带漂移的多重混合分数布朗运动的精确有限样本推断

中文摘要 AI 辅助

本文研究了一个线性漂移被m个独立分数布朗运动的叠加（具有已知的Hurst参数和共同尺度）所扰动，并在N个等距时间点上观测该过程。此类模型的推断通常是渐近的；我们证明在此处推断是精确的。我们以闭式形式推导了漂移θ和尺度α²的最大似然估计量，并获得了它们的精确有限样本联合分布：θ̂是高斯分布，Nα̂²/α²服从自由度为N-1的卡方分布，且两者相互独立。由于该分布不依赖于任何模型参数，我们推导出对于每个N≥2，无论Hurst向量如何，都具有精确水平的Student和卡方置信区间及检验。我们还证明了这些估计量是一致最小方差无偏的，其中θ̂在每一个N处都达到Cramér-Rao界，两者都是强相合且渐近正态的，并且漂移估计量在分布上构成一个沿其自身方差尺度运行的布朗运动。一个尖锐的非渐近界表明，漂移的精度由观测窗口的长度决定，而非网格大小，蒙特卡洛研究证实了即使在较小样本量下也具有精确覆盖，并量化了当Hurst向量被错误指定时所损失的内容。

英文摘要

In this paper we study a linear drift perturbed by a superposition of $m$ independent fractional Brownian motions with known Hurst parameters and a common scale, observed at $N$ equidistant times. Inference for such models is usually asymptotic; we show that here it is exact. We derive the maximum likelihood estimators of the drift $θ$ and of the scale $α^{2}$ in closed form and obtain their exact finite-sample joint law: $\widehatθ$ is Gaussian, $N\widehatα^{\,2}/α^{2}$ is chi-square with $N-1$ degrees of freedom, and the two are independent. As this law is free of every model parameter, we deduce Student and chi-square confidence intervals and tests of exact level for every $N\ge2$, whatever the Hurst vector. We also prove that the estimators are uniformly minimum variance unbiased with $\widehatθ$ attaining the Cramér--Rao bound at every $N$, that both are strongly consistent and asymptotically normal, and that the drift estimators form, in law, a Brownian motion run along their own variance scale. A sharp non-asymptotic bound shows that the accuracy of the drift is governed by the length of the observation window and not by the mesh, and a Monte Carlo study confirms exact coverage, even at small sample sizes, and quantifies what is lost when the Hurst vector is misspecified.

发表机构

• Imam Abdulrahman Bin Faisal University（伊玛姆阿卜杜勒拉赫曼·本·费萨尔大学）

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