Semi-Nonparametric Modeling of Densities on the Unit Interval, with Applications to Censored Mixed Proportional Hazard Models and Ordered Probability Models∗

In this paper I propose to estimate densities with possibly restricted support semi-nonparametrically via semi-nonparametric (SNP) estimation of densities on the unit interval. The latter will be done similarly to the approach of Gallant and Nychka (1987), but instead of using Hermite polynomials I propose to use orthonormal Legendre polynomials on the unit interval. This approach will be applied to the mixed proportional hazard (MPH) model, where the duration involved is only observed within brackets, and the distribution of the unobserved heterogeneity is modeled semi-nonparametrically. Under mild conditions the MPH model is nonparametrically identified. It appears that the identification conditions involved also apply to generalized ordered probability models. I will set forth conditions such that for both types of models the SNP maximum likelihood estimators are consistent. ∗This is a preliminary and incomplete paper. Please do not quote. †I am grateful to Aleksandr Vashchilko for pointing out an error in a previous version of this paper.