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TIME SERIES RESIDUALS WITH APPLICATION TO PROBABILITY DENSITY ESTIMATION
Author(s) -
Robinson P. M.
Publication year - 1987
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.1467-9892.1987.tb00445.x
Subject(s) - mathematics , estimator , series (stratigraphy) , probability density function , kernel density estimation , invertible matrix , fourier series , density estimation , kernel (algebra) , gaussian , multivariate kernel density estimation , statistics , combinatorics , mathematical analysis , kernel method , variable kernel density estimation , pure mathematics , paleontology , physics , quantum mechanics , artificial intelligence , computer science , support vector machine , biology
. A linear stationary and invertible process y t models the second‐order properties of T observations on a discrete time series, up to finitely many unknown parameters θ. Two estimators of the residuals or innovations ɛ t of y t are presented, based on a θ estimator which is root‐ T consistent with respect to a wide class of ɛ t distributions, such as a Gaussian estimator. One sets unobserved y t equal to their mean, the other treats y t as a circulant and may be best computed via two passes of the fast Fourier transform. The convergence of both estimators to ɛ t is investigated. We apply the estimated ɛ t to estimate the probability density function of ɛ t . Kernel density estimators are shown to converge uniformly in probability to the true density. A new sub‐class of linear time series models is motivated.