On the influence of sampling and observation times on estimation of the bandwidth parameter of a Gauss-Markov process

The statistical problem of estimating the bandwidth parameter of a Gauss-Markov (GM) 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. The second-order Taylor expansion produces better bias and variance results than a first-order one does. The distribution of the estimator is also discussed. According to the gradient change of the variance, a key result is that three sample size regions, which are termed finite, large, and very large, corresponding to substantial, gradual, and very slight decrease in the 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 tradeoff between the variance decrease and sampling rate results in a practical guideline for choosing sampling rate. These results are applied to the prediction problems of a time invariant GM process to show their value.