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August 2005 Semiparametric estimation for stationary processes whose spectra have an unknown pole
Javier Hidalgo
Ann. Statist. 33(4): 1843-1889 (August 2005). DOI: 10.1214/009053605000000318


We consider the estimation of the location of the pole and memory parameter, λ0 and α, respectively, of covariance stationary linear processes whose spectral density function f(λ) satisfies f(λ)∼C|λ−λ0|−α in a neighborhood of λ0. We define a consistent estimator of λ0 and derive its limit distribution Zλ0. As in related optimization problems, when the true parameter value can lie on the boundary of the parameter space, we show that Zλ0 is distributed as a normal random variable when λ0∈(0,π), whereas for λ0=0 or π, Zλ0 is a mixture of discrete and continuous random variables with weights equal to 1/2. More specifically, when λ0=0, Zλ0 is distributed as a normal random variable truncated at zero. Moreover, we describe and examine a two-step estimator of the memory parameter α, showing that neither its limit distribution nor its rate of convergence is affected by the estimation of λ0. Thus, we reinforce and extend previous results with respect to the estimation of α when λ0 is assumed to be known a priori. A small Monte Carlo study is included to illustrate the finite sample performance of our estimators.


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Javier Hidalgo. "Semiparametric estimation for stationary processes whose spectra have an unknown pole." Ann. Statist. 33 (4) 1843 - 1889, August 2005.


Published: August 2005
First available in Project Euclid: 5 August 2005

zbMATH: 1078.62098
MathSciNet: MR2166564
Digital Object Identifier: 10.1214/009053605000000318

Primary: 62M15
Secondary: 62G05 , 62G20

Keywords: Gaussian processes , long-memory processes , spectral density estimation

Rights: Copyright © 2005 Institute of Mathematical Statistics


Vol.33 • No. 4 • August 2005
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