Bmix: Binomial mixture estimation via Kiefer Wolfowitz MLE

In REBayes: Empirical Bayes Estimation and Inference

Binomial mixture estimation via Kiefer Wolfowitz MLE

Description

Interior point solution of Kiefer-Wolfowitz NPMLE for mixture of binomials

Usage

Bmix(x, k, v = 300, collapse = TRUE, weights = NULL, unique = FALSE, ...)

Arguments

x	Count of "successes" for binomial observations
k	Number of trials for binomial observations
v	Grid Values for the mixing distribution defaults to equal spacing of length v on [eps, 1- eps], if v is scalar.
collapse	Collapse observations into cell counts.
weights	replicate weights for x obervations, should sum to 1
unique	option to check unique of reported solution
...	Other arguments to be passed to KWDual to control optimization

Details

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.

Value

An object of class density with components:

• xgrid midpoints of evaluation of the mixing density

• yfunction values of the mixing density at x

• gestimates of the mixture density at the distinct data values

• logLikLog Likelihood value at the estimate

• dyBayes rule estimates of binomial probabilities for distinct data values

• uniqueFlag indicating whether the solution is unique

• statusexit code from the optimizer

Author(s)

R. Koenker

References

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.

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