Sequential model selection method for nonparametric autoregression

Abstract

In this paper for the first time the nonparametric autoregression estimation problem for the quadratic risks is considered. To this end we develop a new adaptive sequential model selection method based on the efficient sequential kernel estimators proposed by Arkoun and Pergamenshchikov (2016). Moreover, we develop a new analytical tool for general regression models to obtain the non asymptotic sharp oracle inequalities for both usual quadratic and robust quadratic risks. Then, we show that the constructed sequential model selection procedure is optimal in the sense of oracle inequalities.MSC: primary 62G08, secondary 62G05

Consistent model selection criteria and goodness-of-fit test for common time series models

“…(i) In this part, we first show that for any k ∈ N, On the other hand, when θ = θ * , e 2 t (θ * ) = ξ 2 t for any t ∈ Z and since E[ξ 2 0 ] = 1, we deduce that e 2 t (θ * ) − 1 t is a sequence of centred iid random variables with variance μ 4 − 1 with μ 4 = E[ξ 4 0 ]. In such as case, the asymptotic behavior of the covariograms is well known and we deduce: (7.36) with I k the (K × K) identity matrix.…”

“…[35] considers model selection for density estimation under mixing conditions and derived oracle inequalities of the slope heuristic procedure ( [14] or [6]); whereas [3] develop oracle inequalities for model selection for weakly dependent time series forecasting. Recently, [46] have considered the model selection for ARMA time series with trend, and proved the consistency of BIC for the detrended residual sequence, while [4] developed oracle inequalities of sequential model selection method for nonparametric autoregression. [26] pointed out that most existing model selection procedure cannot simultaneously enjoy consistency and (asymptotic) efficiency.…”

“…1. In practice the constant μ 4 and the columns of the matrix J K (m * ) (see (7.34)) involved in V (θ * , m * ) are estimated by the correspondent sample average; they are respectively μ 4 = 1 n n t=1 ( e t ( m)) 4 and…”

Adaptive efficient robust sequential analysis for autoregressive big data models

“…It should be noted, that for such models this constant is calculated for the first time. Then, using the weighted least square method and sharp non asymptotic oracle inequalities from [4] we provide the efficiency property in the minimax sense for the proposed estimation procedure, i.e. we establish, that the upper bound for its risk coincides with the obtained lower bound.…”

“…Moreover, it turned out that only the sequential methods can provide the adaptive estimation for autoregressive models. That is why in this paper we use the adaptive sequential procedures from [4] for the efficient estimation, which we study for the quadratic risks…”