Bias in calc_bias should be checked
I am adding this issue to remind myself to look into where the bias factor
bias = Neff / (Neff - 1)
in calc_bias()
comes from and if it's correct.
This seems to have originally come from Andrew Matas
https://stochastic-alog.ligo.org/aLOG/index.php?callRep=340004
and he mentions that “the correction N/(N-1) arises due a standard result when estimating the standard deviation from data”.
I think here he is referring to the fact that if the sample variance is calculated using \bar{x}
in place of \mu
, a factor of 1/(N-1)
must be used in order to obtain an unbiased estimator. I’m not sure that it’s applicable in this situation, since even though the paper is using the sample means P1, P2, for a \chi^2_\nu
distribution the sample mean is an unbiased estimator of \nu
, and since variance is 2\nu
, 2\bar{x}
is an unbiased estimator for the variance.
It won't make much difference to the overall bias but I would nevertheless like to check if it's correct.