要性。此外，本文还通过yuima软件包展示了计算机实现方法，并给出数值算例。