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Modelling N- and W-shaped hazard rate functions without mixing distributions

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  • M Bebbington
  • C D Lai
  • D N P Murthy
  • R Zitikis

Abstract

The presence of non-conforming components instead of, or in addition to, the usual assembly errors results in N- or W-shaped hazard rate (HR) functions rather than the usual bathtub (i.e. U-shaped) ones. Although there have been numerous models for bathtub-shaped HR functions, N- and W-shaped HR functions are usually modelled using mixtures of two or more distributions. While this approach does sometimes lead to tidy interpretation, there can be a degree of overparameterization, with consequent problems in stability and fitting. For this reason, the present paper revisits the natural approach of modelling N- and W-shaped HR functions using polynomial functions of degree three or four. Although the non-negativity of the hazard rate function becomes non-trivial, this ensures a minimal number of parameters. The polynomial approach also allows the use of a parametric model without imposing a particular shape of hazard rate function on the data, which usually requires a non-parametric approach. The possible hazard rate shapes obtainable are characterized, and detailed formulae for local minima and maxima of the functions provided. The performance of the models is compared to that of several generalizations of the Weibull distribution, with promising results.

Suggested Citation

  • M Bebbington & C D Lai & D N P Murthy & R Zitikis, 2009. "Modelling N- and W-shaped hazard rate functions without mixing distributions," Journal of Risk and Reliability, , vol. 223(1), pages 59-69, March.
    • Handle: RePEc:sae:risrel:v:223:y:2009:i:1:p:59-69
    DOI: 10.1243/1748006XJRR219
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    References listed on IDEAS

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    1. Ming-Yen Cheng & Peter Hall & Dongsheng Tu, 2006. "Confidence bands for hazard rates under random censorship," Biometrika, Biometrika Trust, vol. 93(2), pages 357-366, June.
    2. R. Jiang & D. N. P. Murthy, 1998. "Mixture of Weibull distributions—parametric characterization of failure rate function," Applied Stochastic Models and Data Analysis, John Wiley & Sons, vol. 14(1), pages 47-65, March.
    3. Jiang, R. & Murthy, D.N.P., 2009. "Impact of quality variations on product reliability," Reliability Engineering and System Safety, Elsevier, vol. 94(2), pages 490-496.
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