Extremum Sieve Estimation in k-out-of-n Systems

This article considers the non parametric estimation of absolutely continuous distribution functions of independent lifetimes of non identical components in k-out-of-n systems, 2 ⩽ k ⩽ n, from the observed “autopsy” data. In economics, ascending “button” or “clock” auctions with n heterogeneous bidders with independent private values present 2-out-of-n systems. Classical competing risks models are examples of n-out-of-n systems. Under weak conditions on the underlying distributions, the estimation problem is shown to be well-posed and the suggested extremum sieve estimator is proven to be consistent. This article considers the sieve spaces of Bernstein polynomials which allow to easily implement constraints on the monotonicity of estimated distribution functions.