38,822 research outputs found
Asymptotics of the discrete log-concave maximum likelihood estimator and related applications
The assumption of log-concavity is a flexible and appealing nonparametric
shape constraint in distribution modelling. In this work, we study the
log-concave maximum likelihood estimator (MLE) of a probability mass function
(pmf). We show that the MLE is strongly consistent and derive its pointwise
asymptotic theory under both the well- and misspecified setting. Our asymptotic
results are used to calculate confidence intervals for the true log-concave
pmf. Both the MLE and the associated confidence intervals may be easily
computed using the R package logcondiscr. We illustrate our theoretical results
using recent data from the H1N1 pandemic in Ontario, Canada.Comment: 21 pages, 7 Figure
Testing k-monotonicity of a discrete distribution. Application to the estimation of the number of classes in a population
We develop here several goodness-of-fit tests for testing the k-monotonicity
of a discrete density, based on the empirical distribution of the observations.
Our tests are non-parametric, easy to implement and are proved to be
asymptotically of the desired level and consistent. We propose an estimator of
the degree of k-monotonicity of the distribution based on the non-parametric
goodness-of-fit tests. We apply our work to the estimation of the total number
of classes in a population. A large simulation study allows to assess the
performances of our procedures.Comment: 32 pages, 8 figure
On Convex Least Squares Estimation when the Truth is Linear
We prove that the convex least squares estimator (LSE) attains a
pointwise rate of convergence in any region where the truth is linear. In
addition, the asymptotic distribution can be characterized by a modified
invelope process. Analogous results hold when one uses the derivative of the
convex LSE to perform derivative estimation. These asymptotic results
facilitate a new consistent testing procedure on the linearity against a convex
alternative. Moreover, we show that the convex LSE adapts to the optimal rate
at the boundary points of the region where the truth is linear, up to a log-log
factor. These conclusions are valid in the context of both density estimation
and regression function estimation.Comment: 35 pages, 5 figure
Large Scale Variational Bayesian Inference for Structured Scale Mixture Models
Natural image statistics exhibit hierarchical dependencies across multiple
scales. Representing such prior knowledge in non-factorial latent tree models
can boost performance of image denoising, inpainting, deconvolution or
reconstruction substantially, beyond standard factorial "sparse" methodology.
We derive a large scale approximate Bayesian inference algorithm for linear
models with non-factorial (latent tree-structured) scale mixture priors.
Experimental results on a range of denoising and inpainting problems
demonstrate substantially improved performance compared to MAP estimation or to
inference with factorial priors.Comment: Appears in Proceedings of the 29th International Conference on
Machine Learning (ICML 2012
A probabilistic interpretation of set-membership filtering: application to polynomial systems through polytopic bounding
Set-membership estimation is usually formulated in the context of set-valued
calculus and no probabilistic calculations are necessary. In this paper, we
show that set-membership estimation can be equivalently formulated in the
probabilistic setting by employing sets of probability measures. Inference in
set-membership estimation is thus carried out by computing expectations with
respect to the updated set of probability measures P as in the probabilistic
case. In particular, it is shown that inference can be performed by solving a
particular semi-infinite linear programming problem, which is a special case of
the truncated moment problem in which only the zero-th order moment is known
(i.e., the support). By writing the dual of the above semi-infinite linear
programming problem, it is shown that, if the nonlinearities in the measurement
and process equations are polynomial and if the bounding sets for initial
state, process and measurement noises are described by polynomial inequalities,
then an approximation of this semi-infinite linear programming problem can
efficiently be obtained by using the theory of sum-of-squares polynomial
optimization. We then derive a smart greedy procedure to compute a polytopic
outer-approximation of the true membership-set, by computing the minimum-volume
polytope that outer-bounds the set that includes all the means computed with
respect to P
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