Pull distribution for toyMC with low number of entries

Roofit and RooStats
1

Dear Experts,
here is an example of simple code that makes toy MC studies
(I copied from here and there from the tutorials)

The p.d.f. is a simple Gaussian

As you can see, if you set the number of entries of the Gaussian, let’s say to 100,
the pulls for the sigma are significantly not centered

Instead if you increase the number of entries of the Gaussian, let’s say to 500,
the pulls for the sigma are nicely normally distributed

My question is: why the distribution of the pulls for the parameter sigma are
not always normally distributed regardless the number of entries ?

Is there a somehow a way to make them become normally distributed even
when the toy MC has a low number of entries ?

Thanks,

2

Hi,
I think this is expected. The ML estimate is biased for finite statistics and it is only asymptotic unbiased. If you have a low number of events in your sample, then can happen that you get biased results for some parameters, especially the most-non linear ones.

Best Regards

Lorenzo