Why your model parameter confidences might be too optimistic. Unbiased estimation of the inverse covariance matrix
Abstract
Aims:The maximumlikelihood method is the standard approach to obtain model fits to observational data and the corresponding confidence regions. We investigate possible sources of bias in the loglikelihood function and its subsequent analysis, focusing on estimators of the inverse covariance matrix. Furthermore, we study under which circumstances the estimated covariance matrix is invertible.
Methods: We perform MonteCarlo simulations to investigate the behaviour of estimators for the inverse covariance matrix, depending on the number of independent data sets and the number of variables of the data vectors.
Results: We find that the inverse of the maximumlikelihood estimator of the covariance is biased, the amount of bias depending on the ratio of the number of bins (data vector variables), p, to the number of data sets, n. This bias inevitably leads to an  in extreme cases catastrophic  underestimation of the size of confidence regions. We report on a method to remove this bias for the idealised case of Gaussian noise and statistically independent data vectors. Moreover, we demonstrate that marginalisation over parameters introduces a bias into the marginalised loglikelihood function. Measures of the sizes of confidence regions suffer from the same problem. Furthermore, we give an analytic proof for the fact that the estimated covariance matrix is singular if p>n.
 Publication:

Astronomy and Astrophysics
 Pub Date:
 March 2007
 DOI:
 10.1051/00046361:20066170
 arXiv:
 arXiv:astroph/0608064
 Bibcode:
 2007A&A...464..399H
 Keywords:

 methods: analytical;
 methods: data analysis;
 gravitational lensing;
 Astrophysics;
 High Energy Physics  Experiment;
 High Energy Physics  Phenomenology;
 Mathematical Physics;
 Mathematics  Mathematical Physics
 EPrint:
 6 pages, 3 figures, A&