The Annals of Statistics

Efficiency improvements in inference on stationary and nonstationary fractional time series

P. M. Robinson

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We consider a time series model involving a fractional stochastic component, whose integration order can lie in the stationary/invertible or nonstationary regions and be unknown, and an additive deterministic component consisting of a generalized polynomial. The model can thus incorporate competing descriptions of trending behavior. The stationary input to the stochastic component has parametric autocorrelation, but innovation with distribution of unknown form. The model is thus semiparametric, and we develop estimates of the parametric component which are asymptotically normal and achieve an M-estimation efficiency bound, equal to that found in work using an adaptive LAM/LAN approach. A major technical feature which we treat is the effect of truncating the autoregressive representation in order to form innovation proxies. This is relevant also when the innovation density is parameterized, and we provide a result for that case also. Our semiparametric estimates employ nonparametric series estimation, which avoids some complications and conditions in kernel approaches featured in much work on adaptive estimation of time series models; our work thus also contributes to methods and theory for nonfractional time series models, such as autoregressive moving averages. A Monte Carlo study of finite sample performance of the semiparametric estimates is included.

Article information

Ann. Statist., Volume 33, Number 4 (2005), 1800-1842.

First available in Project Euclid: 5 August 2005

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84]
Secondary: 62F11 62G10: Hypothesis testing 62J05: Linear regression

Fractional processes efficient semiparametric estimation adaptive estimation nonstationary processes series estimation M-estimation


Robinson, P. M. Efficiency improvements in inference on stationary and nonstationary fractional time series. Ann. Statist. 33 (2005), no. 4, 1800--1842. doi:10.1214/009053605000000354.

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