Shape Constrained Kernel PDF and PMF Estimation

Abstract

We consider shape constrained kernel-based probability density function (PDF) and probability mass function (PMF) estimation. Our approach is of widespread potential applicability and includes, separately or simultaneously, constraints on the PDF (PMF) function itself, its integral (sum), and derivatives (finite-differences) of any order. We also allow for pointwise upper and lower bounds (i.e., inequality constraints) on the PDF and PMF in addition to more popular equality constraints, and the approach handles a range of transformations of the PDF and PMF including, for example, logarithmic transformations (which allows for the imposition of log-concave or log-convex constraints that are popular with practitioners). Theoretical underpinnings for the procedures are provided. A simulation-based comparison of our proposed approach with those obtained using Grenander-type methods is favourable to our approach when the DGP is itself smooth. As far as we know, ours is also the only smooth framework that handles PDFs and PMFs in the presence of inequality bounds, equality constraints, and other popular constraints such as those mentioned above. An implementation in R exists that incorporates constraints such as monotonicity (both increasing and decreasing), convexity and concavity, and log-convexity and log-concavity, among others, while respecting finite-support boundaries via explicit use of boundary kernel functions.