Semiparametric Estimation for Stationary Processes Whose Spectra Have an Unknown Pole

We consider the estimation of the location of the pole and memory parameter, amp;#955;lt;supgt;0lt;/supgt; and amp;#945; respectively, of covariance stationary linear processes whose spectral density function f(amp;#955;) satisfies f(amp;#955;) amp;#8764; C|amp;#955; amp;#8722; amp;#955;lt;supgt;0lt;/supgt;|lt;supgt;amp;#8722;amp;#945;lt;/supgt; in a neighbourhood of amp;#955;lt;supgt;0lt;/supgt;. We define a consistent estimator of amp;#955;lt;supgt;0lt;/supgt; and derive its limit distribution Zlt;subgt;amp;#955;lt;supgt;0lt;/supgt;lt;/subgt; . As in related optimization problems, when the true parameter value can lie on the boundary of the parameter space, we show that Zlt;subgt;amp;#955;lt;supgt;0lt;/supgt;lt;/subgt; is distributed as a normal random variable when amp;#955;lt;supgt;0lt;/supgt; amp;#8712; (0, amp;#960;), whereas for amp;#955;lt;supgt;0lt;/supgt; = 0 or amp;#960;, Zlt;subgt;amp;#955;lt;supgt;0lt;/supgt;lt;/subgt; is a mixture of discrete and continuous random variables with weights equal to 1/2. More specifically, when amp;#955;lt;supgt;0lt;/supgt; = 0, Zlt;subgt;amp;#955;lt;supgt;0lt;/supgt;lt;/subgt; is distributed as a normal random variable truncated at zero. Moreover, we describe and examine a two-step estimator of the memory parameter amp;#945;, showing that neither its limit distribution nor its rate of convergence is affected by the estimation of amp;#955;lt;supgt;0lt;/supgt;. Thus, we reinforce and extend previous results with respect to the estimation of amp;#945; when amp;#955;lt;supgt;0lt;/supgt; is assumed to be known a priori. A small Monte Carlo study is included to illustrate the finite sample performance of our estimators