离散状态空间上遍历马尔可夫链核的统计估计
Statistical Estimation of Ergodic Markov Chain Kernel over Discrete State Space
摘要：
我们研究了从一个长的状态观测序列估计离散状态马尔可夫链核参数的统计复杂性。在有限的情况下，我们的特点（模对数因子）的极大极小样本估计的复杂性相对于运营商的无穷范数，而在可数无限的情况下，我们分析的问题相对于一个自然的条目明智的范数来自全变差。我们表明，在这两种情况下，样本的复杂性是由未知链的混合性质，其中，在有限状态的情况下，有已知的有限样本估计完全经验的置信区间。
英文摘要：
We investigate the statistical complexity of estimating the parameters of a discrete-state Markov chain kernel from a single long sequence of state observations. In the finite case, we characterize (modulo logarithmic factors) the minimax sample complexity of estimation with respect to the operator infinity norm, while in the countably infinite case, we analyze the problem with respect to a natural entry-wise norm derived from total variation. We show that in both cases, the sample complexity is governed by the mixing properties of the unknown chain, for which, in the finite-state case, there are known finite-sample estimators with fully empirical confidence intervals.
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