The Annals of Statistics

Semiparametric estimation for stationary processes whose spectra have an unknown pole

Javier Hidalgo

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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.

Article information

Ann. Statist., Volume 33, Number 4 (2005), 1843-1889.

First available in Project Euclid: 5 August 2005

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62M15: Spectral analysis
Secondary: 62G05: Estimation 62G20: Asymptotic properties

Spectral density estimation long-memory processes Gaussian processes


Hidalgo, Javier. Semiparametric estimation for stationary processes whose spectra have an unknown pole. Ann. Statist. 33 (2005), no. 4, 1843--1889. doi:10.1214/009053605000000318.

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