On curve estimation under order restrictions

Abstract

Robust regression is of interest in many problems where assumptions of a parametric function may be inadequate. In this thesis, we study regression problems where the assumptions concern only whether the curve is increasing or decreasing. Examples in economics and public health are given. In a forthcoming paper, the estimation methods presented here will be the basis for likelihood based surveillance systems for detecting changes in monotonicity. Maximum likelihood estimators are thus derived. Distributions belonging to the regular exponential family, for example the normal and Poisson distributions, are considered. The approach is semiparametric, since the regression function is nonparametric and the family of distributions is parametric. In Paper I, “Unimodal Regression in the Two-parameter Exponential Family with Constant or Known Dispersion Parameter”, we suggest and study methods based on the restriction that the curve has a peak (or, equivalently, a trough). This is of interest for example in turning point detection. Properties of the method are described and examples are given. The starting point for Paper II, “Semiparametric Estimation of Outbreak Regression”, was the situation at the outbreak of a disease. A regression may be constant before the outbreak. At the onset, there is an increase. We construct a maximum likelihood estimator for a regression which is constant at first but then starts to increase at an unknown time. The consistency of the estimator is proved. The method is applied to Swedish influenza data and some of its properties are demonstrated by a simulation study.