Asymptotic properties of the maximum likelihood estimator for Hidden Markov Models indexed by binary trees

Julien WeibelIDP, CERMICS
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Abstract

We consider hidden Markov models indexed by a binary tree where the hidden state space is a general metric space. We study the maximum likelihood estimator (MLE) of the model parameters based only on the observed variables. In both stationary and non-stationary regimes, we prove strong consistency and asymptotic normality of the MLE under standard assumptions. Those standard assumptions imply uniform exponential memorylessness properties of the initial distribution conditional on the observations. The proofs rely on ergodic theorems for Markov chain indexed by trees with neighborhood-dependent functions.
以二叉树为索引的隐马尔可夫模型最大似然估计器的渐近特性
我们考虑了以二叉树为索引的隐马尔可夫模型,其中的隐状态空间是一般度量空间。我们研究了仅基于观测变量的模型参数最大似然估计(MLE)。在静态和非静态状态下,我们证明了标准假设下 MLE 的强一致性和渐近正态性。这些标准假设意味着初始分布在观测值条件下的均匀指数无记忆特性。证明依赖于马尔可夫链的遍历定理,这些马尔可夫链由具有邻域函数的树索引。
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