Supplementary material to the paper:Grendár, M. and Špitalský, V., Multinomial and Empirical Likelihood under convex constraints: directions of recession, Fenchel duality, perturbations. arXiv:1408.5621 [math.ST], Aug 2014.
The primal problem of multinomial likelihood maximization
restricted to a convex closed subset of the probability
simplex is studied. Contrary to widely held belief, a solution
of this problem may assign a positive mass to an outcome with
zero count. Related flaws in the simplified Lagrange and
Fenchel dual problems, which arise because the recession
directions are ignored, are identified and corrected.
A solution of the primal problem can be obtained by the PP
(perturbed primal) algorithm, that is, as the limit of a
sequence of solutions of perturbed primal problems. The PP
algorithm may be implemented by the simplified Fenchel dual.
The results permit us to specify linear sets and data such
that the empirical likelihood-maximizing distribution exists
and is the same as the multinomial likelihood-maximizing
distribution. The multinomial likelihood ratio reaches, in
general, a different conclusion than the empirical likelihood
Implications for minimum discrimination information, compositional data analysis, Lindsay geometry, bootstrap with auxiliary information, and Lagrange multiplier tests are discussed.
An R code and data to reproduce the numerical examples can be found in here.
M. Grendár, Multinomial
and empirical likelihood under convex constraints, a
talk at Andrej Pazman's seminar, FMPH Comenius University,
Bratislava, September 29, and October 13, 2014.
M. Grendár, Contingency
tables and the PP algorithm, a talk at the Institute of
Mathematics, UPJS, Kosice, June 3, 2015. -- Valuable feedback
from participants of the seminar is gratefully acknowledged.