Characterizations of distributions based on recurrence relations for single and product moments of generalized order statistics has attracted the attention of many researchers. As generalized order statistics (GOS) provide a unifying approach to models of ordered random variables, we establish here some characterizations on absolutely continuous distributions based on GOS which contain and strengthen several known results in this regard. Because we do not impose restrictions on the model parameters (as done in the most of previous studies), our findings yield new results for various useful models of ordered random variables including k-record values, sequential order statistics, and progressively Type-II censored order statistics with an arbitrary
censoring plan. Characterizations of Weibull - Weibull distribution based on generalized order statistics will be presented in this paper.

An interesting method of adding a new parameter to an existing G distribution has been proposed by Bourguignon et al. (2014). The resulting distribution, known as the Weibull generated distribution, includes the original distribution as a special case and gives more flexibility to model various types of data.