10,859 research outputs found
Predicting extreme VaR: Nonparametric quantile regression with refinements from extreme value theory
This paper studies the performance of nonparametric quantile regression as a tool to predict Value at Risk (VaR). The approach is flexible as it requires no assumptions on the form of return distributions. A monotonized double kernel local linear estimator is applied to estimate moderate (1%) conditional quantiles of index return distributions. For extreme (0.1%) quantiles, where particularly few data points are available, we propose to combine nonparametric quantile regression with extreme value theory. The out-of-sample forecasting performance of our methods turns out to be clearly superior to different specifications of the Conditionally Autoregressive VaR (CAViaR) models.Value at Risk, nonparametric quantile regression, risk management, extreme value theory, monotonization, CAViaR
Uniform Bahadur Representation for Nonparametric Censored Quantile Regression: A Redistribution-of-Mass Approach
Censored quantile regressions have received a great deal of attention in the literature. In a linear setup, recent research has found that an estimator based on the idea of “redistribution-of-mass” in Efron (1967, Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, vol. 4, pp. 831–853, University of California Press) has better numerical performance than other available methods. In this paper, this idea is combined with the local polynomial kernel smoothing for nonparametric quantile regression of censored data. We derive the uniform Bahadur representation for the estimator and, more importantly, give theoretical justification for its improved efficiency over existing estimation methods. We include an example to illustrate the usefulness of such a uniform representation in the context of sufficient dimension reduction in regression analysis. Finally, simulations are used to investigate the finite sample performance of the new estimator
Nonparametric inference of quantile curves for nonstationary time series
The paper considers nonparametric specification tests of quantile curves for
a general class of nonstationary processes. Using Bahadur representation and
Gaussian approximation results for nonstationary time series, simultaneous
confidence bands and integrated squared difference tests are proposed to test
various parametric forms of the quantile curves with asymptotically correct
type I error rates. A wild bootstrap procedure is implemented to alleviate the
problem of slow convergence of the asymptotic results. In particular, our
results can be used to test the trends of extremes of climate variables, an
important problem in understanding climate change. Our methodology is applied
to the analysis of the maximum speed of tropical cyclone winds. It was found
that an inhomogeneous upward trend for cyclone wind speeds is pronounced at
high quantile values. However, there is no trend in the mean lifetime-maximum
wind speed. This example shows the effectiveness of the quantile regression
technique.Comment: Published in at http://dx.doi.org/10.1214/09-AOS769 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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
has supplementary materials
Bayesian Quantile Regression for Single-Index Models
Using an asymmetric Laplace distribution, which provides a mechanism for
Bayesian inference of quantile regression models, we develop a fully Bayesian
approach to fitting single-index models in conditional quantile regression. In
this work, we use a Gaussian process prior for the unknown nonparametric link
function and a Laplace distribution on the index vector, with the latter
motivated by the recent popularity of the Bayesian lasso idea. We design a
Markov chain Monte Carlo algorithm for posterior inference. Careful
consideration of the singularity of the kernel matrix, and tractability of some
of the full conditional distributions leads to a partially collapsed approach
where the nonparametric link function is integrated out in some of the sampling
steps. Our simulations demonstrate the superior performance of the Bayesian
method versus the frequentist approach. The method is further illustrated by an
application to the hurricane data.Comment: 26 pages, 8 figures, 10 table
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