Nonparametric maximum likelihood estimation of distribution functions when they cross

Nonparametrie maximum likelihood estimation of two general distribution functions is studied when they cross once. The restriction of having a crossing point on two distribution functions turns out to be a dual problem of unimodal ordering. The isotonic regression procedure for unimodal ordering is employed to estimate two distribution functions under the restriction. The proposed estimator has strong consistency. A real data is analyzed for illustrative purpose.