Lifelength in a random environment

Let T denote the lifelength of a component, the distribution of which, conditional on a parameter y, is known. It is supposed that y is a realization of nonnegative stochastic process {Y,t[greater-or-equal, slanted]0}, reflecting random variation in the environment in which the component is operating. Conditions are determined under which the distribution of T given {Y(t),t[greater-or-equal, slanted]0} is IFR, DFR, IFRA, DFRA, NBU or NWU. Further, conditions are determined under which, if the conditional distributions of T1 and T2, the lifelengths of two components, are ordered stochastically or in variability, a similar order holds for the corresponding marginal distributions.