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TheoH: A New High-Confidence, Hybrid Statistic That Improves on the Allan Diviation



David A. Howe


TheO1 (short terminology for theorectical variance #1) is derived based on psi_y2 (t,T), a variance that separates the Allan variance's sampling or observation interval form its averagin (tau) of sampled y(t), the fractional-frequency measurements in a data run of length T. Every permissible squared difference of sampled and averaged y (t) is then averaged to compute Theo1 resulting in exceptional confidence. Unfortunately, to be an unbiased Allan estimator, Theo1 requires a data run of length 3T. Since it can only operate on T, the data run at hand, or one third of 3T-long data, Theo1 is regarded as a short time , or windowed, measure of variance with modest bias relative to the Allan variance. Although biased, Theo1 mimics all statistical properties of the Allan variance and works to total observation intervals of tau that are 3/4 the length of the data run T, or 50% longer than that possible using the Allan variance while retaining superior confidence. The notion of a completely Allan-compatible statistic with substantially improved properties motivates a bias-removed version of Theo1 called TheoBR by which ultimately constructured TheoH (H to indicate high confidence and or hybrid Allan and TheoBR functions). TheoH combines the Allan deviation in short term confidence and 50% beyond that possible using the Allan deviation alone.


Allan deviation, confidence, frequency stability, frequency standard, long-term, noise, oscillator, spectrum, Theo1


Howe, D. (2006), TheoH: A New High-Confidence, Hybrid Statistic That Improves on the Allan Diviation, Metrologia, [online], (Accessed April 24, 2024)
Created August 1, 2006, Updated February 17, 2017