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.
"Semiparametric estimation for stationary processes whose spectra have an unknown pole." Ann. Statist. 33 (4) 1843 - 1889, August 2005. https://doi.org/10.1214/009053605000000318