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

翻译：提议对一个参数的计数分布进行适当测试,以有限的第二秒为限定时间。测试统计数据来自该模型在无效假设下产生概率功能的函数的1美元1美元距离,以及随机变量实际产生数据的函数的距离,后者属于适当的大类替代数据。测试统计数据有相当简单的形式,在无效假设下正常分配,允许直接执行测试。此外,该测试与属于该类的替代分布一致,但对于与无效假设下零概率不同的所有替代分布也一致。因此,对试验的使用也提出了建议,并对不属于该类的替代数据进行调查。测试的有限样本特性通过广泛的模拟研究进行评估。