The statistical uncertainty associated with the weighted mean frequency in optical frequency comb comparison

The weighted mean is widely used in combining data sets of experimental measurements with a weight proportional to the value of the data number divided by the sample variance in a conventional method. However, this standard procedure is not appropriate for obtaining the weighted mean frequency of a phase-stabilized signal with white phase noise, since the data are autocorrelated. The autocorrelation is obtained in the case of white phase noise and a new weighting method is proposed. Using this, the uncertainty associated with the weighted mean frequency of a phase-stabilized signal with white phase noise is given. The effect of counter dead-time is also discussed. The result is significantly different from the conventional method, so that the weight is proportional to the square of data number divided by the sample variance, and as a result the uncertainty of the estimated mean is proportional to the sample standard deviation divided by the data number. In an optical frequency comb comparison, frequency combs are phase-locked to a common frequency reference and the frequency difference between optical frequency combs shows white phase noise properties. Thus, it is expected that this new weighting method can be utilized in frequency comparisons of optical frequency combs, settling existing controversy, and providing a way of achieving lower uncertainty.