Multilevel maximum likelihood estimation with application to covariance matrices

ABSTRACT

The asymptotic variance of the maximum likelihood estimate is proved to decrease when the maximization is restricted to a subspace that contains the true parameter value. Maximum likelihood estimation allows a systematic fitting of covariance models to the sample, which is important in data assimilation. The hierarchical maximum likelihood approach is applied to the spectral diagonal covariance model with different parameterizations of eigenvalue decay, and to the sparse inverse covariance model with specified parameter values on different sets of nonzero entries. It is shown computationally that using smaller sets of parameters can decrease the sampling noise in high dimension substantially.

Keywords:

• Fisher information
• High dimension
• Hierarchical maximum likelihood
• Nested parameter spaces
• Spectral diagonal covariance model
• Sparse inverse covariance model

MATHEMATICS SUBJECT CLASSIFICATION:

• 62H12
• 62F12

Acknowledgements

This work was partially supported by the the Czech Science Foundation (GACR) under grant 13-34856S and by the U.S. National Science Foundation under grants DMS-1216481 and ICER-1664175.

Notes

1 In this paper, by sample covariance we mean the maximum likelihood estimate of covariance matrix using the norming constant N as opposed to the unbiased estimate with norming constant (N − 1).

Additional information

Funding

Grantová Agentura české Republiky [13-34856S]; National Science Foundation [DMS-1216481]; National Science Foundation [ICER-1664175].