The Cash Statistic and Forward Fitting

We have investigated Forward Fitting of low count-rate profiles using both the Cash Statistic and the usual Chi-Square Statistic.

Cash Statistic Peculiarities

The Cash Statistic (Cash, W., ApJ 228, 939, 1979) is essentially the same as the Chi-Square Statistic for counts/bin greater than about 10, but it is definitely not the same when the counts/bin become of the order of 1. To see this, look at this plot of the Cash Statistic

versus the Chi-Square Statistic for counts/bin equal to 1 (red) , 2 (green) , 4 (blue) , 8 (yellow) , and 16 (black) .
(These were created by using 10,000 Poisson-distributed pseudo-random numbers with means of 1,2,4,8, and 16.)

The first thing to note is that the Chi-Square Statistic is always scattered about 1.0, but the Cash Statistic can be as large as 1.2 for counts/bin=1. The second thing to note is that the slope of the scatter plots seem to change from slopes less than 1 for small counts to unity at large counts.

What happens for even smaller count rates?

The next figure shows scatterplots of the two statistics for counts/bin equal to 0.3 (red), 0.4 (green), 0.5 (blue), 0.7 (yellow), and 0.9 (black).

Perhaps not surprisingly, the departure of the Cash Statistic becomes greater as the count rates become smaller. But the Cash Statistic becomes much smaller than 1.0 for counts/bin < 0.5. This is an unavoidable consequence of small-number probabilities.

But is Cash correct for Forward Fitting?

To see the effects of the two statistics on forward fitting, we have looked at the process of fitting the flux of a source to a count rate profile that is about half zeroes using the two different statistical measures. The 9 count-rate profiles are shown below:

The topmost time profile (for the finest collimator) was fitted to an array of model profiles of a point source with a continuum of fluxes. For the purposes of this test, the location of the source was restricted to the simulated position. The chi-square and Cash statistics were computed for each model, and the results are plotted below as a function of the ratio of the model flux to the observed flux. (The observed flux is obtained from the mean of the observed count profile.)

The solid curve for the Cash Statistic shows a minimum at Flux Ratio = 1, while the dashed curve for the Chi-Square Statistic shows a minimum around Flux Ratio = 1.5.

This is shown again for counts/bin =10, 1, and 0.2. The thick curves are for the Cash Statistic , and the thin curves are for the Chi-Square Statistic. When the count rates are high, as illustrated by the two highest parabolic curves, there isn't much difference between the two statistics. The lowest two curves show that at very low count rates. the Chi-Square Statistic is spectacularly bad

Ed Schmahl
Last modified: Mon Aug 1 10:31:21 EDT 2005