Some Developments in the Theory of Shape Constrained Inference

Abstract

Shape constraints enter in many statistical models. Sometimes these constraints emerge naturally from the origin of the data. In other situations, they are used to replace parametric models by more versatile models retaining qualitative shape properties of the parametric model. In this paper, we sketch a part of the history of shape constrained statistical inference in a nutshell, using landmark results obtained in this area. For this, we mainly use the prototypical problems of estimating a decreasing probability density on [0, ∞) and the estimation of a distribution function based on current status data as illustrations.

Modeling the cell-type specific mesoscale murine connectome with anterograde tracing experiments

“…Ours is not the first cross-validation based model averaging method Gao, Zhang, Wang, and Zou (2016), but our use of shape-constrained estimator in target-encoded feature space is novel and fundamentally different from Nadaraya-Watson estimators that use an optimization method for selecting the weights (Saul & Roweis, 2003). The properties of this estimator and its relation to estimators fit using an optimization algorithm are therefore a possible future avenue of research (Groeneboom & Jongbloed, 2018;Salha & El Shekh Ahmed, n.d.). Since the impact of the virus depends on the particular injection region, a deep model such as Lotfollahi, Naghipourfar, Theis, and Alexander Wolf (2019) could be appropriate, provided enough data was available, but our sample size seems too low to utilize a fixed or mixed effect model generative model.…”

Bootstrap‐Based Inference for Cube Root Asymptotics

“…In a seminal paper , Kim and Pollard () studied estimators exhibiting “cube root asymptotics.” These estimators not only have a non‐standard rate of convergence, but also have the property that, rather than being Gaussian, their limiting distributions are of Chernoff () type; that is, the non‐Gaussian limiting distribution is that of the maximizer of a Gaussian process. Kim and Pollard's results cover not only celebrated examples such as the maximum score estimator of Manski () and the isotonic density estimator of Grenander (), but also more contemporary estimators arising in examples related to classification problems in machine learning (Mohammadi and van de Geer ()), nonparametric inference under shape restrictions (Groeneboom and Jongbloed ()), massive data M ‐estimation framework (Shi, Lu, and Song ()), and maximum score estimation in high‐dimensional settings (Mukherjee, Banerjee, and Ritov ()). Moreover, Seo and Otsu () recently generalized Kim and Pollard () to allow for n ‐varying objective functions ( n denotes the sample size), further widening the applicability of cube‐root‐type asymptotics.…”

Grenander functionals and Cauchy's formula

“…In contrast with the result (1.1), we get for a sample from a decreasing density f on [0, ∞) at a point t ∈ (0, ∞), where f is differentiable and f (t) < 0, the following result, due to Prakasa Rao in [16]: where Z = argmax t {W (t)−t 2 }, that is: Z is the (almost surely unique) location of the maximum of two-sided Brownian motion minus the parabola y(t) = t 2 . For further details, see, e.g., [10] and [11].…”

“…It was introduced in Grenander (1957), where it was proved that it is the left‐continuous slope of the least concave majorant of the empirical distribution function. Some properties and limit results are discussed in Groeneboom and Jongbloed (2014) and also in Groeneboom and Jongbloed (2018) in the special issue on nonparametric inference under shape constraints of the journal Statistical Science . The Grenander estimator is shown in Figure 1 for a sample of size n =100 from the uniform distribution on [0,1].…”

“…In contrast with the result , we get for a sample from a decreasing density f on [0,∞) at a point t ∈(0,∞), where f is differentiable and f ′ ( t )<0, the following result, due to Prakasa Rao in Prakasa Rao (1969): where Z =argmax t { W ( t )− t 2 }, that is, Z is the (almost surely unique) location of the maximum of two‐sided Brownian motion minus the parabola y ( t )= t 2 . For further details, see, for example, Groeneboom and Jongbloed (2014) and Groeneboom and Jongbloed (2018).…”