Determination of Mathematical Expectations of Two Weighting Statistics for Ensemble Time Scales

Abstract The Bureau International des Poids et Mesures uses the ALGOS algorithm to generate the Echelle Atomique Libre. The original and new ALGOS weighting algorithms employ two different statistics to calculate the weights, resulting in different experimental phenomena. These two statistics are believed to be the keys to analyzing these experimental phenomena, because the statistical characteristics (e.g. the mathematical expectations and variances) of the statistics completely determine the statistical characteristics of the weights, and subsequently the experimental phenomena. The core work conducted in this study is to derive the analytical expressions of the mathematical expectations of these two statistics under ideal conditions and to validate them empirically. This paper provides a theoretical analysis of these expressions and discusses the relationships between them and the mathematical expectations of the Allan variances. These expressions theoretically reveal the reasons behind the experimental phenomena, especially why hydrogen masers are assigned lower weights in the original ALGOS weighting algorithm but higher weights in the new one. According to the derivation process, these analytical expressions are universally applicable and can also be applied to time-keeping laboratories. In the future, this work can be extended to determine the theoretical statistic and weight performances of different types of frequency standards quantitatively for time-keeping laboratories under both ideal and non-ideal conditions, providing a theoretical basis for improving the design of ensemble time scale algorithms. In the appendix, a method of obtaining the experimental variances and probability density functions (PDFs) of these two statistics is additionally proposed. The two-sample Kolmogorov–Smirnov test is presented to compare the experimental and theoretical PDFs of an example.