Efficient pointwise estimation based on discrete data in ergodic nonparametric diffusions

A truncated sequential procedure is constructed for estimating the drift coefficient at a given state point based on discrete data of ergodic diffusion process. A nonasymptotic upper bound is obtained for a pointwise absolute error risk. The optimal convergence rate and a sharp constant in the bounds are found for the asymptotic pointwise minimax risk. As a consequence, the efficiency is obtained of the proposed sequential procedure.Comment: Published at http://dx.doi.org/10.3150/14-BEJ655 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Adaptive efficient analysis for big data ergodic diffusion models

We consider drift estimation problems for high dimension ergodic diffusion processes in nonparametric setting based on observations at discrete fixed time moments in the case when diffusion coefficients are unknown. To this end on the basis of sequential analysis methods we develop model selection procedures, for which we show non asymptotic sharp oracle inequalities. Through the obtained inequalities we show that the constructed model selection procedures are asymptotically efficient in adaptive setting, i.e. in the case when the model regularity is unknown. For the first time for such problem, we found in the explicit form the celebrated Pinsker constant which provides the sharp lower bound for the minimax squared accuracy normalized with the optimal convergence rate. Then we show that the asymptotic quadratic risk for the model selection procedure asymptotically coincides with the obtained lower bound, i.e this means that the constructed procedure is efficient. Finally, on the basis of the constructed model selection procedures in the framework of the big data models we provide the efficient estimation without using the parameter dimension or any sparse conditions

Эффективное оценивание больших данных эргодических диффузионных процессов

Изучаем задачу оценивания эргодического диффузионного процесса со сносом S(y) = ψ0(y) +Pq j=1 tajψj(y) и волатильностью b(y), где ψj , j ≤ q, — линейно независимые функции, taj , b(ot), q — неизвестные параметры.Наблюдения доступны в дискретные моменты времени размера N <q. Болеетого,переходим к непараметрической настройке.Задача состоит в том,чтобы оценить снос S(y), y ∈ [a, b], при этом волатильность b(ot) является мешающим параметром.Путем последовательного анализа строятся оценки ˆ S для S, которые позволяют нам моделировать процедуры выбора и доказывать оракульные неравенства, обеспечивающие асимптотическую эффективность

Точные оракульные неравенства для задач оценивания сноса по дискретным данным

In this paper we consider the non parametric drift estimation problem for the ergodic diffusion processes on the basis of the observations in the fixed discrete time moments in the case when the diffusion coefficients are unknown. Using the truncated sequentia