Nonparametric estimation of density and hazard rate functions with shape restrictions

Abstract

Methods for nonparametric maximum likelihood estimation of probability distributions are presented, with assumptions concerning the smoothness and shape. In particular, the decreasing density is considered, as well as constraints on the hazard function including increasing, convex or bathtub-shaped, and increasing and convex. Regression splines are used to formulate the problem in terms of convex programming, and iteratively re-weighted least squares cone projection algorithms are proposed. The estimators obtain the convergence rate r=(p+1)/(2p+3) where p is the degree of the polynomial spline. The method can be used with right-censored data. These methods are applied to real and simulated data sets to illustrate the small sample properties of the estimators and to compare with existing nonparametric estimators.

Keywords:

• monotone failure rate
• convex failure rate
• regression splines
• right censoring
• decreasing density

Mathematics Subject Classification Codes :

• 62N01
• 62N05
• 62G07