A practical implementation of weighted kernel density estimation for handling shape constraints

Abstract

The weighted kernel density estimator is an attractive option for shape-restricted density estimation, because it is simple, familiar, and potentially applicable to many different shape constraints. Despite this, no reliable software implementation has appeared since the method's original proposal in 2002. We found that serious numerical and practical difficulties arise when attempting to implement the method. We overcame these difficulties and in the process discovered that the weighted method and our own recently proposed method-controlling the shape of a kernel density using an adjustment curve-can be unified in a single computational framework. This article describes our findings and introduces the R package scdensity, which can be used to easily obtain density estimates that are unimodal, bimodal, symmetric, and more.

Modality-Constrained Density Estimation via Deformable Templates

“…However, the method is ad hoc, and asymptotic convergence of the estimates, although seen empirically, is not guaranteed. Very recently, Wolters and Braun (2018) introduced a technique that solves the limitations of the approach in Hall and Huang (2002). Specifically, they provide an algorithm to find a constrained estimate that is the nearest to an unconstrained kernel density estimate (under the integrated squared error loss function), and that can handle up to bimodal constraints.…”

“…The shape constraint is fully captured in the initial template itself. As a result, the subsequent estimation of the transformation is independent of the constraint, providing much greater stability in practical performance with respect to higher modality constraints than methods such as Wolters and Braun (2018).…”

“…However, there are some clear advantages to our approach. First, note that the bimodality constraints in Wolters and Braun (2018) are satisfied only on a pre-specified finite grid. As a result, the final estimate has spurious modes violating the shape constraint, and thus technically does not belong in the correct shape class; the ability to violate the constraints probably explains the slightly better L 2 errors for its estimates.…”

“…Finally, we emphasize that the shape constraints appear directly in the estimation procedure of Wolters and Braun (2018). This makes the constrained estimation and the nested search for critical points increasingly complex as the modality is increased and makes the approach ill-equipped to handle higher modality constraints.…”

Constrained Sampling from a Kernel Density Estimator to Generate Scenarios for the Assessment of Automated Vehicles

“…We will propose an algorithm to sample parameters from a KDE while the parameters are subject to a linear equality constraint. Our work differs from [17], [18] because these works consider (shape) constraints on the estimated pdf. The proposed algorithm can be regarded as a generalization of the conditional density estimation in [19].…”