# Implementation of test statistic \tilde{q}_mu

Dear experts,

I have a question about the \tilde{q}_mu in background-only hypothesis.

We know \tilde{q}_mu is the ratio between the conditional maximum likelihood (mu is fixed at 0 for the b-only hypothesis) and the global maximum likelihood. Also we always have the one-side constraint 0<= \hat{mu} <= mu in obtaining the global maximum likelihood. For b-only hypothesis, since the mu is fixed at 0, do I understand correctly that then the \hat{mu} is actually always fixed at 0 and not floated ?
Would you point to me the corresponding implementation of \tilde{q}_mu in RooStat ?

Thanks!
Javier

Hi Javier,

You should not confuse the definition of the test statistics and the way the pseudo-experiments are generated. For the b-only hypothesis, mu is fixed at 0 only when generating pseudo-experiments, but in the evaluation of the test statistics, mu is left to float in the fit.
If you look at the definition given in the paper defining q mu tilde
(see arxiv.org/pdf/1007.1727v3.pdf )
mu’=0 but mu is floating for the b-only case and mu_hat is the best estimated from the fit.

In RooStats we actually use q_mu instead of q_mu_tilde. It is computed in the class RooStat::ProfileLikelikelihoodTestStat when the flag fLimitType == oneSided

The asymptotic approximation of q_mu_tilde is used in the AsymptoticCalculator when the flag fUseQTilde=1. See
root.cern.ch/root/html/src/RooS … tml#Rua8gE

Lorenzo

Dear Lorenzo

Thanks a lot for the nice answers! I see now.

Best Regards,
Javier