The statistical problem of estimating the bandwidth parameter of a Gauss-Markov process from a realization of fixed and finite duration T at selectable sampling interval Δ is addressed in this paper. As the observation time, T, is fixed and finite, the variance of estimated autocorrelation and continuous-time parameter does not
vanish as Δ approaches 0. This necessitates a second order Taylor expansion in deriving the parameter estimator bias and variance, which produces significantly more accurate bias and variance results than a first order one does. Using likelihood ratio methods, we also show that even the large sample distributions of β estimator are better modeled by a gamma than by a normal form. According to the gradient change of the variance, a key result is that three sample regions, which are termed finite, large and very large, corresponding to substantial, gradual, and very slight decrease in variance of the parameter estimator respectively, are quantified. In terms of analysis BW, the three regions are (-23,-35), (-35,-55) and (-55,-∞) dB. The characterization of the trade off between the variance decrease and sampling rate results in a practical guideline for choosing sampling rate. To demonstrate the practical value of our results, we apply them to the noise prediction problems of a time invariant GM processes. Using moment generating functions, we are able to arrive at explicit and accurate relations between, the set of variables (Δ, T, β) and m-step prediction performance. In particular, we show that prediction performance is highly robust with respect to estimation accuracy of β. This is significant, in that it allows one to use a surprisingly small observation time, T , and still achieve nearly optimal performance associated with perfect knowledge of β.