On the Efficiency of Estimates of Trend in the Ornstein Uhlenbeck Process

Abstract

The problem is that of estimating the trend of a normal process when the trend function is known up to a finite number of coefficients. That is, $$y_t = x_t + f(t),\qquad 0 \leqq t \leqq T,$$ where $x_t$ is a normal process with mean zero and covariance function $$E\lbrack X_u, X_v\rbrack = C(u, v)$$ and $$f(t) = k_{1\phi_1}(t) + \cdots + k_s\phi_s(t).$$ The $\phi_i(t)$ are known functions and the $k_i$ are to be estimated. The standard procedure in such a case is to derive the estimates by the maximum likelihood method. However, if the covariance function $C(u, v)$ is not completely known, this is usually impossible, and it is essential to find an alternative procedure. The method of least squares has been proposed by Mann [1]. The estimates obtained by this method are independent of $C(u, v)$ and have the additional advantage of being easily computed. Mann and Moranda [2] showed that for the Ornstein Uhlenbeck process the asymptotic efficiency of the least square estimate relative to the maximum likelihood estimate is one, in the special case that the $\phi_i(t)$ are polynomials or trigonometric polynomials. Mann defines the efficiency $\bar e(T)$ of an estimate $\bar f(t)$ $$\bar e(T) = \frac{E\Big\lbrack\int^T_0 \lbrack\hat f(t) - f(t)\rbrack^2 dt\Big\rbrack}{E\Big\lbrack\int^T_0 \lbrack\hat f(t) - f(t)\rbrack^2 dt\Big\rbrack},$$ where $\hat f(t)$ is the maximum likelihood estimate. For the cases that shall be of particular interest--the Ornstein Uhlenbeck process with $\hat f(t)$ a linear unbiased estimate--Mann and Moranda [2] have shown that $\bar e(T) \leqq 1$. In the present paper the asymptotic efficiency of the least square estimates will be computed for a wider class of functions $\phi_i(t)$. It will be shown that except for a special case just slightly broader than the one treated by Mann and Moranda, the asymptotic efficiency is actually less than one. Thus except for this special case, the least square estimates could be improved upon. An alternative estimate $\bar k_i(\alpha)$ is proposed. It will be shown that for $a \geqq \beta$, where $\beta$ is the true correlation parameter in the Ornstein Uhlenbeck process, the estimates $\bar k_i(\alpha)$ are asymptotically more efficient than the least square estimates, and in fact as $\alpha \rightarrow \beta$ from above the efficiency increases (strictly) to one.