Distributions of the maximum likelihood and minimum contrast estimators associated with the fractional Ornstein–Uhlenbeck process

Abstract

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 $H$ is known and is in $[1/2, 1)$. 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 $H$ and the sampling span $M$. 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.

Appendix

Proof of Theorem 1

It follows from the fractional version of Girsanov’s theorem that

$$\begin{aligned} m(\theta _1,\theta _2)&= \text{ E }\left[ \exp \left\{ \theta _1 \int \limits _0^M Q_H^\alpha (t) dZ_H^\alpha (t)+\theta _2 \int \limits _0^M \left\{ Q_H^\alpha (t)\right\} ^2 dv_H(t)\right\} \right] \\&= \text{ E }\left[ \exp \left\{ \theta _1 \int \limits _0^M Q_H^\beta (t) dZ_H^\beta (t)+\theta _2 \int \limits _0^M \left\{ Q_H^\beta (t)\right\} ^2 dv_H(t)\right\} \right. \nonumber \\&\quad \times \left. \frac{d \mu _{Y_H^\alpha }}{d \mu _{Y_H^\beta }}(Y_H^\beta )\right] \\&= \text{ E }\left[ \exp \left\{ (\theta _1+\alpha -\beta ) \int \limits _0^M Q_H^\beta (t) dZ_H^\beta (t) \right. \right. \\&\quad \left. \left. +\left( \theta _2-\frac{\alpha ^2-\beta ^2}{2}\right) \int \limits _0^M \left\{ Q_H^\beta (t)\right\} ^2 dv_H(t)\right\} \right] \nonumber \\&= \text{ E }\left[ \exp \left\{ (\theta _1+\alpha -\beta ) \int \limits _0^M Q_H^\beta (t) dZ_H^\beta (t) \right\} \right] \\&= e^{M(\beta -\alpha -\theta _1)/2} \text{ E }\left[ \exp \left\{ \kappa Z_H^\beta (M) \int \limits _0^M \, t^{2H-1} \, dZ_H^\beta (t)\right\} \right] \!\!, \end{aligned}$$

where $\kappa =(\theta _1+\alpha -\beta )\eta _H/(4(1-H))$ and we have put $\beta =\sqrt{\alpha ^2-2 \theta _2}$. Then it follows from Kleptsyna and Le Breton (2002) that this last quantity yields

$$\begin{aligned} m(\theta _1,\theta _2)&= e^{M(\beta -\alpha -\theta _1)/2} \left( \frac{4 \sin \pi H}{\pi \beta M}\right) ^{1/2} \left[ \left( 1+\frac{\beta -\alpha -\theta _1}{\beta } e^{z} \sinh z\right) ^2 I_{-H}(z)I_{H-1}(z)\right. \\&\quad \left. - \left( 1-\frac{\beta -\alpha -\theta _1}{\beta } e^{z} \cosh z\right) ^2 I_{1-H}(z)I_{H}(z)\right] ^{-1/2} \\&= e^{M(-\alpha -\theta _1)/2} \left( \frac{4 \sin \pi H}{\pi M}\right) ^{1/2} \left[ \frac{1}{\beta }\left( \beta \cosh z-(\alpha +\theta _1) \sinh z\right) ^2 I_{-H}(z)I_{H-1}(z)\right. \\&\quad \left. -\, \frac{1}{\beta }\left( \beta \sinh z-(\alpha +\theta _1) \cosh z\right) ^2 I_{1-H}(z)I_{H}(z)\right] ^{-1/2}\!\!, \\ \end{aligned}$$

where $z=\beta M/2$. Then, using the relation

$$\begin{aligned} I_H(z) I_{H-1}(z)-I_{1-H}(z) I_H(z)=\frac{4 \sin \pi H}{2 \pi z}, \end{aligned}$$

it can be verified that $m(\theta _1,\theta _2)$ is given by (35), which establishes Theorem 1.

Proof of Theorem 2

When $\alpha =0$, the distibution function of $\tilde{\alpha }_{MLE}$ is defined by

$$\begin{aligned} F_{MLE}(x)&= \frac{1}{2}+\frac{1}{\pi } \int \limits _0^\infty \, \frac{1}{\theta } \text{ Im } \big [m(-i \theta , i \theta x)\big ] \, d\theta \nonumber \\&= \frac{1}{2}+\frac{1}{\pi } \int \limits _0^\infty \, \frac{1}{u} \text{ Im } \big [m(-i u/M, i ux/M)\big ] \, du, \end{aligned}$$

(48)

where

$$\begin{aligned} m(-i u/M, i u x/M)&= e^{i u/2}\left[ \left( 1-\frac{i u}{2 M x}\right) \cosh ^2 z+ u \frac{\sinh 2 z}{2 z} \right. \\&\quad \left. +\, \frac{\pi }{4 \sin \pi H}\left( 2zI_{1-H}(z) I_H(z)+\frac{u^2}{2 z} I_{-H}(z) I_{H-1}(z)\right) \right] ^{-1/2}, \\ z&= \frac{1}{2} \sqrt{-2 i u M x}\!. \end{aligned}$$

Then, it follows from the form of the integrand that $x$ is always coupled with $M$ as $M \times x$. Thus we can confirm the statement in the theorem.

As for the MCE, we have

$$\begin{aligned} F_{MCE}(x)&= \frac{1}{2}+\frac{1}{\pi } \int \limits _0^\infty \, \frac{1}{\theta } \text{ Im } \left[ e^{i M \theta /2} \, m(0, i \theta x)\right] \, d\theta \nonumber \\&= \frac{1}{2}+\frac{1}{\pi } \int \limits _0^\infty \, \frac{1}{u} \text{ Im } \left[ e^{i u/2} \, m(0, i u x/M)\right] \, du, \end{aligned}$$

(49)

where

$$\begin{aligned} m(0, i u x/M)&= e^{i u/2}\left[ \cosh ^2 z+\frac{\pi }{4 \sin \pi H} 2z I_{1-H}(z) I_H(z)\right] ^{-1/2}\!. \end{aligned}$$

The same reasoning as before applies here, which establishes Theorem 2.

Proof of Theorem 3

Let $x_\gamma (H,M)$ be the $100 \gamma \%$ point of the distribution of $\tilde{\alpha }_{MLE}$ for $\alpha =0$ under $H$ and $M$. Then the power of the test at the level $\gamma$ is computed as

$$\begin{aligned} P\left( \tilde{\alpha }_{MLE} < x_\gamma (H,M)\left| \alpha <0\right) \right.&= \frac{1}{2}+\frac{1}{\pi } \int \limits _0^\infty \, \frac{1}{\theta } \text{ Im }\left[ \phi (\theta ;x_\gamma (H,M))\right] \, d\theta , \\&= \frac{1}{2}+\frac{1}{\pi } \int \limits _0^\infty \, \frac{1}{u} \text{ Im }\left[ \phi (u/M;x_\gamma (H,M))\right] \, du, \end{aligned}$$

where

$$\begin{aligned} \phi (u/M;x_\gamma (H,M))&= m(-i u/M, i u x_\gamma (H,m)/M) \\&= e^{i u-\alpha M} \left[ \left( 1+\frac{(i u-\alpha M)^2}{4 z^2}\right) \cosh ^2 z +(i u-\alpha M) \frac{\sinh 2z}{2 z} \right. \\&\quad \left. +\, \frac{\pi }{4 \sin \pi H}\left\{ 2 z I_{1-H}(z) I_H(z)-\frac{(i u-\alpha M)^2}{2 z} I_{-H}(z) I_{H-1}(z)\right\} \right] ^{-1/2}\!\!\!, \\ z&= \frac{1}{2}\left( (\alpha M)^2-2 i u M x_\gamma (H,M)\right) ^{1/2}. \end{aligned}$$

Noting that $M x_\gamma (H,M)$ is independent of $M$ because of Theorem 2, it is seen from the form of the c.f. that the power depends only on $\alpha M$. We can also prove this fact for the power of the test based on $\hat{\alpha }_{MCE}$, which establishes Theorem 3.