Nonparametric and shape restricted statistics

Nonparametric curve estimation is primarily concerned with understanding the structure (trend/pattern) in the data without making strong parametric assumptions on its form. Examples are nonparametric estimation procedures for a regression function, a failure rate, or a probability density. Shape restricted curve estimation by-passes the dependence on tuning parameters and seeks to develop estimates that are fully automated in situations where prior knowledge on the shape of the function is available. Such shape constraints arise naturally in numerous applications, e.g., utility/production functions are known to be non-decreasing and concave, distribution functions are known to be non-decreasing, sometimes regression curves or failure rates are known to be monotone/unimodal, etc.\(\)

Researchers

Prof.dr.ir. G. (Geurt) Jongbloed

Prof.dr. P. (Piet) Groeneboom

Dr. H.P. (Rik) Lopuhaä

Books

2014 - Piet Groeneboom, Geurt Jongbloed - Cambridge University Press

Nonparametric Estimation under Shape Constraints: Estimators Algorithms and Asymptotics

This book treats the latest developments in the theory of order-restricted inference, with special attention to nonparametric methods and algorithmic aspects. Among the topics treated are current status and interval censoring models, competing risk models, and deconvolution. Methods of order restricted inference are used in computing maximum likelihood estimators and developing distribution theory for inverse problems of this type. The authors have been active in developing these tools and present the state of the art and the open problems in the field. The earlier chapters provide an introduction to the subject, while the later chapters are written with graduate students and researchers in mathematical statistics in mind. Each chapter ends with a set of exercises of varying difficulty. The theory is illustrated with the analysis of real-life data, which are mostly medical in nature.

Key Publications

2023 - Thomas van der Jagt, Geurt Jongbloed, Martina Vittorietti - Electronic Journal of Statistics

Stereological determination of particle size distributions for similar convex bodies

Consider an opaque medium which contains 3D particles. All particles are convex bodies of the same shape, but they vary in size. The particles are randomly positioned and oriented within the medium and cannot be observed directly. Taking a planar section of the medium we obtain a sample of observed 2D section profile areas of the intersected particles. In this paper the distribution of interest is the underlying 3D particle size distribution for which an identifiability result is obtained. Moreover, a nonparametric estimator is proposed for this size distribution. The estimator is proven to be consistent and its performance is assessed in a simulation study.

2012 - Cécile Durot, Vladimir N. Kulikov, Hendrik P. Lopuhaä - The Annals of Statistics

The Limit Distribution of the \(L_\infty\)-error of Grenander-type Estimators

Let \(f\) be a nonincreasing function defined on \([0,1]\). Under standard regularity conditions, we derive the asymptotic distribution of the supremum norm of the difference between \(f\) and its Grenander-type estimator on sub-intervals of \([0,1]\). The rate of convergence is found to be of order \((n/ \log n)^{−1/3}\) and the limiting distribution to be Gumbel.

1999 - Piet Groeneboom, Gerard Hooghiemstra, Hendrik P. Lopuhaä - The Annals of Statistics

Asymptotic Normality of the \(L_1\) Error of the Grenander Estimator

Groeneboom introduced a jump process that can be used (among other things) to study the asymptotic properties of the Grenander estimator of a monotone density. In this paper we derive the asymptotic normality of a suitably rescaled version of the \(L_1\) error of the Grenander estimator, using properties of this jump process.