We are interested in assessing the order of a finite-state Hidden Markov Model (HMM) with the only two assumptions that the transition matrix of the latent Markov chain has full rank and that the density functions of the emission distributions are linearly independent. We introduce a new procedure for estimating this order by investigating the rank of some well-chosen integral operator which relies on the distribution of a pair of consecutive observations. This method circumvents the usual limits of the spectral method when it is used for estimating the order of an HMM: it avoids the choice of the basis functions; it does not require any knowledge of an upper-bound on the order of the HMM (for the spectral method, such an upper-bound is defined by the number of basis functions); it permits to easily handle different types of data (including continuous data, circular data or multivariate continuous data) with a suitable choice of kernel. The method relies on the fact that the order of the HMM can be identified from the distribution of a pair of consecutive observations and that this order is equal to the rank of some integral operator (\emph{i.e.} the number of its singular values that are non-zero). Since only the empirical counter-part of the singular values of the operator can be obtained, we propose a data-driven thresholding procedure. An upper-bound on the probability of overestimating the order of the HMM is established. Moreover, sufficient conditions on the bandwidth used for kernel density estimation and on the threshold are stated to obtain the consistency of the estimator of the order of the HMM. The procedure is easily implemented since the values of all the tuning parameters are determined by the sample size.

翻译：本文旨在评估有限状态隐马尔可夫模型（HMM）的阶数，仅需满足两个假设：潜在马尔可夫链的转移矩阵满秩，且发射分布的密度函数线性无关。我们提出了一种新方法，通过研究某个精心选取的积分算子的秩来估计该阶数，该算子基于连续两次观测的联合分布。该方法克服了谱方法在估计HMM阶数时的常见局限：无需选择基函数；不要求预先知道HMM阶数的上界（谱方法中该上界由基函数数量决定）；通过适当选择核函数，可轻松处理不同类型的数据（包括连续数据、圆形数据或多变量连续数据）。该方法基于以下事实：HMM的阶数可由连续两次观测的联合分布唯一确定，且该阶数等于某个积分算子的秩（即其非零奇异值的数量）。由于仅能获得算子奇异值的经验对应量，我们提出了一种数据驱动的阈值选择程序。本文建立了HMM阶数被高估概率的上界，并给出了确保阶数估计量一致性的充分条件——包括核密度估计带宽和阈值的选取条件。由于所有调优参数值均由样本量决定，该程序易于实现。