2,474 research outputs found
Bayesian multivariate mixed-scale density estimation
Although continuous density estimation has received abundant attention in the
Bayesian nonparametrics literature, there is limited theory on multivariate
mixed scale density estimation. In this note, we consider a general framework
to jointly model continuous, count and categorical variables under a
nonparametric prior, which is induced through rounding latent variables having
an unknown density with respect to Lebesgue measure. For the proposed class of
priors, we provide sufficient conditions for large support, strong consistency
and rates of posterior contraction. These conditions allow one to convert
sufficient conditions obtained in the setting of multivariate continuous
density estimation to the mixed scale case. To illustrate the procedure a
rounded multivariate nonparametric mixture of Gaussians is introduced and
applied to a crime and communities dataset
Conditional density estimation in a regression setting
Regression problems are traditionally analyzed via univariate characteristics
like the regression function, scale function and marginal density of regression
errors. These characteristics are useful and informative whenever the
association between the predictor and the response is relatively simple. More
detailed information about the association can be provided by the conditional
density of the response given the predictor. For the first time in the
literature, this article develops the theory of minimax estimation of the
conditional density for regression settings with fixed and random designs of
predictors, bounded and unbounded responses and a vast set of anisotropic
classes of conditional densities. The study of fixed design regression is of
special interest and novelty because the known literature is devoted to the
case of random predictors. For the aforementioned models, the paper suggests a
universal adaptive estimator which (i) matches performance of an oracle that
knows both an underlying model and an estimated conditional density; (ii) is
sharp minimax over a vast class of anisotropic conditional densities; (iii) is
at least rate minimax when the response is independent of the predictor and
thus a bivariate conditional density becomes a univariate density; (iv) is
adaptive to an underlying design (fixed or random) of predictors.Comment: Published in at http://dx.doi.org/10.1214/009053607000000253 the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Empirical Bayes conditional density estimation
The problem of nonparametric estimation of the conditional density of a
response, given a vector of explanatory variables, is classical and of
prominent importance in many prediction problems since the conditional density
provides a more comprehensive description of the association between the
response and the predictor than, for instance, does the regression function.
The problem has applications across different fields like economy, actuarial
sciences and medicine. We investigate empirical Bayes estimation of conditional
densities establishing that an automatic data-driven selection of the prior
hyper-parameters in infinite mixtures of Gaussian kernels, with
predictor-dependent mixing weights, can lead to estimators whose performance is
on par with that of frequentist estimators in being minimax-optimal (up to
logarithmic factors) rate adaptive over classes of locally H\"older smooth
conditional densities and in performing an adaptive dimension reduction if the
response is independent of (some of) the explanatory variables which,
containing no information about the response, are irrelevant to the purpose of
estimating its conditional density
Aggregation and long memory: recent developments
It is well-known that the aggregated time series might have very different
properties from those of the individual series, in particular, long memory. At
the present time, aggregation has become one of the main tools for modelling of
long memory processes. We review recent work on contemporaneous aggregation of
random-coefficient AR(1) and related models, with particular focus on various
long memory properties of the aggregated process
Recent advances in directional statistics
Mainstream statistical methodology is generally applicable to data observed
in Euclidean space. There are, however, numerous contexts of considerable
scientific interest in which the natural supports for the data under
consideration are Riemannian manifolds like the unit circle, torus, sphere and
their extensions. Typically, such data can be represented using one or more
directions, and directional statistics is the branch of statistics that deals
with their analysis. In this paper we provide a review of the many recent
developments in the field since the publication of Mardia and Jupp (1999),
still the most comprehensive text on directional statistics. Many of those
developments have been stimulated by interesting applications in fields as
diverse as astronomy, medicine, genetics, neurology, aeronautics, acoustics,
image analysis, text mining, environmetrics, and machine learning. We begin by
considering developments for the exploratory analysis of directional data
before progressing to distributional models, general approaches to inference,
hypothesis testing, regression, nonparametric curve estimation, methods for
dimension reduction, classification and clustering, and the modelling of time
series, spatial and spatio-temporal data. An overview of currently available
software for analysing directional data is also provided, and potential future
developments discussed.Comment: 61 page
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