Interior point solution of Kiefer-Wolfowitz NPMLE for mixture of binomials
Bmix(x, k, v = 300, collapse = TRUE, weights = NULL, unique = FALSE, ...)An object of class density with components:
grid midpoints of evaluation of the mixing density
function values of the mixing density at x
estimates of the mixture density at the distinct data values
Log Likelihood value at the estimate
Bayes rule estimates of binomial probabilities for distinct data values
Flag indicating whether the solution is unique
exit code from the optimizer
Count of "successes" for binomial observations
Number of trials for binomial observations
Grid Values for the mixing distribution defaults to equal spacing of length v on [eps, 1- eps], if v is scalar.
Collapse observations into cell counts.
replicate weights for x obervations, should sum to 1
option to check unique of reported solution
Other arguments to be passed to KWDual to control optimization
R. Koenker
The predict method for Bmix 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).
When the number of trials is small the NPMLE may be non-unique. This happens
when there exists a vector \(v\) in the unit simplex of \(R^m\)
such that \(Av = f\) where \(f = (n_0/n , ... , n_k/n)\) the observed frequencies,
and A is the k by m matrix with typical element $$C(k,x) p_j^x (1-p_j)^{k - x}.$$
If there exists such a solution, it follows that the maximal likelihood value is attained by any Ghat
such that $$p_j = \int C(k,j) p^j (1-p)^{k-j} dGhat (p) = n_j/n,$$ for j = 0, ... , k.
There will be many such solutions, but by the Caratheodory theorem any one of them can be expressed
as a linear combination of no more than k extreme points of the constraint set.
In contrast, when there are no solutions
inside the simplex satisfying the equation, then the NPMLE is the unique projection onto the boundary
of that set. To facilitate checking this condition if the check parameter is TRUE, the
linear program is feasible and the unique component is returned as TRUE if
the program is infeasible, and FALSE is returned otherwise. This check is restricted to
settings in which k is fixed, and collapse is TRUE. See Robbins (1956, p 161) for
some further discussion of the binomial mixture model and a very clever alternative approach to
prediction.
Kiefer, J. and J. Wolfowitz Consistency of the Maximum Likelihood Estimator in the Presence of Infinitely Many Incidental Parameters Ann. Math. Statist. 27, (1956), 887-906.
Koenker, R and I. Mizera, (2013) ``Convex Optimization, Shape Constraints, Compound Decisions, and Empirical Bayes Rules,'' JASA, 109, 674--685.
Robbins, H. (1956) An Empirical Bayes Approach to Statistics, 3rd Berkeley Symposium.
Koenker, R. and J. Gu, (2017) REBayes: An R Package for Empirical Bayes Mixture Methods, Journal of Statistical Software, 82, 1--26.