Closure of the NBUE (New Better Than Used in Expectation) and DMRL (Decreasing Mean Residual Life) Classes under Formation of Parallel Systems.

Abstract

Abstract : The class of new better than used in expectation (NBUE) life distributions arises quite naturally in considering maintenance and replacement policies. The class of life distributions with decreasing mean residual life (DMRL) is receiving a good deal of current attention, especially in biostatistics. It is well known that the NBUE class and the DMRL class are not closed under the formation of general coherent systems with independent components. In fact a series system with independent and identically distributed (i.i.d.) NBUE components may not have an NBUE life distribution. However, the question as to whether or not the NBUE and the DMRL properties are preserved under the formation of parallel system with i.i.d. units has remained open. In this note we show that the class of NBUE life distributions and the class of differentiable DMRL life distributions are both closed under the formation of parallel systems with i.i.d. units. Keywords: random variables; distribution functions.