This paper considers nonparametric estimation of absolutely continuous distribution functions of lifetimes of non-identical components in k-out-of-n systems from the observed ‘autopsy’ data. In economics, ascending ‘button’ or ‘clock’ auctions with n heterogeneous bidders present 2-out-of-n systems. Classical competing risk models are examples of k-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. The paper illustrates the suggested estimation method by using sieve spaces of Bernstein polynomials which allow an easy implementation of constraints on the monotonicity of estimated distribution functions.