AMANDA-II 2000-2006

John Kelley, UW-Madison, May 2008

Our observable in both the new physics and conventional hypotheses
is the zenith-angle / energy spectrum of atmospheric muon neutrinos. As an energy
proxy, we use the common energy-correlated observable *Nch*, or number of
channels / OMs hit during an event. To quantify agreement /
disagreement with a given hypothesis, we use a **binned likelihood
analysis** comparing the cosine of the Pandel-reconstructed zenith angle
and the *Nch* distribution with that predicted by a set of new physics or
conventional parameters that we wish to test.

Feldman and Cousins' paper
(physics/9711021)
describes how to build central confidence intervals using a likelihood
ratio as a test statistic, and gives a relevant example involving
conventional neutrino oscillations. The
downside is that they do not discuss the problem of how to incorporate
systematic errors. We address this by using an extension recommended by
Feldman in
a Fermilab
colloquium called the *profile construction method*.

The *profile likelihood* incorporates systematic errors /
nuisance parameters by choosing the "worst-case" set of nuisance
parameters for a given hypothesis. It is already used in the 'MINOS'
error methods of MINUIT; Feldman's method simply applies it to the frequentist
construction.

We refer the interested reader to our writeup of this procedure here (PDF).

A note on binning: in general, finer binning is better for a likelihood analysis, to the limit of an unbinned analysis using a continuous PDF; this increase in sensitivity is shown in the following figure.

In practice, though, binning more finely than makes sense for the detector (for example, exceeding the angular resolution) could potentially introduce artifacts from data/MC disagreement. Since the increase in sensitivity is not dramatic after 10x10, we choose the following for the range and binning for our observables:

Parameter |
Low |
High |
# of bins |

cos(Zenith) |
-1.0 |
0.0 |
10 |

Nch |
20 |
120 |
10 |

With regards to the range of the observables chosen: cos(Zenith) is
chosen to use only the events below the horizon. The lower end
of *Nch* is set by the multiplicity trigger. We cut off the upper
end of the *Nch* distribution after we drop to the level of a few
events at the final cut level -- beyond this, any small high-multiplicity
background contribution could significantly distort the likelihood. Expected
distributions are shown in the simulation section.