This paper derives second-order expansions for the distributions of the Whittle and proﬁle plug-in maximum likelihood estimators of the fractional diﬀerence parameter in the ARFIMA(0, d ,0) with unknown mean and variance. Both estimators are shown to be second-order pivotal. This extends earlier ﬁndings of Lieberman and Phillips (2001), who derived expansions for the Gaussian maximum likelihood estimator under the assumption that the mean and variance are known. One implication of the results is that the parametric bootstrap upper one-sided conﬁdence interval provides an o ( n -1 ln n ) improvement over the delta method. For statistics that are not second-order pivotal, the improvement is generally only of the order o ( n -1/ 2 ln n ).
Lieberman, Offer and Phillips, Peter C.B., "Expansions for Approximate Maximum Likelihood Estimators of the Fractional Difference Parameter" (2004). Cowles Foundation Discussion Papers. 1753.