On the robustness of maximum likelihood estimators for the Poisson INAR(1) model

This article examines the robustness of maximum likelihood estimators in Poisson integer-valued autoregressive processes of order one, which model count time series with autocorrelation and Poisson-distributed innovations. While maximum likelihood estimation is widely used to estimate the autocorrelation parameter α and the mean parameter λ, its sensitivity to outliers and model deviations remains insufficiently understood. We use influence function theory to quantify the local impact of infinitesimal contamination on the estimators and compute gross error sensitivity to assess worst-case sensitivity. The methodology connects robust statistical ideas with count time series modeling by providing explicit score expressions and estimating Fisher information by simulation under stationarity. Visual diagnostics, including heatmaps, three-dimensional sensitivity surfaces, and summary tables, identify parameter regimes where the estimators become vulnerable to contamination. The study provides practical guidance and diagnostics for applied researchers in the social sciences and epidemiology.