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Optimal convergence rates in non‐parametric regression with fractional time series errors
Author(s) -
Feng Yuanhua,
Beran Jan
Publication year - 2013
Publication title -
journal of time series analysis
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.576
H-Index - 54
eISSN - 1467-9892
pISSN - 0143-9782
DOI - 10.1111/j.1467-9892.2012.00811.x
Subject(s) - mathematics , invertible matrix , series (stratigraphy) , polynomial regression , rate of convergence , parametric statistics , regression , polynomial , regression function , convergence (economics) , linear regression , statistics , mathematical analysis , pure mathematics , paleontology , channel (broadcasting) , economic growth , electrical engineering , economics , biology , engineering
Consider the estimation of g ( ν ) , the ν th derivative of the mean function, in a fixed‐design non‐parametric regression model with stationary time series errors ξ i . We assume that , ξ i are obtained by applying an invertible linear filter to iid innovations, and the spectral density of ξ i has the form as λ → 0 with constants c f > 0 and α ∈ (−1,1). Under regularity conditions, the optimal convergence rate of is shown to be with r = (1 − α )( k − ν )/(2 k +1 − α ). This rate is achieved by local polynomial fitting. Moreover, in spite of including long memory and antipersistence, the required conditions on the innovation distribution turn out to be the same as in non‐parametric regression with iid errors.