Umix: NPMLE for Uniform Scale Mixtures

In REBayes: Empirical Bayes Estimation and Inference

NPMLE for Uniform Scale Mixtures

Description

Kiefer-Wolfowitz Nonparametric MLE for Uniform Scale Mixtures

Usage

Umix(x, ...)

Arguments

x	Data: Sample Observations
...	other parameters to pass to KWDual to control optimization

Details

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.

Value

An object of class density with components:

x	points of evaluation on the domain of the density
y	estimated mass at the points x of the mixing density
g	the estimated mixture density function values at x
logLik	Log likelihood value at the proposed solution
status	exit code from the optimizer

Author(s)

Jiaying Gu and Roger Koenker

References

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.

REBayes documentation built on Aug. 19, 2023, 5:10 p.m.