检验多维伊藤半鞅的现货协方差矩阵的秩
Testing the rank of the spot covariance matrix of a multidimensional Itô semi-martingale
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中文总结 AI 辅助
利用高频观测开发统计检验，针对多维伊藤半鞅现货协方差矩阵最大秩，通过重新中心化协方差估计器考虑适应漂移，检验原假设，导出临界值，分析功效，模拟结果显示该检验比经典方法功效更高。
中文摘要 AI 辅助
本文利用高频观测数据，针对连续时间\(\mathbb{R}^d\)值伊藤半鞅\(X(t)\)的确定性瞬时（或现货）协方差矩阵的最大秩，开发了一种统计检验方法，特别关注适应漂移的影响。我们通过引入重新中心化的协方差估计器，明确考虑了适应漂移过程的存在，而不是仅依赖二阶矩估计器。在此估计器的基础上，我们针对局部备择假设检验原假设：即对于所有\(t\)，现货协方差矩阵的秩至多为\(r<d\)，其中第\((r + 1)\)个特征值大于某个消失的信号检测率。临界值在非渐近框架中导出，可能会受到潜在漂移的显著影响。然而，功效分析建立了分离率的渐近一致性，其取决于漂移和现货协方差矩阵的赫尔德正则性，以及原假设下的潜在谱隙\(\underline{\lambda}_r \geq 0\)。模拟结果表明，与基于经典二阶矩的程序相比，基于协方差的检验在更广泛的备择假设范围内具有更高的功效。
英文摘要
This work develops a statistical test for the maximal rank of the deterministic instantaneous (or spot) covariance matrix of a continuous-time $\mathbb{R}^d$-valued Itô semi-martingale $X(t)$ using high-frequency observations with a particular focus on the impact of an adapted drift. We explicitly account for the presence of an adapted drift process, which, as our results demonstrate, cannot be neglected, by introducing a re-centred covariance estimator instead of relying solely on a second moment estimator. Building on this estimator, we test the null hypothesis that the rank of the spot covariance matrix is at most $r<d$ for all $t$ against local alternatives in which the $(r+1)$th eigenvalue is greater than some vanishing signal detection rate. Critical values are derived in a non-asymptotic framework and can be significantly affected by a potential drift. However, the power analysis establishes asymptotic consistency for separation rates, which depend on the Hölder regularity of both the drift and the spot covariance matrix, as well as on a potential spectral gap $\underlineλ_r \geq 0$ under the null hypothesis. Simulation results indicate that the covariance-based test achieves higher power across a wider range of alternatives compared to classical second moment-based procedures.
