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Likelihood Analysis in Low Count-Rate Experiments

This paper discusses the use of likelihood analysis and goodness-of-fit methods in determining 85Kr activity in a low count-rate liquid scintillator detector. The analysis includes parameter estimation, uncertainties estimation, and testing for goodness-of-fit.

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Likelihood Analysis in Low Count-Rate Experiments

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  1. Likelihood analysis and goodness-of-fit in low count-rate experiments Aldo Ianni Gran Sasso Laboratory PHYSTAT 05, Oxford 12th-15th Sept.

  2. Outline • The problem • Determine 85Kr activity in a low count-rate Liquid Scintillator detector • The method • Unbinned Likelihood • Results: • Parameters and errors estimation • Goodness-of-fit estimation A. Ianni, LNGS

  3. The problem: • Determine 85Kr activity searching for a delayed coincidence • A possible background from 232Th to be considered: A. Ianni, LNGS

  4. The detector: prototype for a sub-MeV solar neutrino detector (Borexino) • A 4-ton Liquid Scintillator (C9H12) • 103 m3 high purity shielding water • 100 PMTs • Active muon veto • Location: Gran Sasso underground laboratory A. Ianni, LNGS

  5. Data num. of events selected in 555 days = 48 • Selection cuts: • Prompt events, E<0.3MeV • Delayed event, 0.3MeV<E<0.65MeV • Coincidence time, [0.5,6]ms • Vertex of prompt event, r<0.9m A. Ianni, LNGS

  6. Analysis method • A poor data sample asks for: • unbinned likelihood • a MC test for uncertainties estimation • an hypothesis test for unbinned data A. Ianni, LNGS

  7. Likelihood: A. Ianni, LNGS

  8. Results: • Minimization of –lnL • From a Likelihood ratio test it turns out that the parameter b can be neglected (b=0) • Uncertainties estimation • aMLKr =30+11-10 (68.3%) • aMLTh =85+26-24 (68.3%) 85Kr activity is : 14+5-4 decays/day/ton A. Ianni, LNGS

  9. Profiles of the likelihood vs parameters • Minimize –lnL against one parameter changing the other A. Ianni, LNGS

  10. MC estimation of uncertainties • Use best-fit values to generate MC sample distributions (103 experiments in this case) • Fit MC distributions w/ ML • Determine distribution of parameters • Determine errors and correlations from distribution of parameters • aMCKr =30±10 (68.3%) • aMCTh =84±20(68.3%) A. Ianni, LNGS

  11. Goodness-of-fit [1]: unbinned test • w/ a few data sample use an unbinned test • A possibility: Smirnov-Cramér-Von Mises test A. Ianni, LNGS

  12. Goodness-of-fit [2]: MC experiments • Use best-fit values • Determine: • Create a MC distribution for c2 • Determine p-value • Warning: p-value may depend on binning A. Ianni, LNGS

  13. Conclusions • An example of analysis of a data sample from a low count-rate experiment has been shown • Poor data samples handled with unbinned likelihood • Uncertainties and correlations determined with MC method • g.o.f. determined with MC or unbinned tests A. Ianni, LNGS

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