  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.
