Local bilinear multiple-output quantile/depth regression

Abstract

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

Directional quantile regression in R

“…Such concepts have been theoretically investigated in several articles including [8,9,10,16], and [19], their computational side has been successfully addressed in [17] and [18], and their practical applications still grow in number; see, e. g., [15] and [20]. Similar ideas also appear in other articles such as [3,5,6,13,14] and [4].…”

“…The last two ways (mentioned in [17] and [18]) compute the (nested) contours from the innermost one outwards. Figure 2 shows the τ -quantile cuts through several fixed regression values that were obtained by means of the locally constant regression of [8]. They strongly (and rightly) suggest that the observations follow a homoscedastic model with a quadratic trend.…”

Directional quantile regression in Octave (and MATLAB)

“…And even if such models are miraculously specified correctly, there is still no reason to interpret the cuts of the resulting regression quantile regions as conditional multivariate quantiles, maybe except for some very special cases. Such an interpretation would be somehow justified only if the nonparametric regression quantiles of [8] were used, as in Example E.…”

“…. , n. Nevertheless, integer weights may be useful for handling multiple identical observations, and kernel weights asymptotically shrinking to zero lead to the local constant multivariate quantile regression of [8] for p = 1. The weighted case can be transformed to the unweighted one by substitutions Y i := w i Y i and X i := w i X i because the computation does not employ any special information about the first coordinate of X i 's.…”

“…Those concepts have been elaborated in a long series of articles including [8,9,10,16,17,18], and [23], and they have already found some practical applications in [15] and [24]; see also [4,13], and [14] for other articles dealing with similar ideas. As the two concepts generalize the standard quantile regression of [12] and [11] to the multiple-response setup, they are very likely to once become as popular and widespread as their single-response predecessor.…”