Kiefer-Wolfowitz Nonparametric MLE for Uniform Scale Mixtures
Umix(x, ...)An object of class density with components:
points of evaluation on the domain of the density
estimated mass at the points x of the mixing density
the estimated mixture density function values at x
Log likelihood value at the proposed solution
exit code from the optimizer
Data: Sample Observations
other parameters to pass to KWDual to control optimization
Jiaying Gu and Roger Koenker
Kiefer-Wolfowitz MLE for the mixture model \(Y \sim U[0,T], \; T \sim G\)
No gridding is required since mass points of the mixing distribution, \(G\),
must occur at the data points. This formalism is equivalent, as noted by
Groeneboom and Jongbloed (2014) to the Grenander estimator of a monotone
density in the sense that the estimated mixture density, i.e. the marginal
density of \(Y\), is the Grenander estimate, see the remark at the end
of their Section 2.2. See also demo(Grenander). Note that this
refers to the decreasing version of the Grenander estimator, for the
increasing version try standing on your head.
Kiefer, J. and J. Wolfowitz Consistency of the Maximum Likelihood Estimator in the Presence of Infinitely Many Incidental Parameters Ann. Math. Statist. Volume 27, Number 4 (1956), 887-906.
Groeneboom, P. and G. Jongbloed, Nonparametric Estimation under Shape Constraints, 2014, Cambridge U. Press.