Preservation of Some Aging Properties and Stochastic Orders by Weighted Distributions

By assuming that a random variable X possesses an aging property, we provide conditions under which the corresponding weighted version X ₁, with weight function w ₁(·), would also possess this aging property. Similarly, by assuming that two random variables X and Y are ordered with respect to a stochastic order, we provide conditions under which the corresponding weighted versions X ₁ and Y ₂, with weight functions w ₁(·) and w ₂(·), respectively, preserve this stochastic ordering. We also point out fallacies in the similar results claimed by Jain et al. (1989 Jain , K. , Singh , H. , Bagai , I. ( 1989 ). Relations for reliability measures of weighted distributions . Commun. Statist. Theory Meth. 18 : 4393 – 4412 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) and Bartoszewicz and Skolimowska (2006 Bartoszewicz , J. , Skolimowska , M. ( 2006 ). Preservation of classes of life distribution and stochastic orders under weighting . Statist. Probab. Lett. 76 : 587 – 596 .[Crossref] , [Google Scholar]) and correct them.