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In this paper, a new definition of the number of observations near the th order statistics is developed. Then some characterization results for Pareto and some related distributions are established in terms of mass probability function, first moment of these new counting random variables, and using completeness properties of the sequence of functions . Finally, new goodness-of-fit tests based on these new characterizations for Pareto distribution are presented. And the power values of the proposed tests are compared with the power values of well-known tests such as Kolmogorovâ€“Smirnov and Cramer-von Mises tests by Monte Carlo simulations.

Characterization and Goodness-of-Fit Test of Pareto and Some Related Distributions Based on Near-Order Statistics