Local bilinear multiple-output quantile/depth regression

A new quantile regression concept, based on a directional version of Koenker and Bassett's traditional single-output one, has been introduced in [Ann. Statist. (2010) 38 635-669] for multiple-output location/linear regression problems. The polyhedral contours provided by the empirical counterpart of that concept, however, cannot adapt to unknown nonlinear and/or heteroskedastic dependencies. This paper therefore introduces local constant and local linear (actually, bilinear) versions of those contours, which both allow to asymptotically recover the conditional halfspace depth contours that completely characterize the response's conditional distributions. Bahadur representation and asymptotic normality results are established. Illustrations are provided both on simulated and real data.Comment: Published at http://dx.doi.org/10.3150/14-BEJ610 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Asset selection using Factor Model and Data Envelope Analysis - A Quantile Regression approach

With the growing number of stocks and other financial instruments in the investment market, there is always a need for profitable methods of asset selection. The Fama-French three factor model, makes the problem of asset selection easy, by narrowing down the number of parameters, but the usual technique of Ordinary Least Square (OLS), used for estimation of the coefficients of the three factors suffers from the problem of modelling using the conditional mean of the distribution, as is the case with OLS. In this paper, we use the technique of Data Envelopment Analysis (DEA) applied to the Fama-French Three Factor Model, to choose stocks from Dow Jones Industrial Index. We use a more robust technique called as Quantile Regression to estimate the coefficients for the factor model and show that the assets selected using this regression method form a higher return equally weighted portfolio.Asset Selection, Factor Model, DEA, Quantile Regression

Stochastic Equicontinuity in Nonlinear Time Series Models

In this paper I provide simple and easily verifiable conditions under which a strong form of stochastic equicontinuity holds in a wide variety of modern time series models. In contrast to most results currently available in the literature, my methods avoid mixing conditions. I discuss several applications in detail.Comment: 10 page

A Bayesian Approach to Multiple-Output Quantile Regression

This paper presents a Bayesian approach to multiple-output quantile regression. The unconditional model is proven to be consistent and asymptotically correct frequentist confidence intervals can be obtained. The prior for the unconditional model can be elicited as the ex-ante knowledge of the distance of the tau-Tukey depth contour to the Tukey median, the first prior of its kind. A proposal for conditional regression is also presented. The model is applied to the Tennessee Project Steps to Achieving Resilience (STAR) experiment and it finds a joint increase in tau-quantile subpopulations for mathematics and reading scores given a decrease in the number of students per teacher. This result is consistent with, and much stronger than, the result one would find with multiple-output linear regression. Multiple-output linear regression finds the average mathematics and reading scores increase given a decrease in the number of students per teacher. However, there could still be subpopulations where the score declines. The multiple-output quantile regression approach confirms there are no quantile subpopulations (of the inspected subpopulations) where the score declines. This is truly a statement of `no child left behind' opposed to `no average child left behind.

Fast computation of Tukey trimmed regions and median in dimension p>2p>2

Given data in $\mathbb{R}^{p}$, a Tukey $\kappa$-trimmed region is the set of all points that have at least Tukey depth $\kappa$ w.r.t. the data. As they are visual, affine equivariant and robust, Tukey regions are useful tools in nonparametric multivariate analysis. While these regions are easily defined and interpreted, their practical use in applications has been impeded so far by the lack of efficient computational procedures in dimension $p > 2$. We construct two novel algorithms to compute a Tukey $\kappa$-trimmed region, a na\"{i}ve one and a more sophisticated one that is much faster than known algorithms. Further, a strict bound on the number of facets of a Tukey region is derived. In a large simulation study the novel fast algorithm is compared with the na\"{i}ve one, which is slower and by construction exact, yielding in every case the same correct results. Finally, the approach is extended to an algorithm that calculates the innermost Tukey region and its barycenter, the Tukey median