In this paper we study generalized semi-Markov high dimension regression models in continuous time, observed at fixed discrete time moments. The generalized semi-Markov process has dependent jumps and, therefore, it is an extension of the semi-Markov regression introduced in Barbu et al. (Stat Inference Stoch Process 22:187–231, 2019a). For such models we consider estimation problems in nonparametric setting. To this end, we develop model selection procedures for which sharp non-asymptotic oracle inequalities for the robust risks are obtained. Moreover, we give constructive sufficient conditions which provide through the obtained oracle inequalities the adaptive robust efficiency property in the minimax sense. It should be noted also that, for these results, we do not use neither sparse conditions nor the parameter dimension in the model. As examples, regression models constructed through spherical symmetric noise impulses and truncated fractional Poisson processes are considered. Numerical Monte-Carlo simulations confirming the theoretical results are given in the supplementary materials.

在本文中，我们研究了在固定离散时间时刻观察到的连续时间的广义半马尔可夫高维回归模型。广义半马尔可夫过程具有依赖跳跃，因此，它是 Barbu 等人引入的半马尔可夫回归的扩展。（Stat Inference Stoch Process 22:187–231, 2019a）。对于此类模型，我们考虑非参数设置中的估计问题。为此，我们开发了模型选择程序，从中获得了针对稳健风险的尖锐的非渐近预言不等式。此外，我们给出了建设性的充分条件，这些条件通过所获得的预言不等式提供了极小极大意义上的自适应鲁棒效率属性。还应该注意的是，对于这些结果，我们既没有使用稀疏条件，也没有使用模型中的参数维度。例如，考虑通过球对称噪声脉冲和截断分数泊松过程构建的回归模型。补充材料中给出了证实理论结果的数值蒙特卡罗模拟。