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

‘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 m τ 0 segments of y ( t ) to compute Theo1( m , τ 0 ). Although biased relative to the Allan variance, Theo1( m , τ 0 ) 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.