JavaScript is disabled for your browser. Some features of this site may not
work without it.

Likelihood and Exponential Families

Geyer, Charles J. (University of Washington, 1990)

Likelihood and Exponential Families

University of Washington

Thesis or Dissertation

A family of probability densities with respect to a positive Borel measure on a finite-dimensional affine space is standard exponential if the log densities are affine functions. The family is convex if the natural parameter set (gradients of the log densities) is convex. In the closure of the family in the topology of pointwise almost everywhere convergence of densities, the maximum likelihood estimate (MLE) exists whenever the supremum of the log likelihood is finite. It is not defective if the family is convex. The MLE is a density in the original family conditioned on some affine subspace (the support of the MLE) which is determined by the "Phase I" algorithm, a sequence of linear programming feasibility problems. Standard methods determine the MLE in the family conditioned on the support ("Phase II").
An extended-real-valued function on an affine space is generalized affine if it is
both convex and concave. The space of all generalized affine functions is a compact
Hausdorff space, sequentially compact if the carrier is finite-dimensional. A family
of probability densities is a standard generalized exponential family if the log densities are generalized affine. The closure of an exponential family is equivalent to a generalized exponential family.
When the likelihood of an exponential family cannot calculated exactly, it can sometimes be calculated by Monte Carlo using the Metropolis algorithm or the Gibbs sampler. The Monte Carlo log likelihood (the log likelihood in the exponential family
generated by the Monte Carlo empirical distribution) then hypoconverges strongly
(almost surely over sample paths) to the true log likelihood. For a closed convex
family the Monte Carlo approximants to the MLE and all level sets of the likelihood
converge strongly to the truth. For nonconvex families, the outer set limits converge.
These methods are demonstrated by an autologistic model for estimation of relatedness from DNA fingerprint data and by isotonic, convex logistic regression for
the maternal-age-specific incidence of Down’s syndrome, both constrained MLE problems. Hypothesis tests and confidence intervals are constructed for these models using the iterated parametric bootstrap.

Thesis

37918

This is not a copy of the original, which is
in the University of Washington library because the original PostScript was lost and the LaTeX fonts used in 1990 are no longer available so the LaTeX source no longer paginates identically when processed with a modern LaTeX. However the text, tables, and figures are all exactly the same as in the original. No corrections have been made. Therefore, page numbers in this version should not be cited. Use chapter, section, figure, table, and equation numbers instead.

Geyer, Charles J..
(1990).
Likelihood and Exponential Families.
University of Washington.
Retrieved from the University of Minnesota Digital Conservancy,
http://hdl.handle.net/11299/56330.

Content distributed via the University of Minnesota's Digital Conservancy may be subject to additional license and use restrictions applied by the depositor.