Distributions of the maximum likelihood and minimum contrast estimators associated with the fractional Ornstein–Uhlenbeck process
We discuss some inference problems associated with the fractional Ornstein–Uhlenbeck (fO–U) process driven by the fractional Brownian motion (fBm). In particular, we are concerned with the estimation of the drift parameter, assuming that the Hurst parameter <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$H$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>H</mi> </math> </EquationSource> </InlineEquation> is known and is in <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$[1/2, 1)$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mo stretchy="false">[</mo> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math> </EquationSource> </InlineEquation>. Under this setting we compute the distributions of the maximum likelihood estimator (MLE) and the minimum contrast estimator (MCE) for the drift parameter, and explore their distributional properties by paying attention to the influence of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$H$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>H</mi> </math> </EquationSource> </InlineEquation> and the sampling span <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$$M$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>M</mi> </math> </EquationSource> </InlineEquation>. We also deal with the ordinary least squares estimator (OLSE) and examine the asymptotic relative efficiency. It is shown that the MCE is asymptotically efficient, while the OLSE is inefficient. We also consider the unit root testing problem in the fO–U process and compute the power of the tests based on the MLE and MCE. Copyright Springer Science+Business Media Dordrecht 2013
