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Warranty Data Collection and Analysis pp 319–347Cite as

Complex Models for Parametric Analysis of 1-D Warranty Data

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Part of the book series: Springer Series in Reliability Engineering ((RELIABILITY))

Abstract

In Chaps. 11and 12, 1-D warranty claims data were analyzed using standard lifetime distributions under the assumption that the underlying population is homogeneous, that is, there are no quality variations in production and all customers are similar in terms of their usage intensity, operating environment, etc. In real life, however, the assumption of homogeneity may not hold for many products and the underlying population may consist of several subpopulations with the reliability of the items depending on explanatory variables such as operating environment, manufacturing periods, vendors, etc. The parametric approach to analysis of data in these cases requires the use of more complex model formulations. This Chapter presents some of these complex models (competing risk, mixture, AFT, PH and parametric regression models) and the analysis of 1-D warranty data using these models.

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Notes

  1. 1.

    The competing risk model has also been called the compound model, series system model, and multi-risk model in the reliability literature.

  2. 2.

    When all \( F_{k} (t),\;k = 1,2, \ldots ,K \) in (13.7) are either two- or three-parameter Weibull distributions, the model is called a finite Weibull mixture model. The Weibull mixture model has been referred to by many other names, including additive-mixed Weibull distribution, bimodal-mixed Weibull (for a two-fold mixture) mixed-mode Weibull distribution, Weibull distribution of the mixed type, multimodal Weibull distribution, and so forth [16].

  3. 3.

    Many more can be found in [17].

  4. 4.

    The Cox PH model is widely used in biomedical applications, especially in the analysis of clinical trial data and in its application in the reliability context is increasing [11].

  5. 5.

    This scaling is different from that used in the AFT model.

  6. 6.

    This is the most widely used form in the literature on parametric regression, is comparatively simple to apply, and is available in many statistical software packages.

  7. 7.

    Reference [9] presents an algorithm for finding MLEs of the parameters of a Weibull mixture model with right censored data. Reference [1] presents a procedure for finding the MLEs of the parameters of two-fold Weibull mixture models.

  8. 8.

    Two-fold mixture refers to a mixture model with two components, that is, K = 2 in (13.8) and (A.53).

  9. 9.

    The function “mle” given in the “stats4” library of R-language is used to find the MLEs of the parameters. It is very sensitive to initial values of parameters of these models.

  10. 10.

    The life-stress relationship describes a characteristic point or a life characteristic of the distribution from one stress level to another. For example, for the Weibull distribution, the scale parameter, \( \alpha , \) is considered to be life characteristic that is stress dependent and thus the life-stress relationship is assigned to \( \alpha . \) For the exponential and lognormal distributions, the mean life and median life, respectively, are considered to be life characteristics that are stress dependent.

  11. 11.

    These may be accelerating variables or stresses such as use-rate, temperature, voltage, humidity, pressure, etc.

  12. 12.

    These and any other percentiles can be estimated by Minitab.

  13. 13.

    [2] deals with a mixed-Weibull regression model to analyze automotive component warranty data. The distribution function for each subpopulation depends on a vector of covariates that characterize specific operating conditions.

  14. 14.

    Many of the widely used statistical models are either location-scale or log-location-scale families of distributions. The analytical methods developed for these families can be applied easily to any of its members. (For more on these families of distributions, see [14, 15].)

  15. 15.

    This may also be done with S-plus and R-language.

  16. 16.

    ML estimation for a different type of imperfect repair model is discussed in [8, 12].

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Authors and Affiliations

  1. 5401 Katherine Avenue, 91401-4922, Sherman Oaks, Los Angeles, CA, USA

    Wallace R. Blischke

  2. Department of Statistics, Rajshahi University, Rajshahi, Bangladesh

    M. Rezaul Karim

  3. School of Mechanical and Mining Engineering, The University of Queensland, Brisbane, QLD, 4072, Australia

    D. N. Prabhakar Murthy

Authors
  1. Wallace R. Blischke
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  2. M. Rezaul Karim
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  3. D. N. Prabhakar Murthy
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Correspondence to Wallace R. Blischke .

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Blischke, W.R., Rezaul Karim, M., Prabhakar Murthy, D.N. (2011). Complex Models for Parametric Analysis of 1-D Warranty Data. In: Warranty Data Collection and Analysis. Springer Series in Reliability Engineering. Springer, London. https://doi.org/10.1007/978-0-85729-647-4_13

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  • DOI: https://doi.org/10.1007/978-0-85729-647-4_13

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