It is proved in the paper that every possible set of conditions which implies consistency of the maximum likelihood method also implies consistency of Bayes estimates for a large class of prior distributions. The method of the proof is based on a necessary and sufficient condition for the consistency of maximum likelihood estimates which is due to Perlman (1972). Perlman's result is improved as far as certain questions of measurability are concerned.

/pdf/consistency-of-maximum-likelihood-and-bayes-estimates-2u4rhp6n0o.pdf

Consistency of Maximum Likelihood and Bayes Estimates

Distributions with unimodal densities are among the most commonly used in practice. However, for many unimodal distribution families the likelihood functions may be unbounded, thereby leading to inconsistent estimates. The maximum product of spacings (MPS) method, introduced by Cheng and Amin and independently by Ranneby, has been known to give consistent and asymptotically normal estimators in many parametric situations where the maximum likelihood method fails. In this paper, strong consistency theorems for the MPS method are obtained under general conditions which are comparable to the conditions of Bahadur and Wang for the maximum likelihood method. The consistency theorems obtained here apply to both parametric models and some nonparametric models. In particular, in any unimodal distribution family the asymptotic MPS estimator of the underlying unimodal density is shown to be universally L1 consistent without any further conditions (in parametric or nonparametric settings).

Strong Consistency of the Maximum Product of Spacings Estimates with Applications in Nonparametrics and in Estimation of Unimodal Densities

The first part of this paper gives some general consistency theorems for the maximum product of spacings (MPS) method, an estimation method related to maximum likelihood. The second part deals with nonparametric estimation of a concave (convex) distribution and more generally a unimodal distribution, without smoothness assumptions on the densities. The MPS estimator for a distribution function with a monotone density is shown to have a simple explicit representa- tion analogous to the Grenander estimator, and is asymptotically minimax with respect to Kolmogorov-Smirnov type loss. A simple consistent MPS estimator for a unimodal distribution is also discussed.

/pdf/consistency-of-the-maximum-product-of-spacings-method-and-3ma0pw1bpx.pdf

Consistency of the maximum product of spacings method and estimation of a unimodal distribution

SYNOPTIC ABSTRACTIn a previous paper we introduced minimum Kolmogorov distance estimates of distribution parameters and distribution densities for arbitrarily parametrized univariate statistical models. This paper is an extension to the multivariate case. Some results extend straightforwardly, the extension of other ones is quite difficult. It is shown that the Kolmogorov distance parameter estimates are strongly consistent if the parameter space metric is topologically weaker than the metric induced by the Kolmogorov distance of distributions from the statistical model. If the parameter space metric is locally uniformly bounded above by the induced metric then the estimates are consistent of order n−½ Similar results are proved for the Kolmogorov distance estimates of densities from parametrized families. In this case the consistency is considered in the L1-norm.

Minimum Kolmogorov Distance Estimates for Multivariate Parametrized Families

Starting from LeCam and Schwartz, 1960, it is proved that for experiments invariant under the action of a group, finite distinguishability reduces to a separation property in the parameter space.

D. Landers

Papers

Characterization of the topologies used in the theory of maximum likelihood estimation