Functional estimation of diversity profiles

It is well known that the diversity profile provides a complete picture about the evenness of the relative abundance distribution of an ecological population. This complexity measure is a continuous function evaluated on a suitable grid of values x ≥ 0 that determine the measure's sensitivity to the most dominant species. In this paper, a functional design-based estimation of diversity profiles is developed by applying a framework based on functional data analysis (FDA). These curves, which are positive, decreasing, and convex, can be viewed as constrained functional data. Therefore, a naive direct application of the FDA methodology can be misleading, both theoretically and practically. To tackle this problem, the diversity profile is defined in terms of a differential equation, in such a manner that the function to be estimated is unconstrained. An approximation of the bias and the variance of the estimator is derived using the delta method. The accuracy of the proposed functional constrained estimator is evaluated through a simulation study. The procedure is also applied on a real dataset concerning tree stem diameter diversity.