170 research outputs found
msBP: An R package to perform Bayesian nonparametric inference using multiscale Bernstein polynomials mixtures
msBP is an R package that implements a new method to perform Bayesian multiscale nonparametric inference introduced by Canale and Dunson (2016). The method, based on mixtures of multiscale beta dictionary densities, overcomes the drawbacks of Pólya trees and inherits many of the advantages of Dirichlet process mixture models. The key idea is that an infinitely-deep binary tree is introduced, with a beta dictionary density assigned to each node of the tree. Using a multiscale stick-breaking characterization, stochastically decreasing weights are assigned to each node. The result is an infinite mixture model. The package msBP implements a series of basic functions to deal with this family of priors such as random densities and numbers generation, creation and manipulation of binary tree objects, and generic functions to plot and print the results. In addition, it implements the Gibbs samplers for posterior computation to perform multiscale density estimation and multiscale testing of group differences described in Canale and Dunson (2016)
Confidence bands for convex median curves using sign-tests
Suppose that one observes pairs , , ..., ,
where are fixed numbers, and are
independent random variables with unknown distributions. The only assumption is
that for some unknown convex function . We
present a confidence band for this regression function using suitable
multiscale sign-tests. While the exact computation of this band requires
steps, good approximations can be obtained in steps. In
addition the confidence band is shown to have desirable asymptotic properties
as the sample size tends to infinity.Comment: Published at http://dx.doi.org/10.1214/074921707000000283 in the IMS
Lecture Notes Monograph Series
(http://www.imstat.org/publications/lecnotes.htm) by the Institute of
Mathematical Statistics (http://www.imstat.org
Multiscale inference about a density
We introduce a multiscale test statistic based on local order statistics and
spacings that provides simultaneous confidence statements for the existence and
location of local increases and decreases of a density or a failure rate. The
procedure provides guaranteed finite-sample significance levels, is easy to
implement and possesses certain asymptotic optimality and adaptivity
properties.Comment: Version 2 is an extended version (Technical report 56, IMSV, Univ.
Bern) which is referred to in version 3. Published in at
http://dx.doi.org/10.1214/07-AOS521 the Annals of Statistics
(http://www.imstat.org/aos/) by the Institute of Mathematical Statistics
(http://www.imstat.org
Multiscale Bernstein polynomials for densities
Our focus is on constructing a multiscale nonparametric prior for densities.
The Bayes density estimation literature is dominated by single scale methods,
with the exception of Polya trees, which favor overly-spiky densities even when
the truth is smooth. We propose a multiscale Bernstein polynomial family of
priors, which produce smooth realizations that do not rely on hard partitioning
of the support. At each level in an infinitely-deep binary tree, we place a
beta dictionary density; within a scale the densities are equivalent to
Bernstein polynomials. Using a stick-breaking characterization, stochastically
decreasing weights are allocated to the finer scale dictionary elements. A
slice sampler is used for posterior computation, and properties are described.
The method characterizes densities with locally-varying smoothness, and can
produce a sequence of coarse to fine density estimates. An extension for
Bayesian testing of group differences is introduced and applied to DNA
methylation array data
Adaptive goodness-of-fit tests based on signed ranks
Within the nonparametric regression model with unknown regression function
and independent, symmetric errors, a new multiscale signed rank statistic
is introduced and a conditional multiple test of the simple hypothesis
against a nonparametric alternative is proposed. This test is distribution-free
and exact for finite samples even in the heteroscedastic case. It adapts in a
certain sense to the unknown smoothness of the regression function under the
alternative, and it is uniformly consistent against alternatives whose sup-norm
tends to zero at the fastest possible rate. The test is shown to be
asymptotically optimal in two senses: It is rate-optimal adaptive against
H\"{o}lder classes. Furthermore, its relative asymptotic efficiency with
respect to an asymptotically minimax optimal test under sup-norm loss is close
to 1 in case of homoscedastic Gaussian errors within a broad range of
H\"{o}lder classes simultaneously.Comment: Published in at http://dx.doi.org/10.1214/009053607000000992 the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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