Refinements of the Kiefer-Wolfowitz theorem and a test of concavity

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.