This thesis proposes some new model independent or nonparametric methods for estimating the dose-response curve and the effective dosage curve in the context of bioassay. The research problem is also of importance in environmental risk assessment and other areas of health sciences. It is shown in the thesis that our new nonparametric methods while bearing optimal asymptotic properties also exhibit strong finite sample performance. Although our specific emphasis is on bioassay and environmental risk assessment, the methodology developed in this dissertation applies broadly to general order restricted inference.