Very long-term frequency stability: estimation using a special-purpose statistic

We introduce a statistic that can be used for a particularly difficult measurement problem, namely, determining frequency stability for frequency standards and oscillators for averaging times longer than those the traditional Allan deviation can estimate. Theoretical variance #1 ("Theol") has statistical properties that are like "Avar" (Allan variance), with two significant enhancements: (1) it can evaluate frequency stability at longer averaging times than given by the definition of Avar, and (2) it has the highest number of equivalent degrees of freedom (edf) of any estimator of frequency stability. Theol is unbiased relative to Avar for white FM noise, and only moderately biased for the other noises. Given measurements of the time-error function x(t) between two clocks, we have a sequence of time-error samples {x_n : n=1,..., N_x] with a sampling period between adjacent observations given by τ₀. In integer multiples of τ₀, we can obtain an average of fractional-frequency deviates over time τ=mτ0, 1 ≤m≤N_x-1. Theol is given by Theol(m,T0, N_x)=1/0.75(N_x-m)(mT0)² N_x-m/Σ/i=1/2-1/Σ/δ=0 1/(m/-δ)[x_i-x_i-δ+m/2)+(x_i+m^-x_i=δ=m/2)]², for m even, 10 ≤m≤ N_x-1, where frequency stability is evaluated at span or stride T_s=0.75mT₀. This means that the last T_s=0.75(N_x-1)T₀, or an averaging time corresponding to 3/4 of the duration of a data run, or 50 % longer than the longest T- value of Avar.