Statistical biases and very long term time stability analysis

The prediction of very long term time stability is a key issue in various fields, such as time keeping, obviously, but also navigation and spatial applications. This is usually performed by extrapolating the measures obtained by estimators such as the Allan deviation, modified Allan deviation, Hadamard deviation, etc. Unlike the variance estimates, the deviation estimates are severely biased and it is of importance to perform this extrapolation from a fit over the variance estimates, and not over the deviation estimates. However, the fit should be performed on the log-log graph of the estimates, which corresponds to a least-squares minimization of the relative difference between the variance estimates and the fitting curve. But also in this case, a bias exists between the average of the log of the estimates and the log of the true value of the estimated variance. This paper presents the theoretical calculation of this “log-log bias” based on the number of equivalent degrees of freedom of the estimates, shows simulations over a large number of realizations, and provides a reliable method of unbiased logarithmic fit. Extrapolating this fit yields a more confident assessment of the very long term time stability.