Overview

The above figure depicts the use of this methodology in an important problem in cosmology, the estimation of cosmological parameters using observations of the cosmic microwave background radiation (CMB). A complex model relates physical constants of interest to the power spectrum of the variations in temperature of photons that comprise the CMB. The observable CMB is modeled as the realization of a Gaussian process on the sphere. It is customary to work with the spherical harmonic transform; hence, the spectrum gives the power at the different spherical wavenumbers.

In the figure, the green region is a simultaneous 95% minimax regret confidence band for the power spectrum. The data in use are from the recent WMAP experiment, and the dots shown in the plot are the estimated band powers based on the experimental observations. This confidence region cannot be interpreted without its associated parametric model, in this case the six parameter flat power law Lambda-CDM model. The procedure that produced this estimate was designed to maximize the rejection power, to the extent possible, for this particular model. For more background on this model, and results from a Bayesian analysis with the same data, see section three of Spergel, et.al. (2003).

Rejection power (i.e. probability of exclusion from the confidence region) is a function of two spectra: the "true" spectrum and the spectrum being tested for inclusion. It is not possible to form a procedure that simultaneously maximizes this quantity over all possible pairs of spectra. Since it is necessary to sacrifice lower rejection power at some combinations to increase power at other combinations, it is desirable to limit the set of spectra under consideration to those considered physically feasible.

The six parameters are given in the following table. The second and third columns give the minimum and maximum accepted values of the parameters from an initial test of 135,000 parameter combinations (4,116 were accepted). For instance, among the 4,116 accepted spectra the value of the Hubble constant ranged from 53 to 88. The last column gives the values corresponding to the (accepted) example blue spectrum shown in the figure.

Related Papers

Frequently physical scientists seek a confidence set for a parameter whose precise value is unknown, but constrained by theory or previous experiments. The confidence set should exclude parameter values that violate those constraints, but further improvements are possible: We construct minimax expected size and minimax regret confidence procedures. The resulting confidence sets include only values that satisfy the constraints; they have the correct coverage probability; and they minimize a measure of average size. We illustrate these approaches with three examples: estimating the mean of a normal distribution when this mean is known to be bounded, estimating a parameter of a bivariate normal distribution arising in a signal detection problem, and estimating cosmological parameters from MAXIMA-1 observations of the cosmic microwave background radiation. In the first two examples, the new methods are compared with two others: a standard approach adapted to force the estimate to conform to the bounds, and the likelihood-ratio testing approach proposed by Feldman and Cousins (1998). Software that implements the new method efficiently is available online.

The Code

A simple example program which illustrates use of the subroutine for approximating the LFA or LRA in the bounded normal mean problem.

Written and designed by Chad Schafer.

Based heavily on research by Steve Evans, Ben Hansen, and Philip Stark.

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