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Prediction and nonparametric estimation for time series with heavy tails
Author(s) -
HALL PETER,
PENG LIANG,
YAO QIWEI
Publication year - 2002
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 - 10.1111/1467-9892.00266
Subject(s) - mathematics , least absolute deviations , estimator , statistics , series (stratigraphy) , least squares function approximation , generalized least squares , polynomial regression , nonparametric statistics , local regression , linear regression , paleontology , biology
Motivated by prediction problems for time series with heavy‐tailed marginal distributions, we consider methods based on `local least absolute deviations' for estimating a regression median from dependent data. Unlike more conventional `local median' methods, which are in effect based on locally fitting a polynomial of degree 0, techniques founded on local least absolute deviations have quadratic bias right up to the boundary of the design interval. Also in contrast to local least‐squares methods based on linear fits, the order of magnitude of variance does not depend on tail‐weight of the error distribution. To make these points clear, we develop theory describing local applications to time series of both least‐squares and least‐absolute‐deviations methods, showing for example that, in the case of heavy‐tailed data, the conventional local‐linear least‐squares estimator suffers from an additional bias term as well as increased variance.