  Using a matched filter technique, we derive the minimum variance, unbiased
estimator for the equilibrium displacement of a damped harmonic oscillator in
thermal equilibrium when interactions with the thermal bath are the leading
source of noise. We compare the variance in this optimal estimator with the
variance in other, commonly used estimators in the presence of pure thermal
noise and pure white noise. We also compare the variance in these estimators
for a mixture of white and thermal noise. This result has implications for
experimental design and the collection and analysis of data.
