The integer autoregressive (INAR) model is one of the most commonly used models in nonnegative integer-valued time series analysis and is a counterpart to the traditional autoregressive model for continuous-valued time series. To guarantee the integer-valued nature, the binomial thinning operator or more generally the generalized Steutel and van Harn operator is used to define the INAR model. However, the distributions of the counting sequences used in the operators have been determined by the preference of analyst without statistical verification so far. In this paper, we propose a test based on the mean and variance relationships for distributions of counting sequences and a disturbance process to check if the operator is reasonable. We show that our proposed test has asymptotically correct size and is consistent. Numerical simulation is carried out to evaluate the finite sample performance of our test. As a real data application, we apply our test to the monthly number of anorexia cases in animals submitted to animal health laboratories in New Zealand and we conclude that binomial thinning operator is not appropriate.

翻译：整数自回归模型（INAR）是非负整数值时间序列分析中最常用的模型之一，是连续值时间序列中传统自回归模型的对应模型。为保证整数值特性，通常采用二项稀疏算子或更一般的Steutel-van Harn广义算子来定义INAR模型。然而，迄今为止，算子中使用的计数序列的分布一直由分析者根据偏好决定，缺乏统计验证。本文基于计数序列分布与扰动过程的均值-方差关系，提出了一种检验算子合理性的方法。我们证明，该检验方法具有渐近正确尺寸且具有一致性。通过数值模拟评估了该方法在有限样本下的表现。在实际数据应用中，我们将该方法应用于新西兰动物健康实验室提交的动物厌食症月度病例数，结果表明二项稀疏算子并不适用。