Definitions of "total" estimators of common time-domain variances

This paper summarizes the definitions and properties of frequency and time variance estimators based on the "total" approach. These definitions and properties are not available in one document, yet this statistical approach produces variance estimates with the best confidence, as indicated by equivalent degrees of freedom (edf), for the time-domain sample variances most frequently used to analyze the stability of oscillators. There are differences in the total approach as applied to each variance because each variance reports noise levels for a specific range of power-law noise processes. Development of these total definitions has been motivated by the goal of realizing the highest edf and the lowest bias across this range while keeping the algorithms relatively simple. The relevant variances are: (i) Allan variance, σ²_y(τ) (ii) Modified Allan variance, Mod-σ²_y(τ) (iii) Time variance, σ²_x(τ) (iv) Hadamard variance, _Hσ²_y(τ) The Hadamard variance is used in managing clocks of the Global Positioning System (GPS). Specifically we will give precise definitions for sample variances that use the "total" approach and whose corresponding names are: (i) Total variance (ii) Modified Total variance (iii) Time Total variance (iv) Hadamard Total variance