Local bandwidth selection for kernel density estimation in bifurcating Markov chain model

We propose an adaptive estimator for the stationary distribution of a bifurcating Markov Chain on $\mathbb R^d$. Bifurcating Markov chains (BMC for short) are a class of stochastic processes indexed by regular binary trees. A kernel estimator is proposed whose bandwidth is selected by a method inspired by the works of Goldenshluger and Lepski [18]. Drawing inspiration from dimension jump methods for model selection, we also provide an algorithm to select the best constant in the penalty.