4,038 research outputs found
Maximal uniform convergence rates in parametric estimation problems
This paper considers parametric estimation problems with independent, identically nonregularly distributed data. It focuses on rate efficiency, in the sense of maximal possible convergence rates of stochastically bounded estimators, as an optimality criterion, largely unexplored in parametric estimation. Under mild conditions, the Hellinger metric, defined on the space of parametric probability measures, is shown to be an essentially universally applicable tool to determine maximal possible convergence rates. These rates are shown to be attainable in general classes of parametric estimation problems
Asymptotic Properties for Methods Combining Minimum Hellinger Distance Estimates and Bayesian Nonparametric Density Estimates
In frequentist inference, minimizing the Hellinger distance between a kernel
density estimate and a parametric family produces estimators that are both
robust to outliers and statistically efficienty when the parametric model is
correct. This paper seeks to extend these results to the use of nonparametric
Bayesian density estimators within disparity methods. We propose two
estimators: one replaces the kernel density estimator with the expected
posterior density from a random histogram prior; the other induces a posterior
over parameters through the posterior for the random histogram. We show that it
is possible to adapt the mathematical machinery of efficient influence
functions from semiparametric models to demonstrate that both our estimators
are efficient in the sense of achieving the Cramer-Rao lower bound. We further
demonstrate a Bernstein-von-Mises result for our second estimator indicating
that it's posterior is asymptotically Gaussian. In addition, the robustness
properties of classical minimum Hellinger distance estimators continue to hold
Bayes and maximum likelihood for -Wasserstein deconvolution of Laplace mixtures
We consider the problem of recovering a distribution function on the real
line from observations additively contaminated with errors following the
standard Laplace distribution. Assuming that the latent distribution is
completely unknown leads to a nonparametric deconvolution problem. We begin by
studying the rates of convergence relative to the -norm and the Hellinger
metric for the direct problem of estimating the sampling density, which is a
mixture of Laplace densities with a possibly unbounded set of locations: the
rate of convergence for the Bayes' density estimator corresponding to a
Dirichlet process prior over the space of all mixing distributions on the real
line matches, up to a logarithmic factor, with the rate
for the maximum likelihood estimator. Then, appealing to an inversion
inequality translating the -norm and the Hellinger distance between
general kernel mixtures, with a kernel density having polynomially decaying
Fourier transform, into any -Wasserstein distance, , between the
corresponding mixing distributions, provided their Laplace transforms are
finite in some neighborhood of zero, we derive the rates of convergence in the
-Wasserstein metric for the Bayes' and maximum likelihood estimators of
the mixing distribution. Merging in the -Wasserstein distance between
Bayes and maximum likelihood follows as a by-product, along with an assessment
on the stochastic order of the discrepancy between the two estimation
procedures
- …