7,805 research outputs found
LogConcDEAD: An R Package for Maximum Likelihood Estimation of a Multivariate Log-Concave Density
In this article we introduce the R package LogConcDEAD (Log-concave density estimation in arbitrary dimensions). Its main function is to compute the nonparametric maximum likelihood estimator of a log-concave density. Functions for plotting, sampling from the density estimate and evaluating the density estimate are provided. All of the functions available in the package are illustrated using simple, reproducible examples with simulated data.
Nonparametric estimation of multivariate convex-transformed densities
We study estimation of multivariate densities of the form
for and for a fixed monotone function and an unknown
convex function . The canonical example is for ; in this case, the resulting class of densities [\mathcal
{P}(e^{-y})={p=\exp(-g):g is convex}] is well known as the class of log-concave
densities. Other functions allow for classes of densities with heavier
tails than the log-concave class. We first investigate when the maximum
likelihood estimator exists for the class for
various choices of monotone transformations , including decreasing and
increasing functions . The resulting models for increasing transformations
extend the classes of log-convex densities studied previously in the
econometrics literature, corresponding to . We then establish
consistency of the maximum likelihood estimator for fairly general functions
, including the log-concave class and many others. In
a final section, we provide asymptotic minimax lower bounds for the estimation
of and its vector of derivatives at a fixed point under natural
smoothness hypotheses on and . The proofs rely heavily on results from
convex analysis.Comment: Published in at http://dx.doi.org/10.1214/10-AOS840 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Bayesian nonparametric multivariate convex regression
In many applications, such as economics, operations research and
reinforcement learning, one often needs to estimate a multivariate regression
function f subject to a convexity constraint. For example, in sequential
decision processes the value of a state under optimal subsequent decisions may
be known to be convex or concave. We propose a new Bayesian nonparametric
multivariate approach based on characterizing the unknown regression function
as the max of a random collection of unknown hyperplanes. This specification
induces a prior with large support in a Kullback-Leibler sense on the space of
convex functions, while also leading to strong posterior consistency. Although
we assume that f is defined over R^p, we show that this model has a convergence
rate of log(n)^{-1} n^{-1/(d+2)} under the empirical L2 norm when f actually
maps a d dimensional linear subspace to R. We design an efficient reversible
jump MCMC algorithm for posterior computation and demonstrate the methods
through application to value function approximation
A Bayesian nonparametric approach to log-concave density estimation
The estimation of a log-concave density on is a canonical
problem in the area of shape-constrained nonparametric inference. We present a
Bayesian nonparametric approach to this problem based on an exponentiated
Dirichlet process mixture prior and show that the posterior distribution
converges to the log-concave truth at the (near-) minimax rate in Hellinger
distance. Our proof proceeds by establishing a general contraction result based
on the log-concave maximum likelihood estimator that prevents the need for
further metric entropy calculations. We also present two computationally more
feasible approximations and a more practical empirical Bayes approach, which
are illustrated numerically via simulations.Comment: 39 pages, 17 figures. Simulation studies were significantly expanded
and one more theorem has been adde
The MM Alternative to EM
The EM algorithm is a special case of a more general algorithm called the MM
algorithm. Specific MM algorithms often have nothing to do with missing data.
The first M step of an MM algorithm creates a surrogate function that is
optimized in the second M step. In minimization, MM stands for
majorize--minimize; in maximization, it stands for minorize--maximize. This
two-step process always drives the objective function in the right direction.
Construction of MM algorithms relies on recognizing and manipulating
inequalities rather than calculating conditional expectations. This survey
walks the reader through the construction of several specific MM algorithms.
The potential of the MM algorithm in solving high-dimensional optimization and
estimation problems is its most attractive feature. Our applications to random
graph models, discriminant analysis and image restoration showcase this
ability.Comment: Published in at http://dx.doi.org/10.1214/08-STS264 the Statistical
Science (http://www.imstat.org/sts/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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