Abstract
In this paper, a new nonparametric technique is proposed for estimating the shape functions in both fixed-effect and random-effect gamma process models. Existing approaches typically yield nonparametric estimates of the shape functions only at inspection time points, making them unsuitable for extrapolation beyond the observed time range. This limitation reduces their practicality in engineering applications. In contrast, we expand the shape functions using Bernstein bases, and impose appropriate constraints on the associated coefficients to achieve desired properties. This formulation enables efficient extrapolation, allowing reliability to be inferred beyond the final inspection time. The model parameters are estimated using the block coordinate descent algorithm and the expectation maximization algorithm. Subsequently, confidence intervals are constructed via the bootstrap method to quantify uncertainty A Bayesian information criterion is used to select the degree of Bernstein bases in practice. The proposed models and methods are verified through simulation studies, and their applications are illustrated by analyzing a fatigue crack dataset.