Preservation of Some Aging Properties and Stochastic Orders by Weighted Distributions

Abstract

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) and Bartoszewicz and Skolimowska (2006) and correct them.

Keywords:

• DFR
• DMRL
• DRFR
• Failure rate
• IFR
• IMRL
• IRFR
• Log-concave
• Log-convex
• mrl function
• Weight function
• Weighted distribution

Mathematics Subject Classification:

• 60E15
• 62N05

Acknowledgments

The authors are thankful to the anonymous referee for his/her valuable comments. The second author would like to acknowledge the financial assistance received from the C.S.I.R., India for carrying out this research work.