class TBinomialEfficiencyFitter: public TObject

 TBinomialEfficiencyFitter

 Binomial fitter for the division of two histograms.
 Use when you need to calculate a selection's efficiency from two histograms,
 one containing all entries, and one containing the subset of these entries
 that pass the selection, and when you have a parametrization available for
 the efficiency as a function of the variable(s) under consideration.

 A very common problem when estimating efficiencies is that of error estimation:
 when no other information is available than the total number of events N and
 the selected number n, the best estimate for the selection efficiency p is n/N.
 Standard binomial statistics dictates that the uncertainty (this presupposes
 sufficiently high statistics that an approximation by a normal distribution is
 reasonable) on p, given N, is


 However, when p is estimated as n/N, fluctuations from the true p to its
 estimate become important, especially for low numbers of events, and giving
 rise to biased results.

 When fitting a parametrized efficiency, these problems can largely be overcome,
 as a hypothesized true efficiency is available by construction. Even so, simply
 using the corresponding uncertainty still presupposes that Gaussian errors
 yields a reasonable approximation. When using, instead of binned efficiency
 histograms, the original numerator and denominator histograms, a binned maximum
 likelihood can be constructed as the product of bin-by-bin binomial probabilities
 to select n out of N events. Assuming that a correct parametrization of the
 efficiency is provided, this construction in general yields less biased results
 (and is much less sensitive to binning details).

 A generic use of this method is given below (note that the method works for 2D
 and 3D histograms as well):

 {
   TH1* denominator;              // denominator histogram
   TH1* numerator;                // corresponding numerator histogram
   TF1* eff;                      // efficiency parametrization
   ....                           // set step sizes and initial parameter
   ....                           //   values for the fit function
   ....                           // possibly also set ranges, see TF1::SetRange()
   TBinomialEfficiencyFitter* f = new TBinomialEfficiencyFitter(
                                      numerator, denominator);
   Int_t status = f->Fit(eff, "I");
   if (status == 0) {
      // if the fit was successful, display bin-by-bin efficiencies
      // as well as the result of the fit
      numerator->Sumw2();
      TH1* hEff = dynamic_cast<TH1*>(numerator->Clone("heff"));
      hEff->Divide(hEff, denominator, 1.0, 1.0, "B");
      hEff->Draw("E");
      eff->Draw("same");
   }
 }

 Note that this method cannot be expected to yield reliable results when using
 weighted histograms (because the likelihood computation will be incorrect).