Involve: A Journal of Mathematics

  • Involve
  • Volume 5, Number 4 (2012), 379-391.

Theoretical properties of the length-biased inverse Weibull distribution

Jing Kersey and Broderick Oluyede

Full-text: Open access

Abstract

We investigate the length-biased inverse Weibull (LBIW) distribution, deriving its density function, hazard and reverse hazard functions, and reliability function. The moments, moment-generating function, Fisher information and Shannon entropy are also given. We discuss parameter estimation via the method of moments and maximum likelihood, and hypothesis testing for the LBIW and parent distributions.

Article information

Source
Involve, Volume 5, Number 4 (2012), 379-391.

Dates
Received: 7 May 2010
Revised: 1 March 2012
Accepted: 13 April 2012
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.involve/1513733531

Digital Object Identifier
doi:10.2140/involve.2012.5.379

Mathematical Reviews number (MathSciNet)
MR3069042

Zentralblatt MATH identifier
1293.62031

Subjects
Primary: 62E15: Exact distribution theory 62F03: Hypothesis testing 62N05: Reliability and life testing [See also 90B25] 62N01: Censored data models

Keywords
inverse Weibull distribution weighted reliability functions integrable function

Citation

Kersey, Jing; Oluyede, Broderick. Theoretical properties of the length-biased inverse Weibull distribution. Involve 5 (2012), no. 4, 379--391. doi:10.2140/involve.2012.5.379. https://projecteuclid.org/euclid.involve/1513733531


Export citation

References

  • R. Arratia and L. Goldstein, “Size bias, sampling, the waiting time paradox, and infinite divisibility: when is the increment independent?”, 2009, http://bcf.usc.edu/~larry/papers/pdf/csb.pdf.
  • R. Calabria and G. Pulcini, “Confidence limits for reliability and tolerance limits in the inverse Weibull distribution”, Reliability Engineering and System Safety 24:1 (1989), 77–85.
  • R. Calabria and G. Pulcini, “On the maximum likelihood and least-squares estimation in the inverse Weibull distribution”, Statistica Applicata 2:1 (1990), 53–66.
  • R. Calabria and G. Pulcini, “Bayes $2$-sample prediction for the inverse Weibull distribution”, Comm. Statist. Theory Methods 23:6 (1994), 1811–1824.
  • M. E. Ghitany, B. Atieh, and S. Nadarajah, “Lindley distribution and its application”, Math. Comput. Simulation 78:4 (2008), 493–506.
  • R. E. Glaser, “Bathtub and related failure rate characterizations”, J. Amer. Statist. Assoc. 75:371 (1980), 667–672.
  • R. C. Gupta and J. P. Keating, “Relations for reliability measures under length biased sampling”, Scand. J. Statist. 13:1 (1986), 49–56.
  • R. C. Gupta and S. N. U. A. Kirmani, “The role of weighted distributions in stochastic modeling”, Comm. Statist. Theory Methods 19:9 (1990), 3147–3162.
  • N. L. Johnson, S. Kotz, and N. Balakrishnan, Continuous univariate distributions, 1, 2nd ed., Wiley, New York, 1994.
  • A. Z. Keller, M. T. Goblin, and N. R. Farnworth, “Reliability analysis of commercial vehicle engines”, Reliability Engineering 10:1 (1985), 15–25.
  • J. F. Lawless, Statistical models and methods for lifetime data, 2nd ed., Wiley-Interscience, Hoboken, NJ, 2003.
  • B. O. Oluyede, “On inequalities and selection of experiments for length biased distributions”, Probab. Engrg. Inform. Sci. 13:2 (1999), 169–185.
  • G. P. Patil and C. R. Rao, “Weighted distributions and size-biased sampling with applications to wildlife populations and human families”, Biometrics 34:2 (1978), 179–189.
  • H. Rinne, The Weibull distribution: a handbook, CRC Press, Boca Raton, FL, 2009.