Refinements of the Kiefer-Wolfowitz theorem and a test of concavity (Q2008622)

Let \(X_{1},\ldots,X_{n}\) be i.i.d. nonnegative random variables with common concave distribution function \(F\) and density function \(f\). Denote by \(\hat{\mathbb{F}}_{n}\) the Grenander distribution estimator, i.e. the least concave majorant of the empirical distribution function \(\mathbb{F}_{n}\). The author studies the problem of the closeness between \(\hat{\mathbb{F}}_{n}\) and \(\mathbb{F}_{n}\) relative to the uniforn norm \(\left\| \cdot \right\|_{\infty} \). In the case when \(f\) has bounded support \textit{J. Kiefer} and \textit{J. Wolfowitz} [Z. Wahrscheinlichkeitstheor. Verw. Geb. 34, 73--85 (1976; Zbl 0354.62035)], under some additional assumptions, show that \(\hat{\mathbb{F}}_{n}\) in fact is asymptotically equivalent to \(\mathbb{F}_{n}\). In this paper, the author generalizes the asymptotic order results by Kiefer and Wolfowitz to setting with unbounded support and contiguous distributions. From the author's abstract: ``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 \(O\left(n^{-2/3}(\log n)^{2/3} \right) \) 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.''