Competing risks.

Consider a unit which can experience any one of k competing failure types, and suppose that for each unit we observe the time to failure, T , and the type of failure, C ∈ {1, 2, . . . , k}. The case of observing the pair (T, C) is termed “competing risks” in the statistical literature. After considering some examples we review basic notation and theory of competing risks. In particular we consider the latent failure time approach to competing risks in which the k risks are represented by potential failure times T1, . . . , Tk where only the smallest, T = minj Tj , is observed together with its index C = argminj Tj . In reliability studies, the marginal distributions of the Tj are often of primary interest, but are unfortunately non-identifiable in general. Additional, though non-testable, assumptions to obtain identifiability are considered, as are bounds for the marginal distributions given in terms of observable functions. The likelihood function of right censored competing risks data is given and its consequences for both parametric and non-parametric estimation are explained. Extensions of the classical theory of competing risks to more general Markov models and to repairable systems are briefly discussed.