On the exponential formula for reliability
It is usually assumed that the exponential formula commonly used in reliability & survival analysis holds for the case of conditioning on "smooth" external covariates, and is not valid for the ease of internal covariates. Using an example of external shocks affecting an item it is shown that though formally the influence is not. smooth in this case, the corresponding exponential formula still holds. On the other hand, internal covariates do not necessarily lead to the not-absolutely-continuous conditional Cdf. If the internal covariate process specifies the full information on the item's failure process, then the corresponding conditional Cdf is not absolutely continuous. If the observed internal covariate does not provide complete information on the items state, then the corresponding conditional Cdf can be still absolutely continuous, and the exponential formula holds.
external covariates internal covariates mixture of distributions observed failure rate shock process