漂移多子分数布朗运动在离散观测下的精确极大似然推断
Exact maximum likelihood inference for drifted multi-sub-fractional Brownian motion at discrete observation
- Imam Abdulrahman Bin Faisal University（伊玛目阿卜杜勒拉赫曼·本·费萨尔大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文针对离散观测的漂移多子分数布朗运动，证明了完整有限样本似然理论的存在，给出显式极大似然估计量、精确水平的区间与检验，并验证了强相合性、渐近正态性及模拟效果。
AI中文摘要：
子分数布朗运动具有自相似性和长程依赖性，但缺乏平稳增量，因此其增量协方差不是Toeplitz矩阵，也不存在谱密度。我们证明，尽管如此，完整的有限样本似然理论仍然成立。该模型是在$N$个等距时间点上通过$m$个独立且具有已知Hurst指数和共同尺度的子分数布朗运动的叠加观测到的线性趋势。非退化性源于将该过程实现为双侧分数布朗运动的偶部，且趋势和尺度的极大似然估计量是显式的。推断所依赖的统计量是枢轴的，其分布仅取决于样本量，因此在每个$N\ge2$下均可获得精确水平的区间和检验，同时具备完全充分性、最小方差无偏性以及达到Cramér--Rao界。一个显式的方差界给出了强相合性和渐近正态性，模拟结果确认了精确覆盖率和错误指定Hurst向量时的预期效应。
英文摘要：
Sub-fractional Brownian motion is self-similar and long-range dependent but has no stationary increments, so the increment covariance is not Toeplitz and no spectral density is available. We show that a complete finite-sample likelihood theory survives nonetheless. The model is a linear trend observed at $N$ equidistant times through a superposition of $m$ independent sub-fractional Brownian motions with known Hurst indices and a common scale. Nondegeneracy follows from realising the process as the even part of a two-sided fractional Brownian motion, and the maximum likelihood estimators of the trend and of the scale are explicit. The statistics on which inference rests are pivotal, their laws depending on the sample size alone, so intervals and tests of exact level are available at every $N\ge2$, together with complete sufficiency, minimum variance unbiasedness and attainment of the Cramér--Rao bound. An explicit variance bound gives strong consistency and asymptotic normality, and simulations confirm the exact coverage and the predicted effect of a misspecified Hurst vector.
