Reduction in mean residual life in the presence of a constant competing risk

The addition of a constant ‘competing risk’ corresponding to an additional, usually less significant, source of failure, frequently improves the fit in reliability and survival analysis. This is often termed a ‘lift’, as the effect is to increase the hazard rate (HR) function by a constant, which does not, of course, change the shape and hence the turning points of the HR function. However, lifting the HR function does not, in general, mean lowering the corresponding mean residual life (MRL) function by a constant, and so the MRL turning points, unlike those of the HR function are not invariant. The MRL turning points are used in, for example, defining burn-in procedures in reliability engineering, and determining premiums in insurance, Hence, it is of interest to examine the changes in the shape of the MRL function, and in the locations of its turning points, resulting from a lift in the HR function. We discuss these problems in detail, with reference to a number of common distributions in reliability and mortality modeling.