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：提出了一种针对具有有限二阶矩的单参数计数分布的拟合优度检验方法。检验统计量基于原假设模型概率生成函数与实际生成数据的随机变量概率生成函数（当后者属于一个适当的广泛备择类时）的$L^1$距离推导而来。该检验统计量形式较为简洁，在原假设下渐近服从正态分布，从而便于直接实施检验。此外，该检验不仅对属于该类的备择分布具有一致性，还对零概率与原假设不同的所有备择分布具有一致性。因此，本文还针对不在该类中的备择分布提出并研究了该检验的应用。通过广泛的模拟研究评估了检验的有限样本性质。