Semiparametric Estimation for Stationary Processes whose Spectra have an Unknown Pole

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