ThêoH: a hybrid, high-confidence statistic that improves on the Allan deviation

Abstract

‘Theo1’ (short terminology for ‘theoretical variance #1’) is derived based on , a variance that separates the Allan variance's averaging time (τ) from its sampling interval (τ s ), also called ‘stride’, as applied to normalized fractional-frequency y(t). The τ-length average of y(t) is denoted , and every permissible squared difference of -values in is averaged in mτ0 segments of y(t) to compute Theo1(m, τ0). Although biased relative to the Allan variance, Theo1(m, τ0) mimics all statistical properties of the Allan variance and works to very long stride. Its estimator has substantially better confidence compared with the best Allan estimator , called ‘Avar’. To make an Allan-compatible statistic, a novel ‘bias-removed’ version of variance, called ‘’ variance, yields ThêoH variance (‘H’ to indicate high confidence and/or hybrid Allan and functions). ThêoH deviation combines the Allan deviation in short term and deviation in long term, reporting very long-term frequency stability 50% beyond that possible using the Allan deviation alone and with excellent confidence.