There are many characterizations based on functions associated with a continuous distribution that uniquely identify it, which are useful tools for a researcher who is interested in constructing a goodness-of-fit test. In this work, we intend to provide some characterization for symmetric distributions. This class is also so broad, which includes familiar distributions like normal, logistic, student-t, Cauchy. We consider the Farlie-Gumbel-Morgenstern (FGM) family of bivariate distributions and establish several characterization results based on the properties of concomitants of order statistics and record values.

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