Theo1: Characterization of very long-term frequency stability

Theo1 is the first new species of variance that addresses a particularly difficult measurement problem, namely, obtaining reliable estimation of frequency stability for sample periods that are long compared to the length of a data run. Theo1 has statistical properties that are like the Allan variance (Avar), but Theo1 also has two advantages over other estimators of frequency stability: (1) it can evaluate frequency stability at a sample period (tau) of 3/4 the length of a data run, and (2) it presently attains the highest equivalent degrees of freedom (edf) of any estimator of frequency stability including Total-var and overlapping-Avar. Theo1 is unbiased relative to Avar for WHFM noise. Theo1 is biased slightly low with FLFM and RWFM, and we present a formula for a hybrid statistic (TheoH) made up of a combination of Theo1 and Avar in which bias is automatically removed. We explain the sampling function used in Theo1 and show that its frequency response is nearly ideal for extracting power-law noise processes of the types encountered with precision oscillators and clocks. We present results which, for a given data run, show how Theo1 anticipates the levels of frequency stability that are determined by Avar when given a longer data run from the same set of clocks.