Likelihood ratio ordering of order statistics, mixtures and systems

Abstract

Let X = ( X 1 , X 2 , … , X n ) be an exchangeable random vector, and denote X 1 : i = min { X 1 , X 2 , … , X i } and X i : i = max { X 1 , X 2 , … , X i } , 1 ⩽ i ⩽ n . These order statistics represent the lifetimes of the series and the parallel systems, respectively, with component lifetimes X i . In this paper we obtain conditions under which X 1 : i (or X i : i ) decreases (increases) in i in the likelihood ratio (lr) order. An even more general result involving general (that is, not necessary exchangeable) random vectors is also derived for general series (or parallel) systems. We show that the series (parallel) systems are not necessarily lr-ordered even if the components are independent. The likelihood ratio order can be characterized in terms of Glaser's function, defined by η ( t ) = - f ′ ( t ) / f ( t ) where f is the density function. This function is also a very useful tool to study the shape of hazard (or failure) rate and the mean residual life functions (see Glaser, R.E., 1980. Bathtub and related failure rate characterizations. J. Amer. Statist. Assoc. 75 (371), 667–672). It is also useful to study the likelihood ratio ordering and the increasing (or decreasing) likelihood ratio ILR (DLR) class. In this paper we also study properties of Glaser's function of mixtures. Specifically, we study ordering properties, monotonicity and the limiting behaviour. We show that, under some conditions, the limiting behaviour is similar to that of the strongest member (in the likelihood ratio order) of the mixture. We also consider the case of finite negative mixtures (i.e. mixtures which have some negative coefficients) which is applied to study Glaser's function of general coherent systems and order statistics and, in particular, the likelihood ratio ordering of coherent systems. The results are illustrated through a series of examples.