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Confidence Corridors for Multivariate Generalized Quantile Regression
We focus on the construction of confidence corridors for multivariate
nonparametric generalized quantile regression functions. This construction is
based on asymptotic results for the maximal deviation between a suitable
nonparametric estimator and the true function of interest which follow after a
series of approximation steps including a Bahadur representation, a new strong
approximation theorem and exponential tail inequalities for Gaussian random
fields. As a byproduct we also obtain confidence corridors for the regression
function in the classical mean regression. In order to deal with the problem of
slowly decreasing error in coverage probability of the asymptotic confidence
corridors, which results in meager coverage for small sample sizes, a simple
bootstrap procedure is designed based on the leading term of the Bahadur
representation. The finite sample properties of both procedures are
investigated by means of a simulation study and it is demonstrated that the
bootstrap procedure considerably outperforms the asymptotic bands in terms of
coverage accuracy. Finally, the bootstrap confidence corridors are used to
study the efficacy of the National Supported Work Demonstration, which is a
randomized employment enhancement program launched in the 1970s. This article
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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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Threshold quantile autoregressive models
We study in this article threshold quantile autoregressive processes. In particular we propose estimation and inference of the parameters in nonlinear quantile processes when the threshold parameter defining nonlinearities is known for each quantile, and also when the parameter vector is estimated consistently. We derive the asymptotic properties of the nonlinear threshold quantile autoregressive estimator. In addition, we develop hypothesis tests for detecting threshold nonlinearities in the quantile process when the threshold parameter vector is not identified under the null hypothesis. In this case we propose to approximate the asymptotic distribution of the composite test using a p-value transformation. This test contributes to the literature on nonlinearity tests by extending Hansen’s (Econometrica 64, 1996, pp.413-430) methodology for the conditional mean process to the entire quantile process. We apply the proposed methodology to model the dynamics of US unemployment growth after the Second World War. The results show evidence of important heterogeneity associated with unemployment, and strong asymmetric persistence on unemployment growth
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