Multiple-decrement models and corresponding conditional single-decrement models

Standard multiple-decrement models in the actuarial mathematics of life contingencies treat two random variables: the future lifetime T of a status and an integer variable J indicating which of m 'causes' was involved in the failure of the status. In addition, there are usually introduced some 'associated single-decrement models' or 'absolute rates of decrement' that can be used in the construction of the full multiple-decrement model if special assumptions are made such as 'UDD' or 'constant force'. A problem with these associated single- decrement models is that it is sometimes unclear what they actually mean that is, to what sort of decrement they actually refer. Thus it may be difficult to have any sense of how to choose appropriate single-decrement models to use in the construction of a desired multiple decrement model. Another but minor difficulty is that these single-decrement models often fail to behave as do many standard single decrement models in that the probability of decrement may not tend to unity as duration increases. A strength of using associated single-decrement models is that they may be chosen quite freely and will produce a proper multiple-decrement model. This paper introduces the alternative notion of 'corresponding conditional single-decrement models'; these models: 1) do have both an intuitive and a precise probabilistic meaning; 2) do behave as do standard single -decrement models; and 3) can easily be used to construct the full multiple decrement model without any special assumption.