Semiparametric estimation of outbreak regression

A regression may be constant for small values of the independent variable (for example time), but then a monotonic increase starts. Such an “outbreak” regression is of interest for example in the study of the outbreak of an epidemic disease. We give the least square estimators for this outbreak regression without assumption of a parametric regression function. It is shown that the least squares estimators are also the maximum likelihood estimators for distributions in the regular exponential family such as the Gaussian or Poisson distribution. The approach is thus semiparametric. The method is applied to Swedish data on influenza, and the properties are demonstrated by a simulation study. The consistency of the estimator is proved.