Estimation of survival functions subject to order restrictions

Abstract

This contribution is framed in, and is part of, the Durabiltiy of Building Facades studies carried out in the last decade in the Institut d’Estadística i Matemàtica Aplicada a l’Edificació and Laboratori d’ Edificació at the Universitat Politècnica de Catalunya. The goal of this presentation is to incorporate in the analysis of the estimation of the durability of building facades respect the severity degrees (low, medium or high) the order of the events of interest. In other words, we will take into account preliminary estimates of the survival probabilities of previous events (e.g. initial severities) as order constraints for the estimation of survival probabilities for subsequent events of interest (e.g. more advanced severities). Standard statistical analyses for this type of interval censored data are usually conducted by using the R statistical software, however it does not include libraries or packages neither for simultaneous estimation nor for the use of order constraints. For this reason, we will use AMPL and the solver SNOPT to implement the Turnbull's estimator with order restrictions, for the estimation of the survival function of each event of interest taking into account the information given by the survival function of some previous event of interest. The proposed methodology solves the inconsistency problem associated with a separate estimation of the respective survival probabilities. As a illustration the methodology will be applied to a simulated dataset. As a theoretical contribution, a conjecture on Turnbull's estimator Theorem under order restrictions will be proposed.