Miroslav Šiman

Computing multiple-output regression quantile regions

A procedure relying on linear programming techniques is developed to compute (regression) quantile regions that have been defined recently. In the location case, this procedure allows for computing halfspace depth regions even beyond dimension two. The corresponding algorithm is described in detail, and illustrations are provided both for simulated and real data. The efficiency of a Matlab implementation of the algorithm is also investigated through extensive simulations.

On directional multiple-output quantile regression

This paper sheds some new light on projection quantiles. Contrary to the sophisticated set analysis used in Kong and Mizera (2008) [13], we adopt a more parametric approach and study the subgradient conditions associated with these quantiles. In this setup, we introduce Lagrange multipliers which can be interpreted in various interesting ways, in particular in a portfolio optimization context. The corresponding projection quantile regions were already shown to coincide with the halfspace depth ones in Kong and Mizera (2008) [13], but we provide here an alternative proof (completely based on projection quantiles) that has the advantage of leading to an exact computation of halfspace depth regions from projection quantiles. Above all, we systematically consider the regression case, which was barely touched in Kong and Mizera (2008) [13]. We show in particular that the regression quantile regions introduced in Hallin, Paindaveine, and Siman (2010) [6,7] can also be obtained from projection (regression) quantiles, which may lead to a faster computation of those regions in some particular cases.

Analyzing growth trajectories.

Growth trajectories play a central role in life course epidemiology, often providing fundamental indicators of prenatal or childhood development, as well as an array of potential determinants of adult health outcomes. Statistical methods for the analysis of growth trajectories have been widely studied, but many challenging problems remain. Repeated measurements of length, weight and head circumference, for example, may be available on most subjects in a study, but usually only sparse temporal sampling of such variables is feasible. It can thus be challenging to gain a detailed understanding of growth patterns, and smoothing techniques are inevitably needed. Moreover, the problem is exacerbated by the presence of large fluctuations in growth velocity during early infancy, and high variability between subjects. Existing approaches, however, can be inflexible because of a reliance on parametric models, require computationally intensive methods that are unsuitable for exploratory analyses, or are only capable of examining each variable separately. This article proposes some new nonparametric approaches to analyzing sparse data on growth trajectories, with flexibility and ease of implementation being key features. The methods are illustrated using data on participants in the Collaborative Perinatal Project.

ON EXACT COMPUTATION OF SOME STATISTICS BASED ON PROJECTION PURSUIT IN A GENERAL REGRESSION CONTEXT

It is shown in detail how recent advances in multiple-output and projectional quantile regression open the door to exact computation of many inferential statistics based on projection pursuit. This is also illustrated on a few examples including new regression generalizations of multivariate skewness, kurtosis, and projection depth.

Precision Index in the Multivariate Context

General multivariate quantiles are employed to extend the classic univariate process precision index to the multivariate context under very mild conditions. Using halfspace depth regions for this purpose is especially recommended because it leads to both computational simplicity and natural generalizations to the tool-wear setup thanks to some recent advances in multiple-output and projectional quantile regression. A few examples are included to illustrate how the methodology might work in practice.

Local bilinear multiple-output quantile/depth regression

A new quantile regression concept, based on a directional version of Koenker and Bassetts 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 responses conditional distributions. Bahadur representation and asymptotic normality results are established. Illustrations are provided both on simulated and real data.

Multivariate Process Capability Indices: A Directional Approach

We propose a unified, universal, natural, and very intuitive way how to obtain new multivariate and tool wear extensions of univariate process capability indices by means of projection pursuit. We also illustrate the methodology in detail of the popular precision and accuracy indices, generalize the latter in a few different ways in the same spirit, add some personal insight, discuss the computational issues involved, and demonstrate the advantages of our approach in a small data example.

On elliptical quantiles in the quantile regression setup

This article defines a meaningful concept of elliptical location quantile with the aid of quantile regression, discusses its basic properties, and suggests its extension to a general regression framework through a locally constant nonparametric approach.

On Kendall's Autocorrelations

This brief article extends the theory of sample Kendalls autocorrelations by providing their exact variances at lags higher than one under the null hypothesis of randomness, by introducing and investigating their weighted modifications, and by numerical demonstration of these results and their usefulness.

Local Constant and Local Bilinear Multiple-Output Quantile Regression

A new quantile regression concept, based on a directional version of Koenker and Bassett’s traditional single-output one, has been introduced in [Hallin, Paindaveine and iSiman, Annals of Statistics 2010, 635-703] for multiple-output regression problems. The polyhedral contours provided by the empirical counterpart of that concept, however, cannot adapt to nonlinear and/or heteroskedastic dependencies. This paper therefore introduces local constant and local linear versions of those contours, which both allow to asymptotically recover the conditional halfspace depth contours of the response. In the multiple-output context considered, the local linear construction actually is of a bilinear nature. Bahadur representation and asymptotic normality results are established. Illustrations are provided both on simulated and real data.