McMaster University

Graduate Program in Statistics



STATISTICS SEMINAR



SPEAKER:
S. P. Mukherjee,
Calcutta University, India
Date :Tuesday September 11, 2001.
Time : 3:30pm
Address Burke Science Building
Room: 101
TITLE:
Equilibrium Distributions and Dominance Relations
ABSTRACT:
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
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References
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Department of Mathematics and Statistics
Graduate Program in Statistics

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Last updated on September 20, 2001