Equilibrium distributions were introduced as limiting distributions of
forward or backward recurrence times in the context of Renewal Theory.
Thus, they can be regarded as limiting distributions of used life or
remaining life (upto or beyond a given point) in Reliability Analysis. We
can also look upon these distributions as particular cases of weighted
distributions, where the reciprocal of failure rate is chosen as the
weight. It is interesting to note that the original and equilibrium
distributions are identical only for the exponential case. Equilibrium
distributions behave like any life distribution and we can examine the
failure rate or mean remaining life or similar other functions used in
the study of ageing properties for an equilibrium distribution. In fact,
the failure rate of the equilibrium distribution is the reciprocal of the
mean remaining life of the original distribution.
One can define bivariate or multivariate equilibrium distributions, not
uniquely of course. One can also define equilibrium distributions of
higher orders. In the context of life distribution comparisons (for two
life or failure time random variables X and Y having p.d.f.s f and g
respectively), we talk of several dominance relations viz. those in terms
of failure rate, likelihood, failure rate average, mean remaining life,
expectation, etc. besides stochastic dominance.
Dominance relations between the original and the equilibrium
distributions can imply different ageing properties of the original
distribution (F). Thus, we have
| Dominance Relation | | Consequence |
|---|
| LR | | F is IFR |
| FR | | F is DMRL |
| FRA or d | | F is NBUE |
| MRL | | F is DVRL |
| E | | C.V. of F<1 |
