ABSTRACT We propose an adaptive estimator for the stationary distribution of a bifurcating Markov Chain on . Bifurcating Markov chains (BMC for short) are a class of stochastic processes indexed by regular binary trees. A kernel estimator is proposed whose bandwidths are selected by a method inspired by the works of Goldenshluger and Lepski [(2011), ‘Bandwidth Selection in Kernel Density Estimation: Oracle Inequalities and Adaptive Minimax Optimality’, The Annals of Statistics 3: 1608–1632). Drawing inspiration from dimension jump methods for model selection, we also provide an algorithm to select the best constant in the penalty. Finally, we investigate the performance of the method by simulation studies and application to real data.

中文翻译：

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分叉马尔可夫链模型中核密度估计的局部带宽选择

摘要 我们为 上的分叉马尔可夫链的平稳分布提出了一种自适应估计器。分叉马尔可夫链（BMC）是一类由规则二叉树索引的随机过程。提出了一种核估计器，其带宽的选择方法受 Goldenshluger 和 Lepski 的作品启发 [(2011)，“核密度估计中的带宽选择：Oracle 不等式和自适应极小极大优化”，统计年鉴 3：1608-1632 ）。从模型选择的维度跳跃方法中汲取灵感，我们还提供了一种算法来选择惩罚中的最佳常数。最后，我们通过模拟研究和实际数据的应用来研究该方法的性能。