(Following section 32.3.2.1 in [PDG-Stat].)

Find intervals for each value of the parameter such that

Here is the p.d.f. of the estimator .

and should depend on monotonically. The functions are invertible: implies .

Then (see figure)

(See section 32.3.2.2 in [PDG-Stat].)

- We pick the edges of the region to have a particular difference log-likelihood from the best fit point, .
- The right to use varies as a function of the best fit parameters. So you need a "map" of the correct to use when checking whether each value of should be in or out of the region.
- If the parameter space has a boundary, it's no problem, the "map" adjusts the log-likelihood level to encompass enough space.
- Note that the "difference in log-likelihood" is simply the log of the Neyman-Pearson test statistic hypothesis test. The "Feldman-Cousins" technique is equivalent to doing a two-hypothesis test for each possible value of , rejecting points where the hypothesis "the true value is " would be rejected at the significance level.
- In general, the p.d.f. of the log-likelihood is generated via MC for each setting of each parameter, just like the hypothesis test.

The procedure for building up the "map" of cutoffs is simply to build the p.d.f.s for on a grid of values of the parameters. For each point on the map, find the value below which a fraction of the distribution lies.

Procedure for building up map of cutoffs for all theta:

Make an array to store the results

Loop over values of :

build the p.d.f. of for this parameter (see below)

sum up the p.d.f. to find the

store for this parameter in the array

The procedure for building up a p.d.f. is essentially identical to that for building up the p.d.f.s of the 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 :

loop M times:

simulate a dataset using the hypothesis

fit the dataset

calculate at the best fit point

calculate at the true value

"fill" histogram using

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)

This is simplicity itself:

given of the best fit point and your map from above,

loop over the points in the "map":

evaluate for your real data at that parameter to find

is 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 must be in the range and the mean must be positive.

- Note this is a good example of where the full Feldman-Cousin's treatment is usually necessary: non-Gaussian model, limits on parameters.
- Just follow the Feldman-Cousins procedure.
- Initially, use a grid of 10 divisions from 0 to 1 for , 10 divisions over the range for .
- Print out the map of to make sure it is reasonable. (Should be for the 90% CL.)
- Print out or draw the confidence region.
- After getting this to work, make the grid finer if you like.

[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 ( http://pdg.lbl.gov/2009/reviews/rpp2009-rev-statistics.pdf ). |