Overdispersion is a common phenomenon in count or frequency responses in Poisson models. For example, number of car accidents on a highway during a year period. A similar phenomenon is observed in electric power systems, where cascading failures often follows some distribution with inflated zero. When the response contains an excess amount of zeros, zero-inflated Poisson (ZIP) is the most favourable model. However, during the data collection process, some of the covariates cannot be accessed directly or are measured with error among numerous disciplines. To the best of our knowledge, little existing work is available in the literature that tackles the population heterogeneity in the count response while some of the covariates are measured with error. With the increasing popularity of such outcomes in modern studies, it is interesting and timely to study zero-inflated Poisson models in which some of the covariates are subject to measurement error while some are not. We propose a flexible partial linear single index model for the log Poisson mean to correct bias potentially due to the error in covariates or the population heterogeneity. We derive consistent and locally efficient semiparametric estimators and study the large sample properties. We further assess the finite sample performance through simulation studies. Finally, we apply the proposed method to a real data application and compare with existing methods that handle measurement error in covariates.

在泊松模型中，过度分散是计数或频率响应中的常见现象。例如，一年中高速公路上的车祸数量。在电力系统中观察到类似的现象，其中级联故障通常遵循零膨胀的某种分布。当响应中包含过多的零时，零膨胀泊松（ZIP）是最合适的模型。但是，在数据收集过程中，某些协变量无法直接访问，或者在许多学科中都无法正确测量。据我们所知，文献中很少有现有的工作可以解决计数响应中的种群异质性，而某些协变量的测量则带有误差。随着这类结果在现代研究中的日益普及，研究零膨胀泊松模型是有趣且及时的，其中一些协变量会出现测量误差，而有些则不会。我们为对数泊松均值提出了一个灵活的局部线性单指数模型，以纠正可能由于协变量或总体异质性错误而引起的偏差。我们导出一致且局部有效的半参数估计量，并研究大样本属性。我们通过仿真研究进一步评估有限样品性能。最后，我们将提出的方法应用于实际数据应用，并与处理协变量中的测量误差的现有方法进行比较。我们为对数泊松均值提出了一个灵活的局部线性单指数模型，以纠正可能由于协变量或总体异质性错误而引起的偏差。我们导出一致且局部有效的半参数估计量，并研究大样本属性。我们通过仿真研究进一步评估有限样品性能。最后，我们将提出的方法应用于实际数据应用，并与处理协变量中的测量误差的现有方法进行比较。我们为对数泊松均值提出了一个灵活的局部线性单指数模型，以纠正可能由于协变量或总体异质性错误而引起的偏差。我们导出一致且局部有效的半参数估计量，并研究大样本属性。我们通过仿真研究进一步评估有限样品性能。最后，我们将提出的方法应用于实际数据应用，并与处理协变量中的测量误差的现有方法进行比较。