标题： 最小二乘型估计特定隐藏马尔可夫链的转移密度 显示英文标题

标题： Least squares type estimation of the transition density of a particular hidden Markov chain

摘要： 本文研究如下隐马尔可夫链模型： $Y_i=X_i+\epsilon_i$，$i=1,...,n+1$，其中 $(X_i)$ 是一个实值平稳马尔可夫链，$(\epsilon_i)_{1\leq i\leq n+1}$ 是一个具有已知分布且与序列 $(X_i)$ 独立的噪声。 我们通过最小化一种原始对比度来构建转移密度的估计量，该对比度利用了问题的回归特性。该估计量是从一组投影估计量中通过模型选择方法选出的。 对于普通光滑噪声，评估了 $L^2$-风险及其收敛速度，并通过一些模拟验证了方法的有效性。 我们在 Besov 球类上得到了一致的风险界。 此外，我们的估计过程无需关于真实转移函数正则性的先验知识。 最后，我们的估计器避免了商数估计器的缺点。 显示英文摘要

摘要： In this paper, we study the following model of hidden Markov chain: $Y_i=X_i+\epsilon_i$, $i=1,...,n+1$ with $(X_i)$ a real-valued stationary Markov chain and $(\epsilon_i)_{1\leq i\leq n+1}$ a noise having a known distribution and independent of the sequence $(X_i)$. We present an estimator of the transition density obtained by minimization of an original contrast that takes advantage of the regressive aspect of the problem. It is selected among a collection of projection estimators with a model selection method. The $L^2$-risk and its rate of convergence are evaluated for ordinary smooth noise and some simulations illustrate the method. We obtain uniform risk bounds over classes of Besov balls. In addition our estimation procedure requires no prior knowledge of the regularity of the true transition. Finally, our estimator permits to avoid the drawbacks of quotient estimators.