統計学輪講（第１８回）

日時　　  2001年7月24日（火）　15時00分〜16時40分
場所　　  経済学部5階視聴覚室
講演者　　Prof. S. P. Mukherjee(University of Calcutta)
演題　    Equilibrium  Distributions and  Dominance Relations

概要　
  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