Class 13.5 (0xD.8): Feldman-Cousins confidence regions

How do we choose confidence intervals (Neyman construction)

(Following section in [PDG-Stat].)

The "Feldman-Cousins" method for generating "unified" confidence intervals

(See section in [PDG-Stat].)

The Feldman-Cousins procedure in detail

The procedure for building up the "map" of \Delta \log L cutoffs is simply to build the p.d.f.s for \Delta \log L on a grid of values of the parameters. For each point on the map, find the value (\Delta \log L)_{\alpha} below which a fraction (1-\alpha) of the distribution lies.

Procedure for building up map of \Delta \log L cutoffs for all theta:
Make an array to store the results
Loop over values of \theta:
build the p.d.f. of \Delta \log L for this parameter (see below)
sum up the p.d.f. to find the (\Delta \log L)_{\alpha}
store (\Delta \log L)_{\alpha} for this parameter in the array

The procedure for building up a \Delta \log L p.d.f. is essentially identical to that for building up the p.d.f.s of the L_\text{max} for a significance test, except for the quantity evaluated. (Contrast the steps below to Class 0x0B example 4.)

Procedure for building up p.d.f. of \Delta \log L :
loop M times:
simulate a dataset using the hypothesis
fit the dataset
calculate \log L_\text{max} at the best fit point
calculate \log L at the true value
"fill" histogram using \Delta \log L = \log L_\text{max}-\log L
Procedure for simulating a dataset:
Loop N times:
generate random variable x according to the model p.d.f. for x
(see class notes on MC simulation, use inverse distribution method)
store x in vector of doubles to be used as dataset
(instead of reading x from a file)

Drawing the confidence region

This is simplicity itself:

given \log L_\text{max} of the best fit point and your map from above,
loop over the points in the "map":
evaluate \log L for your real data at that parameter to find \Delta \log L
is \Delta \log L > (\Delta \log L)_{\alpha} at this point in the map?
if yes, point is excluded -- clear pixel and/or print '.' on screen
if no, point is included -- set pixel and/or print '*' on screen


Build the 90%-CL and 99%-CL confidence regions for the same exponential + background of the assignment from class 11 (aka class 0x0B), with the restriction that the background parameter b must be in the range 0 \leq b < 1 and the mean \mu must be positive.

[KamLAND2008]KamLAND Collaboration, "Precision Measurement of Neutrino Oscillation Parameters with KamLAND", Phys.Rev.Lett.100:221803,2008; arXiv:0801.4589v3 [hep-ex].
[DZero2010]D0 Collaboration, "Evidence for an anomalous like-sign dimuon charge asymmetry", Submitted to Phys. Rev. D, 2010; Fermilab-Pub-10/114-E; arXiv:1005.2757v1 [hep-ex].
[PDG-Stat]"Statistics", G. Cowan, in Review of Particle Physics, C. Amsler et al., PL B667, 1 (2008) and 2009 partial update for the 2010 edition ( ).