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