Asymptotic theory for nonparametric testing of $k$-monotonicity in discrete distributions

In shape-constrained nonparametric inference, it is often necessary to perform preliminary tests to verify whether a probability mass function (p.m.f.) satisfies qualitative constraints such as monotonicity, convexity, or in general $k$-monotonicity. In this paper, we are interested in nonparametric testing of $k$-monotonicity of a finitely supported discrete distribution. We consider a unified testing framework based on a natural statistic which is directly derived from the very definition of $k$-monotonicity. The introduced framework allows us to design a new consistent method to select the unknown knot points that are required to consistently approximate the limit distribution of several test statistics based either on the empirical measure or the shape-constrained estimators of the p.m.f. We show that the resulting tests are asymptotically valid and consistent for any fixed alternative. Additionally, for the test based solely on the empirical measure, we study the asymptotic power under contiguous alternatives and derive a quantitative separation result that provides sufficient conditions to achieve a given power. We employ this test to design an estimator for the largest parameter $k \in \mathbb N_0$ such that the p.m.f. is $j$-monotone for all $j = 0, \ldots, k$, and show that the estimator is different from the true parameter with probability which is asymptotically smaller than the nominal level of the test. A detailed simulation study is performed to assess the finite sample performance of all the proposed tests, and applications to several real datasets are presented to illustrate the theory.