Keming Yu

Local Linear Quantile Regression

Abstract In this article we study nonparametric regression quantile estimation by kernel weighted local linear fitting. Two such estimators are considered. One is based on localizing the characterization of a regression quantile as the minimizer of E{pp (Y — a)|X = x}, where ρp is the appropriate “check” function. The other follows by inverting a local linear conditional distribution estimator and involves two smoothing parameters, rather than one. Our aim is to present fully operational versions of both approaches and to show that each works quite well; although either might be used in practice, we have a particular preference for the second. Our automatic smoothing parameter selection method is novel; the main regression quantile smoothing parameters are chosen by rule-of-thumb adaptations of state-of-the-art methods for smoothing parameter selection for regression mean estimation. The techniques are illustrated by application to two datasets and compared in simulations.

Bayesian quantile regression

The paper introduces the idea of Bayesian quantile regression employing a likelihood function that is based on the asymmetric Laplace distribution. It is shown that irrespective of the original distribution of the data, the use of the asymmetric Laplace distribution is a very natural and effective way for modelling Bayesian quantile regression. The paper also demonstrates that improper uniform priors for the unknown model parameters yield a proper joint posterior. The approach is illustrated via a simulated and two real data sets.

Quantile regression: applications and current research areas

Summary. Quantile regression offers a more complete statistical model than mean regression and now has widespread applications. Consequently, we provide a review of this technique. We begin with an introduction to and motivation for quantile regression. We then discuss some typical application areas. Next we outline various approaches to estimation. We finish by briefly summarizing some recent research areas.

Bayesian Mode Regression

Abstract We consider the problem of accounting for model uncertainty in linear regression models. Conditioning on a single selected model ignores model uncertainty, and thus leads to the underestimation of uncertainty when making inferences about quantities of interest. A Bayesian solution to this problem involves averaging over all possible models (i.e., combinations of predictors) when making inferences about quantities of interest. This approach is often not practical. In this article we offer two alternative approaches. First, we describe an ad hoc procedure, “Occams window,” which indicates a small set of models over which a model average can be computed. Second, we describe a Markov chain Monte Carlo approach that directly approximates the exact solution. In the presence of model uncertainty, both of these model averaging procedures provide better predictive performance than any single model that might reasonably have been selected. In the extreme case where there are many candidate predictors but ...

A Three-Parameter Asymmetric Laplace Distribution and Its Extension

ABSTRACT In this article, a new three-parameter asymmetric Laplace distribution and its extension are introduced. This includes as special case the symmetric Laplace double-exponential distribution. The distribution has established a direct link to estimation of quantile and quantile regression. Properties of the new distribution are presented. Application is made to a flood data modeling example.

Likelihood-Based Local Linear Estimation of the Conditional Variance Function

We consider estimation of mean and variance functions with kernel-weighted local polynomial fitting in a heteroscedastic nonparametric regression model. Our preferred estimators are based on a localized normal likelihood, using a standard local linear form for estimating the mean and a local log-linear form for estimating the variance. It is important to allow two bandwidths in this problem, separate ones for mean and variance estimation. We provide data-based methods for choosing the bandwidths. We also consider asymptotic results, and study and use them. The methodology is compared with a popular competitor and is seen to perform better for small and moderate sample sizes in simulations. A brief example is provided.

Bayesian adaptive Lasso quantile regression

Recently, variable selection by penalized likelihood has attracted much research interest. In this paper, we propose adaptive Lasso quantile regression (BALQR) from a Bayesian perspective. The method extends the Bayesian Lasso quantile regression by allowing different penalization parameters for different regression coefficients. Inverse gamma prior distributions are placed on the penalty parameters. We treat the hyperparameters of the inverse gamma prior as unknowns and estimate them along with the other parameters. A Gibbs sampler is developed to simulate the parameters from the posterior distributions. Through simulation studies and analysis of a prostate cancer dataset, we compare the performance of the BALQR method proposed with six existing Bayesian and non-Bayesian methods. The simulation studies and the prostate cancer data analysis indicate that the BALQR method performs well in comparison to the other approaches.

Local linear spatial quantile regression

Copyright @ 2009 International Statistical Institute / Bernoulli Society for Mathematical Statistics and Probability.

Nonparametric prediction by conditional median and quantiles

Two smooth nonparametric conditional median predictors, based on double kernel and local constant kernel methods, are defined for time series. Consistency and asymptotic normality are obtained for both of them. An extension to pth conditional quantiles is proposed in order to get predictive intervals. A rule-of-thumb selection for the smoothing parameters is developed. We illustrate the technique with a simulated sample and we apply it to a real-data analysis.

A comparison of local constant and local linear regression quantile estimators

Two popular nonparametric conditional quantile estimation methods, local constant fitting and local linear fitting, are compared. We note the relative lack of differences in results between the two approaches. While maintaining the expected preference for the local linear version, the arguments in favour are relatively slight, at least in the interior, and not as compelling as may be thought. The main differences between the approaches lie at the boundaries.