# Refinements of the Kiefer-Wolfowitz Theorem and a Test of Concavity

## TL;DR

This paper refines the Kiefer-Wolfowitz theorem for concave distribution functions, develops a new test for concavity, and analyzes the estimator's convergence properties under various concavity conditions.

## Contribution

It extends existing results on the Grenander estimator to non-strictly concave functions and introduces a new test for concavity with asymptotic level control and power.

## Key findings

- Supremum distance between estimator and empirical distribution is of order $O(n^{-2/3}( ext{log} n)^{2/3})$
- Test of concavity has asymptotic pointwise level control
- Test demonstrates consistency and local power against alternatives

## Abstract

This paper studies estimation of and inference on a distribution function $F$ that is concave on the nonnegative half line and admits a density function $f$ with potentially unbounded support. When $F$ is strictly concave, we show that the supremum distance between the Grenander distribution estimator and the empirical distribution may still be of order $O(n^{-2/3}(\log n)^{2/3})$ almost surely, which reduces to an existing result of Kiefer and Wolfowitz when $f$ has bounded support. We further refine this result by allowing $F$ to be not strictly concave or even non-concave and instead requiring it be "asymptotically" strictly concave. Building on these results, we then develop a test of concavity of $F$ or equivalently monotonicity of $f$, which is shown to have asymptotically pointwise level control under the entire null as well as consistency under any fixed alternative. In fact, we show that our test has local size control and nontrivial local power against any local alternatives that do not approach the null too fast, which may be of interest given the irregularity of the problem. Extensions to settings involving testing concavity/convexity/monotonicity are discussed.