On the Estimation of Regression Coefficients of a Continuous Parameter Time Series with a Stationary Residual
Chiang Tse–Pei · Theory of Probability and Its Applications · 1959
Let $y(t) = x(t) + m(t)$ be a complex probability process with mean value \[ {\bf E}y(t) = m(t) = \sum\limits_{ u = 1}^S {\gamma _ u \varphi _ u (t)} \](where $\varphi _ u (t)$, $ u = 1,2, \cdots s$, are given functions of t, and $\gamma _ u $, $ u = 1,2, \cdots s$ are unkownconstants) and with stationary residual $x(t)$:\[ E(x(u)\overline {x(v)} ) = r(u - v).\]It is required to estimate the vector $\gamma = \{ \gamma _1 ,\gamma _2 , \cdots ,\gamma _s \} $ on the basis of one realization of the process $y(t)$ on the finite segment $ - T \leqq t \leqq T$. In the case being considered the estimate of the vector $\gamma $ by the method of least squares obtained by minimizing the quadratic form\[ \int_{ - T}^T {\left| {y(t) - \sum\limits_{ u = 1}^S {c_ u \varphi _ u (t)} } \right|^2 dt} \] is equal to\[ {}_L C_T = \Phi ^{ - 1} \left( {\begin{array}{*{20}c} {\int_{ - T}^T {y(t)} \overline {\varphi _1 (t)} dt} \\ \vdots \\ {\int_{ - T}^T {y(t)} \overline {\varphi _s (t)} dt} \\ \end{array} } \right), \]where\[ \Phi = \left[ {\Phi _{\mu u } } \right]_{1 \leqq \mu , u \leqq s} ,\quad \Phi _{\mu u } = \int_{ - T}^T {\overline {\varphi _\mu (t)} \varphi _ u (t)dt} .\]This estimate is unbiased. In Section 2 of this article it is shown that in our case there exists a best linear unbiased estimate ${}_0 C_T $ (i.e. a linear unbiased estimate with a least dispersion matrix). In Section 3 the asymptotic behavior of dispersion matrices ${\bf E}({}_L C_T - \gamma )\overline {({}_L C_T - \gamma )'} $ and ${\bf E}({}_0 C_T - \gamma )\overline {({}_0 C_T - \gamma )'} $ (the prime signifies a Hermitean-conjugate matrix) is considered. If (i) the stationary process $x(t)$ is regular, (ii) the functions $\varphi _ u (t)$ are trigonometric, (iii) the spectral density of the process $x(t)$ is continuous in certain separate points, then the main terms of the matrices ${\bf E}({}_L C_T - \gamma )\overline {({}_L C_T - \gamma )'} $ and ${\bf E}({}_0 C_T - \gamma )\overline {({}_0 C_T - \gamma )'} $ as $T \to \infty $ coincide with each other. In other words, under the above-mentioned conditions, the asymptotic efficiency of the estimate ${}_L C_T $ is proved in the class of linear unbiased estimates.