Kiefer Wolfowitz Nonparametric MLE for Gaussian Location Mixtures
GLmix(x, v = 300, sigma = 1, hist = FALSE, histm = 300, weights = NULL, ...)An object of class density with components:
points of evaluation on the domain of the density
estimated function values at the points v, the mixing density
the estimated mixture density function values at x
Log likelihood value at the proposed solution
prediction of mean parameters for each observed x value via Bayes Rule
exit code from the optimizer
Data: Sample Observations
Undata: Grid Values defaults equal spacing of with v bins, when v is a scalar
scale parameter of the Gaussian noise, may take vector values of length(x)
If TRUE then aggregate x to histogram bins, when sigma is vector valued this option is inappropriate unless there are only a small number of distinct sigma values.
histogram bin boundaries, equally spacing with histm
bins when scalar.
replicate weights for x obervations, should sum to 1
other parameters to pass to KWDual to control optimization
Roger Koenker
Kiefer Wolfowitz MLE as proposed by Jiang and Zhang for
the Gaussian compound decision problem. The histogram option is intended
for large problems, say n > 1000, where reducing the sample size dimension
is desirable. When sigma is heterogeneous and hist = TRUE the
procedure tries to do separate histogram binning for distinct values of
sigma, however this is only feasible when there are only a small
number of distinct sigma. By default the grid for the binning is
equally spaced on the support of the data. This function does the normal
convolution problem, for gamma mixtures of variances see GVmix, or
for mixtures of both means and variances TLVmix.
The predict method for GLmix objects will compute means, medians or
modes of the posterior according to whether the Loss argument is 2, 1
or 0, or posterior quantiles if Loss is in (0,1).
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.
Jiang, Wenhua and Cun-Hui Zhang General maximum likelihood empirical Bayes estimation of normal means Ann. Statist., Volume 37, Number 4 (2009), 1647-1684.
Koenker, R and I. Mizera, (2013) ``Convex Optimization, Shape Constraints, Compound Decisions, and Empirical Bayes Rules,'' JASA, 109, 674--685.
Koenker, R. and J. Gu, (2017) REBayes: An R Package for Empirical Bayes Mixture Methods, Journal of Statistical Software, 82, 1--26.