Title: Asymptotic tests for monotonicity and convexity of a probability mass function Show Chinese title

Title: 概率质量函数的单调性和凸性渐近检验

Abstract: 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. Show Chinese abstract

Abstract: 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（p.m.f.）是否满足诸如单调性、凸性或一般$k$-单调性的定性约束。 本文我们感兴趣的是检验一个紧支集上的p.m.f. 的$k$-单调性，并且我们将重点放在单调性和凸性上；即$k \in \{1,2\}$。 我们考虑了新的检验程序，这些程序直接来源于$k$-单调性的定义，并且完全基于经验分布函数，以及基于经验分布函数在$k$-单调p.m.f.类上的投影的检验。 引入检验统计量的渐近行为被推导出来，并且进行了模拟研究以评估所有提出的检验在有限样本下的性能。 还展示了实际数据集的应用来说明理论。