In this paper, we explore the modified Greenwood statistic, which, in contrast to the classical Greenwood statistic, is properly defined for random samples from any distribution. The classical Greenwood statistic, extensively examined in the existing literature, has found diverse and interesting applications across various domains. Furthermore, numerous modifications to the classical statistic have been proposed. The modified Greenwood statistic, as proposed and discussed in this paper, shares several key properties with its classical counterpart. Emphasizing its stochastic monotonicity within three broad classes of distributions - namely, generalized Pareto, $\alpha-$stable, and Student's t distributions - we advocate for the utilization of the modified Greenwood statistic in testing scenarios. Our exploration encompasses three distinct directions. In the first direction, we employ the modified Greenwood statistic for Gaussian distribution testing. Our empirical results compellingly illustrate that the proposed approach consistently outperforms alternative goodness-of-fit tests documented in the literature, particularly exhibiting superior efficacy for small sample sizes. The second considered problem involves testing the infinite-variance distribution of a given random sample. The last proposition suggests using the modified Greenwood statistic for testing of a given distribution. The presented simulation study strongly supports the efficiency of the proposed approach in the considered problems. Theoretical results and power simulation studies are further validated by real data analysis.

翻译：本文探讨了修正格林伍德统计量，与经典格林伍德统计量不同，该统计量可适用于任意分布随机样本的恰当定义。经典格林伍德统计量已在现有文献中得到广泛研究，并在不同领域有着多样且有趣的应用。此外，现有研究还提出了对该经典统计量的多种修正方案。本文提出并讨论的修正格林伍德统计量，与经典版本共享若干关键性质。通过强调其在广义帕累托分布、$\alpha$-稳定分布和t分布这三类广泛分布类中的随机单调性，我们主张将修正格林伍德统计量应用于检验场景。本研究从三个不同方向展开探索：首先，将修正格林伍德统计量应用于高斯分布检验；实证结果有力表明，所提方法始终优于文献记载的替代拟合优度检验，尤其在样本量较小时表现出更优效能。第二个问题涉及检验给定随机样本的无限方差分布。最后一个方案建议将修正格林伍德统计量用于检验特定分布。仿真研究充分支持所提方法在所述问题中的有效性。理论结果和功效仿真研究进一步通过真实数据分析得到验证。