Smooth Estimation of a Lifetime Distribution with Competing Risks by Using Regular Interval Observations: Application to Cocoa Fruits Growth

Abstract

In the estimation of a lifetime distribution from regular interval-censored data with an additional censoring variable, we focus on the case where (contrary to the actuarial method) both events (interest and censorship) can occur on a given individual in the same interval and, thus, are observed simultaneously. Specifically, we consider a population where individuals pass through a finite number of successive stages during their growth and are threatened by a disease. First, we estimate the lifetime and time-to-disease distributions in each developmental stage from such censored data. Using data that were recorded on a cohort of individuals followed over a long period of time, we propose a non-parametric, yet continuously differentiable and piecewise quadratic polynomial, estimator for the survival function of each of these distributions. We applied it to estimate, from weekly field observations in Mbankomo (Cameroon), the lifetime and time-to-disease distributions of cocoa fruits in each of their three developmental stages before maturity. It is found that on average a healthy cocoa fruit spends only 2 1 2 weeks in its first stage (cherelle), compared with nearly 9 weeks as a young pod and 7 1 2 weeks as an adult pod. In a second phase, however, adapting our methodology to competing risks estimation, we observed that, owing to the severe rate of attacks, the fruits' effective lifetime expectancy in farmland is much shorter. Indeed, in that part of Cameroon, the cumulative risk of an attack on cocoa fruits in farmland, especially by pod rot disease, far outweighs their chances of reaching maturity.