On the Smoothness Properties of the Best Linear Unbiased Estimate of a Stochastic Process Observed with Noise

Abstract

Suppose $x(t)$ is a vector stochastic process generated by a first order differential equation and $f(t)$ is a linear combination of the elements of $x(t)$. Functionals of $x(t)$ are observed with noise. We obtain the smoothness properties of the best linear unbiased estimate of $f(t)$, and those of its derivatives that exist. In addition we obtain the smoothness properties of their mean squared errors.