Modified Greenwood statistic and its application for statistical testing

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.