Semiparametric estimation for stationary processes whose spectra have an unknown pole

Abstract

We consider the estimation of the location of the pole and memory parameter, λ^0 and α, respectively, of covariance stationary linear processes whose spectral density function f(λ) satisfies f(λ)\sim C| λ-λ^0| ^{-α} in a neighborhood of λ^0. We define a consistent estimator of λ^0 and derive its limit distribution Z_{λ^0}. As in related optimization problems, when the true parameter value can lie on the boundary of the parameter space, we show that Z_{λ^0} is distributed as a normal random variable when λ^0\in (0,π), whereas for λ^0=0 or π, Z_{λ^0} is a mixture of discrete and continuous random variables with weights equal to 1/2. More specifically, when λ^0=0, Z_{λ^0} 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 λ^0. Thus, we reinforce and extend previous results with respect to the estimation of αwhen λ^0 is assumed to be known a priori. A small Monte Carlo study is included to illustrate the finite sample performance of our estimators.