Abstract: We present an approach for estimating shape-constrained kernel-based probability density functions (PDFs) and probability mass functions (PMFs) that includes 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. Furthermore the approach handles a range of transformations of the PDFs and PMFs including, for example, logarithmic transformations, which allow us to impose log-concave or log-convex constraints. We also provide the theoretical underpinnings for the procedures. The results of a simulation-based comparison between our proposed approach and those Grenander-type methods favor our approach when the data-generating process is smooth. To the best of our knowledge, ours is also the only smooth framework that handles PDFs and PMFs in the presence of inequality bounds, equality constraints, and other popular constraints. An implementation in R 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 by using boundary kernel functions.

Key words and phrases: Kernel density estimation, probability density function, probability mass function, shape constraints.