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 testing $k$-monotonicity of a compactly supported p.m.f. and we put our main focus on monotonicity and convexity; i.e., $k \in \{1,2\}$. We consider new testing procedures that are directly derived from the definition of $k$-monotonicity and rely exclusively on the empirical measure, as well as tests that are based on the projection of the empirical measure on the class of $k$-monotone p.m.f.s. The asymptotic behaviour of the introduced test statistics is derived and a simulation study is performed to assess the finite sample performance of all the proposed tests. Applications to real datasets are presented to illustrate the theory.

翻译：在形状约束的非参数推断中，通常需要进行初步检验以验证概率质量函数是否满足单调性、凸性或一般意义上的$k$-单调性等定性约束。本文主要研究紧支撑概率质量函数的$k$-单调性检验，重点关注单调性与凸性情形（即$k \in \{1,2\}$）。我们提出两种新的检验方法：一种直接基于$k$-单调性的定义且完全依赖经验测度；另一种基于经验测度在$k$-单调概率质量函数类上的投影。本文推导了所提出检验统计量的渐近性质，并通过模拟研究评估了所有检验方法在有限样本下的表现。最后通过实际数据应用展示了理论方法的实践价值。