267 research outputs found
On nonparametric estimation of a mixing density via the predictive recursion algorithm
Nonparametric estimation of a mixing density based on observations from the
corresponding mixture is a challenging statistical problem. This paper surveys
the literature on a fast, recursive estimator based on the predictive recursion
algorithm. After introducing the algorithm and giving a few examples, I
summarize the available asymptotic convergence theory, describe an important
semiparametric extension, and highlight two interesting applications. I
conclude with a discussion of several recent developments in this area and some
open problems.Comment: 22 pages, 5 figures. Comments welcome at
https://www.researchers.one/article/2018-12-
Semiparametric inference in mixture models with predictive recursion marginal likelihood
Predictive recursion is an accurate and computationally efficient algorithm
for nonparametric estimation of mixing densities in mixture models. In
semiparametric mixture models, however, the algorithm fails to account for any
uncertainty in the additional unknown structural parameter. As an alternative
to existing profile likelihood methods, we treat predictive recursion as a
filter approximation to fitting a fully Bayes model, whereby an approximate
marginal likelihood of the structural parameter emerges and can be used for
inference. We call this the predictive recursion marginal likelihood.
Convergence properties of predictive recursion under model mis-specification
also lead to an attractive construction of this new procedure. We show
pointwise convergence of a normalized version of this marginal likelihood
function. Simulations compare the performance of this new marginal likelihood
approach that of existing profile likelihood methods as well as Dirichlet
process mixtures in density estimation. Mixed-effects models and an empirical
Bayes multiple testing application in time series analysis are also considered
Appropriate use of parametric and nonparametric methods in estimating regression models with various shapes of errors
In this paper, a practical estimation method for a regression model is
proposed using semiparametric efficient score functions applicable to data with
various shapes of errors. First, I derive semiparametric efficient score
vectors for a homoscedastic regression model without any assumptions of errors.
Next, the semiparametric efficient score function can be modified assuming a
certain parametric distribution of errors according to the shape of the error
distribution or by estimating the error distribution non-parametrically.
Nonparametric methods for errors can be used to estimate the parameters of
interest or to find an appropriate parametric error distribution. In this
regard, the proposed estimation methods utilize both parametric and
nonparametric methods for errors appropriately. Through numerical studies, the
performance of the proposed estimation methods is demonstrated.Comment: 2 figure
A Survey of Recent Theoretical Results for Time Series Models with GARCH Errors,
This paper provides a review of some recent theoretical results for time series models with GARCH errors, and is directed towards practitioners. Starting with the simple ARCH model and proceeding to the GARCH model, some results for stationary and nonstationary ARMA-GARCH are summarized. Various new ARCH-type models, including double threshold ARCH and GARCH, ARFIMA-GARCH, CHARMA and vector ARMA-GARCH, are also reviewed.
A PRticle filter algorithm for nonparametric estimation of multivariate mixing distributions
Predictive recursion (PR) is a fast, recursive algorithm that gives a smooth
estimate of the mixing distribution under the general mixture model. However,
the PR algorithm requires evaluation of a normalizing constant at each
iteration. When the support of the mixing distribution is of relatively low
dimension, this is not a problem since quadrature methods can be used and are
very efficient. But when the support is of higher dimension, quadrature methods
are inefficient and there is no obvious Monte Carlo-based alternative. In this
paper, we propose a new strategy, which we refer to as PRticle filter, wherein
we augment the basic PR algorithm with a filtering mechanism that adaptively
reweights an initial set of particles along the updating sequence which are
used to obtain Monte Carlo approximations of the normalizing constants.
Convergence properties of the PRticle filter approximation are established and
its empirical accuracy is demonstrated with simulation studies and a marked
spatial point process data analysis.Comment: 21 pages, 5 figure
Estimating a mixing distribution on the sphere using predictive recursion
Mixture models are commonly used when data show signs of heterogeneity and,
often, it is important to estimate the distribution of the latent variable
responsible for that heterogeneity. This is a common problem for data taking
values in a Euclidean space, but the work on mixing distribution estimation
based on directional data taking values on the unit sphere is limited. In this
paper, we propose using the predictive recursion (PR) algorithm to solve for a
mixture on a sphere. One key feature of PR is its computational efficiency.
Moreover, compared to likelihood-based methods that only support finite mixing
distribution estimates, PR is able to estimate a smooth mixing density. PR's
asymptotic consistency in spherical mixture models is established, and
simulation results showcase its benefits compared to existing likelihood-based
methods. We also show two real-data examples to illustrate how PR can be used
for goodness-of-fit testing and clustering.Comment: 26 pages, 8 figures, 2 table
Revisiting consistency of a recursive estimator of mixing distributions
Estimation of the mixing distribution under a general mixture model is a very
difficult problem, especially when the mixing distribution is assumed to have a
density. Predictive recursion (PR) is a fast, recursive algorithm for
nonparametric estimation of a mixing distribution/density in general mixture
models. However, the existing PR consistency results make rather strong
assumptions, some of which fail for a class of mixture models relevant for
monotone density estimation, namely, scale mixtures of uniform kernels. In this
paper, we develop new consistency results for PR under weaker conditions. Armed
with this new theory, we prove that PR is consistent for the scale mixture of
uniforms problem, and we show that the corresponding PR mixture density
estimator has very good practical performance compared to several existing
methods for monotone density estimation.Comment: 27 pages, 3 figure
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