Adaptive efficient estimation for generalized semi-Markov Big Data models

In this paper we study generalized semi-Markov high dimension regression models in continuous time observed in 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, Beltaief and Pergamenshchikov (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 minimax sense. It should be noted also that for these results we do not use either sparse conditions or the parameter dimension in the model. As examples, it is considered regression models constructed through spherical symmetric noise impulses and truncated fractional Poisson processes. Numeric Monte-Carlo simulations confirming the theoretical results are given in the supplementary materials