Inférence statistique asymptotique dans les processus ARMA fractionnaire.

Abstract

Gaussian process (GP) is a stochastic process that has been successfully applied in finance, black-box modeling of biosystems, machine learning, geostatistics, multitask learning or robotics and reinforcement learning. Effectively estimating the spectral density function (SDF) and degree of memory (DOM) of a long-memory stationary GP (LMSGP) is a significant hard problem investigators may face. {This paper gives some new sufficient conditions (NSCs) for improving the lag window estimators (LWEs) of the SDF and DOM for LMSGPs. A comparison study among the behavior of the LWEs under the NSCs, the LWEs without the NSCs and the existing widely used periodogram estimators (PEs) is given. The theoretical and computational justifications show that: the LWEs under the NSCs are better than the LWEs without the NSCs; the LWEs under the NSCs are better than the PEs; the LWEs under the NSCs are asymptotically unbiased and consistent; the asymptotic distributions of the LWEs under the NSCs of the SDF and DOM under the NSCs are chi-square and normal, respectively; the LWE of the DOM under the NSCs has a fast vanishing variance under the regression method; and the LWEs under the NSCs improve the finite sample properties for the regression and local Whittle estimation methods.