Goodness-of-fit test for count distributions with finite second moment

A goodness-of-fit test for one-parameter count distributions with finite second moment is proposed. The statistic derived from the $L_1$-distance of a function probability generating model under null hypothesis and that random variable actually data, when latter belongs to suitable wide class alternatives. has rather simple form it asymptotically normally distributed hypothesis, allowing straightforward implementation test. Moreover, consistent alternative belonging class, but also all whose zero different hypothesis. Thus, use proposed investigated alternatives not in class. finite-sample properties are assessed by means an extensive simulation study.