A bin-free Extended Maximum Likelihood Fit + Feldman-Cousins error analysis Peter Litchfield  A bin free Extended Maximum Likelihood method of fitting oscillation parameters is described  A Feldman-Cousins style error analysis has been developed  Systematic errors are incorporated into the MC experiments comprising the F-C analysis giving error contours with statistical and systematic components

Extended Maximum Likelihood  Described by Roger Barlow; NIM A297,496  Maximum Likelihood with a normalisation condition  The standard maximum likelihood method maximises the likelihood function  where p is the probability density and is normalised to 1, M is the number of events, x is a measured quantity and the a i are parameters to be determined.  The fit thus only fits the shape and says nothing about the number of events

Extended Maximum Likelihood  In Extended Maximum Likelihood p is replaced by the un- normalised quantity P where  The predicted number of events, N, is a function of the fitted parameters.  It can then be shown that  It can also be shown that lnL is maximised for N=M

Extended Maximum Likelihood  In our case the function P is just the extrapolated predicted neutrino measured energy distribution for the given set of parameters.  Strictly P should be a continuous function but with a high statistics MC we can approximate it by the finely binned MC.  So we just sum over the number of predicted MC events N i (E m ) in the bin corresponding to the measured energy E m of each data event  In the plots that follow I use MeV MC bins between 0 and 50 GeV. The bins can be as narrow as the MC warrants.

Comparison Binned v Unbinned Likelihood  Binned likelihood has the standard 500 MeV bins below 10GeV  Unbinned gains at high  m 2 because of the improved resolution on the oscillation dip  Little gain at low  m 2 where there is no data

Feldman-Cousins error analysis  Following the F-C prescription, for each  m 2 -sin 2 2  bin I generate fake experiments with numbers of events Poisson fluctuated about the number predicted by my extrapolation.  For each experiment I select events at random from the full Far detector MC sample, up to the fluctuated number and according to the predicted energy spectrum.  The lnL distribution is calculated on the  m 2 -sin 2 2  grid for each experiment and the  2 difference between the best fit point and the generated point determined.  If say 1000 experiments are generated and fitted, the  2 are sorted and the 900 th  2 from the minimum gives the 90%  2 (  2 90 ) for that grid point.  If the data  2 for that grid point is less than  2 90, that grid point is within the 90% confidence allowed region

Data  2 Surface  2 90 surface F-C results

FC contours

Systematics Analysis  For each fake MC experiment the parameters of the experiment are varied according to a set of systematic errors.  The errors for a given experiment are taken randomly from a uniform distribution between + and – the estimated systematic error.  Notice that CPU time forbids repeating the extrapolation for the > 2.5 billion FC experiments required, so all errors are simulated by varying the selected far MC events.  Systematic parameters can be varied individually or all together. Correlations between systematic parameters are accounted for.  All identified systematics can be included without significant time or complication penalty

Systematics Included 1)Normalisation  The generated event distribution is scaled by a factor randomly selected between 1  )Relative hadronic energy scale  The hadronic energy of the selected far detector events is scaled by a number randomly chosen between 1  for each experiment 3)Muon energy scale  The muon energy is scaled randomly between 1  )Absolute energy scale  I cannot change the energy in the predicted distribution but a change in the absolute scale is equivalent to shifting the predicted oscillation dip in the far detector. The far detector truth energy is shifted by a random amount between  100MeV in calculating the oscillation probability

Systematics included 5)PID cut  The far MC events available at this point in the program have been selected by the PID. At the moment I can only make a one sided cut in the selection. Events with PID with a value randomly selected between 0 and 0.05 above the standard cut are removed from the fake experiments 6)NC background  In the selection of MC events a fraction randomly selected between  50% of true extra NC events are selected

Systematics included 7)Extrapolation error  To try to allow for the extrapolation error I have taken the ratio of the SKZP extrapolation to my extrapolation and scaled the predicted distribution by a random fraction between 0 and 1 of that difference for each experiment

Contours

1D errors -m2-m2 +m2+m2 -sin 2 2  No systematics NC  50%  Energy  0.036% Relative hadronic energy  Absolute hadronic energy  0.1Gev Pid Normalisation  Extrapolation  All systematics No systematics All systematics PRL

Unconstrained contours

To do  More fake experiments to smooth the F-C contours  This analysis just fits the E distribution. The bin-free analysis will be more advantageous for the E  v E shw analysis where the binning of the data is a problem  Extend to the nc and  - data when available