34,544 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-
Consistency of cross validation for comparing regression procedures
Theoretical developments on cross validation (CV) have mainly focused on
selecting one among a list of finite-dimensional models (e.g., subset or order
selection in linear regression) or selecting a smoothing parameter (e.g.,
bandwidth for kernel smoothing). However, little is known about consistency of
cross validation when applied to compare between parametric and nonparametric
methods or within nonparametric methods. We show that under some conditions,
with an appropriate choice of data splitting ratio, cross validation is
consistent in the sense of selecting the better procedure with probability
approaching 1. Our results reveal interesting behavior of cross validation.
When comparing two models (procedures) converging at the same nonparametric
rate, in contrast to the parametric case, it turns out that the proportion of
data used for evaluation in CV does not need to be dominating in size.
Furthermore, it can even be of a smaller order than the proportion for
estimation while not affecting the consistency property.Comment: Published in at http://dx.doi.org/10.1214/009053607000000514 the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Wavelet Estimators in Nonparametric Regression: A Comparative Simulation Study
Wavelet analysis has been found to be a powerful tool for the nonparametric estimation of spatially-variable objects. We discuss in detail wavelet methods in nonparametric regression, where the data are modelled as observations of a signal contaminated with additive Gaussian noise, and provide an extensive review of the vast literature of wavelet shrinkage and wavelet thresholding estimators developed to denoise such data. These estimators arise from a wide range of classical and empirical Bayes methods treating either individual or blocks of wavelet coefficients. We compare various estimators in an extensive simulation study on a variety of sample sizes, test functions, signal-to-noise ratios and wavelet filters. Because there is no single criterion that can adequately summarise the behaviour of an estimator, we use various criteria to measure performance in finite sample situations. Insight into the performance of these estimators is obtained from graphical outputs and numerical tables. In order to provide some hints of how these estimators should be used to analyse real data sets, a detailed practical step-by-step illustration of a wavelet denoising analysis on electrical consumption is provided. Matlab codes are provided so that all figures and tables in this paper can be reproduced
A goodness-of-fit test for parametric and semi-parametric models in multiresponse regression
We propose an empirical likelihood test that is able to test the goodness of
fit of a class of parametric and semi-parametric multiresponse regression
models. The class includes as special cases fully parametric models;
semi-parametric models, like the multiindex and the partially linear models;
and models with shape constraints. Another feature of the test is that it
allows both the response variable and the covariate be multivariate, which
means that multiple regression curves can be tested simultaneously. The test
also allows the presence of infinite-dimensional nuisance functions in the
model to be tested. It is shown that the empirical likelihood test statistic is
asymptotically normally distributed under certain mild conditions and permits a
wild bootstrap calibration. Despite the large size of the class of models to be
considered, the empirical likelihood test enjoys good power properties against
departures from a hypothesized model within the class.Comment: Published in at http://dx.doi.org/10.3150/09-BEJ208 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
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Econometrics: A bird's eye view
As a unified discipline, econometrics is still relatively young and has been transforming and expanding very rapidly over the past few decades. Major advances have taken place in the analysis of cross sectional data by means of semi-parametric and non-parametric techniques. Heterogeneity of economic relations across individuals, firms and industries is increasingly acknowledge and attempts have been made to take them into account either by integrating out their effects or by modeling the sources of heterogeneity when suitable panel data exists. The counterfactual considerations that underlie policy analysis and treatment evaluation have been given a more satisfactory foundation. New time series econometric techniques have been developed and employed extensively in the areas of macroeconometrics and finance. Non-linear econometric techniques are used increasingly in the analysis of cross section and time series observations. Applications of Bayesian techniques to econometric problems have been given new impetus largely thanks to advances in computer power and computational techniques. The use of Bayesian techniques have in turn provided the investigators with a unifying framework where the tasks and forecasting, decision making, model evaluation and learning can be considered as parts of the same interactive and iterative process; thus paving the way for establishing the foundation of the "real time econometrics". This paper attempts to provide an overview of some of these developments
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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