A more general analytical discussion of the causes and magnitude of the bias for an arbitrary model is given in Humphrey, Liu & Buote (2009, ApJ 693, 822). Basically, the order of magnitude of the bias divided by the statistical error should be of order the number of bins divided by the square root of the number of photons. Fits using Pearson’s approximation to chi-square yield a bias of approximately -0.5 times the bias seen with Neyman’s approximation, and the bias on fits using the C-statistic is much smaller than the statistical error (roughly what Aneta’s plot showed).

In summary— don’t use approximations to Chi-square, but use C-statistic if you’re fitting Poisson distributed data!

]]>on the ML estimators. ]]>

2. Sad and glad that it’s already done. The plot and Loredo’s work should go more public. ]]>

Tom L: if you’re reading this, you really should’ve published those notes

The short version: the optimal estimator for the photon index is the maximum

likelihood estimator. If the data are binned, and the total number of counts over

all bins is not fixed (i.e., is a random variable), then the likelihood function is

\Prod_{i=1}^k \frac{ (np_i)^{y_i} }{ y_i ! } e^{-np_i}

where y_i are the number of counts in bin i, n = \sum_i y_i, and p_i is the probability

that a count would be recorded in bin i (and this depends on the distribution

parameter, in this case the photon index).

You can derive the \chi^2 function from this using Stirling’s approximation

(np_i \gtrsim 5 in each bin) and then a Taylor series expansion. Depending on

how you do that expansion, you can derive either \chi^2 with model variance

or \chi^2 with data variance. It’s the combination of Stirling’s approximation

with how you cut off the Taylor expansion that creates the bias term.

Tom Loredo wrote this all up years and years ago, so I take no credit for the

explanation.