Skip to main content
SpringerLink
Log in

On proportional reversed failure rate class

  • Regular Article
  • Published:
Statistical Papers Aims and scope Submit manuscript

Abstract

Motivated by the recent use of the proportional reversed failure rate in economics (rates of increase and elasticity, see Veres-Ferrer and Pavia (Stat Pap 55:275–284, 2014) and in reliability (stochastic comparisons among systems, see Khaledi et al. (J Stat Plan Inference 141:276–286, 2011), in this work, we investigate characterizations and closure properties of the decreasing proportional reversed failure rate (DPRFR) classes for continuous, nonnegative random variables. Among others, we prove that DPRFR distributions are closed under convolutions. In addition, we relate this class of distributions with the class of monotone failure rate, proportional failure rate and likelihood ratio distributions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  • Al-Zahrani B, Stoyanov J (2008) On some properties of life distributions with increasing elasticity and log-concavity. Appl Math Sci 2(48):2349–2361

    MATH  MathSciNet  Google Scholar 

  • Arnold BC, Balakrishnan N, Nagaraja HN (1992) A first course in order statistics. Wiley, New York

    MATH  Google Scholar 

  • Bagnoli M, Bergstrom T (2005) Log-concave probability and its applications. Econ Theory 26:445–469

    Article  MATH  MathSciNet  Google Scholar 

  • Barlow RE, Marshall AW, Proschan F (1963) Properties of probability distributions with monotone hazard rate. Ann Math Stat 34:375–389

    Article  MATH  MathSciNet  Google Scholar 

  • Belzunce F, Candel J, Ruiz JM (1995) Ordering of truncated distributions through concentration curves. Sankhy ā 57(3):375–383

    MATH  MathSciNet  Google Scholar 

  • Belzunce F, Candel J, Ruiz JM (1998) Ordering and asymptotic properties of residual income distributions. Sankhy ā 60(2):331–348

    MATH  MathSciNet  Google Scholar 

  • Block HW, Savits TH, Singh H (1998) The reversed hazard rate function. Probab Eng Inform Sci 12:69–90

    Article  MATH  MathSciNet  Google Scholar 

  • Chandra NK, Roy D (2001) Some results on the reversed hazard rate. Probab Eng Inform Sci 15:95–102

    Article  MATH  MathSciNet  Google Scholar 

  • Chechile RA (2011) Properties of reverse hazard functions. J Math Psychol 55:203–222

    Article  MATH  MathSciNet  Google Scholar 

  • Colombo L, Labrecciosa P (2012) A note on pricing with risk aversion. Eur J Oper Res 216:252–254

    Article  MATH  MathSciNet  Google Scholar 

  • Franco M, Ruiz MC, Ruiz JM (2003) A note on closure of the ILR and DLR classes under formation of coherent systems. Stat Pap 44:279–288

    Article  MATH  MathSciNet  Google Scholar 

  • Glaser RE (1980) Bathtub and related failure rate characterizations. J Am Stat Assoc 75:667–672

    Article  MATH  MathSciNet  Google Scholar 

  • Khaledi B-E, Farsinezhad S, Kochar SC (2011) Stochastic comparisons of order statistics in the scale model. J Stat Plan Inference 141:276–286

    Article  MATH  MathSciNet  Google Scholar 

  • Keilson J, Sumita U (1982) Uniform stochastic ordering and related inequalities. Can J Stat 10:181–198

    Article  MATH  MathSciNet  Google Scholar 

  • Kundu C, Nanda AK, Hu T (2009) A note on reversed hazard rate of order statistics and record values. J Stat Plan Inference 139:1257–1265

    Article  MATH  MathSciNet  Google Scholar 

  • Lariviere MA (2006) A note on probability distributions with increasing generalized failure rates. Oper Res 54(3):602–604

    Article  MATH  MathSciNet  Google Scholar 

  • Lariviere MA, Porteus EL (2001) Selling to a newsvendor: an analysis of price-only contracts. Manuf Serv Oper Manag 3(4):293–305

    Google Scholar 

  • Marshall AW, Olkin I (2007) Life distributions. Springer, New York

    MATH  Google Scholar 

  • Nanda AK, Singh H, Misra N, Paul P (2003) Reliability properties of reversed residual lifetime. Commun Stat Theory Methods 32:2031–2042

    Article  MATH  MathSciNet  Google Scholar 

  • Paul A (2005) A note on closure properties of failure rate distributions. Oper Res 53(4):733–734

    Article  MATH  Google Scholar 

  • Ramos-Romero HM, Sordo-Díaz MA (2001) The proportional likelihood ratio order and applications. Qüestiió 25(2):211–223

    MATH  Google Scholar 

  • Righter R, Shaked M, Shanthikumar JG (2009) Intrinsic aging and classes of nonparametric distributions. Probab Eng Inform Sci 23(4):563–582

    Article  MATH  MathSciNet  Google Scholar 

  • Shaked M, Shanthikumar JG (2007) Stochastic orders. Springer, New York

    Book  MATH  Google Scholar 

  • Singh SK, Maddala GS (1976) A function for size distributions of incomes. Econometrica 44:963–970

    Article  Google Scholar 

  • Veres-Ferrer EJ, Pavia JM (2014) On the relationship between the reversed hazard rate and elasticity. Stat Pap 55:275–284

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

The authors wish to thank the Associate Editor and anonymous referees for their comments which have greatly improved the initial version of this manuscript. This work was partially supported by the Centro de Matemática da Universidade de Coimbra (CMUC), funded by the European Regional Development Fund through the program COMPETE and by the Portuguese Government through the FCT— Fundação para a Ciência e a Tecnologia under the project PEst-C/MAT/UI0324/2013. The research of N. Torrado was supported by the Portuguese Government through the Fundação para a Ciência e a Tecnologia (FCT) under the grant SFRH/BPD/91832/2012 and partially by project of the Spanish Ministry of Education and Science (ECO2012-38442).

Author information

Authors and Affiliations

  1. Department of Mathematics, Centre for Mathematics (CMUC), University of Coimbra, Apartado 3008, EC Santa Cruz, 3001-501, Coimbra, Portugal

    Paulo E. Oliveira & Nuria Torrado

Authors
  1. Paulo E. Oliveira
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nuria Torrado
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nuria Torrado.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Oliveira, P.E., Torrado, N. On proportional reversed failure rate class. Stat Papers 56, 999–1013 (2015). https://doi.org/10.1007/s00362-014-0620-8

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00362-014-0620-8

Keywords

Mathematics Subject Classification

Search

Navigation