Estimation of split-points in binary regression

Let Y=m(X)+ϵ be a regression model with a dichotomous output Y and a step function m with exact one jump at a point θ and two different levels a and b. In the applied sciences the parameter θ is interpreted as a split-point whereas b and 1-a are known as positive and negative predictive value, respectively. We prove n-consistency and a weak convergence type result for a two-step plug-in maximum likelihood estimator of θ. The limit variable is not normal, but a maximizing point of a compound Poisson process on the real line. Estimation of (a,b) yields the usual √n-consistency with normal limit. Both results can be extended to a multivariate weak limit theorem. It allows for the construction of asymptotic confidence intervals for (θ,a,b). The theory is applied to real life data of a large epidemiological study.