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Many existing studies on profile monitoring focus on parametric profiles or normally distributed responses, and usually assume that the design points within different profiles are deterministic. In practice, however, profiles with count responses are common, and different profiles often have different within-profile sample sizes and design points. Furthermore, it is difficult to fit models to complex profiles with multiple explanatory variables either parametrically or nonparametrically. This article aims to monitor small-sample size profiles with count response and arbitrary design using a semiparametric model. Two novel control schemes with dynamic control limits are proposed based on the weighted likelihood ratio test and the weighted  F  test, respectively. Numerical simulations are conducted to investigate the performance of the proposed control charts. The performance between the control chart with constant and dynamic control limits is also compared, and the effect of model misspecification is explored. Finally, a real-data example of automobile warranty claims is presented to illustrate the implementation of the proposed control charts.

Semiparametric control schemes for dynamically monitoring profiles with count data and arbitrary design