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Parametric Approach to the Analysis of 1-D Warranty Data

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Warranty Data Collection and Analysis

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

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Abstract

This chapter discusses some basic ideas behind parametric approaches to the analysis of one dimensional warranty claims data. Various types of one dimensional warranty data are analyzed by a number of commonly used parametric models. The chapter uses the maximum likelihood method to estimate the parameters of the models and some functions of parameters that are of interest in warranty data analysis. The selected model for a given data scenario is used for predicting and drawing inferences on reliability-related quantities. Examples are given to investigate the properties of the methods and models and to compare the results and contrast them with those obtained by the nonparametric approach in Chap. 11.

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Notes

  1. 1.

    See Sects. 8.7 and 10.4.

  2. 2.

    See Appendix D.

  3. 3.

    See also ([23], Chap. 5).

  4. 4.

    The detailed derivation, properties and applications of AIC are given in [1, 2, 7, 24, 26] and references cited therein.

  5. 5.

    Intensity function and cumulative intensity function are defined in Appendix B.

  6. 6.

    The assumption that all customers (or systems) have the same intensity function is a strong assumption and might be inappropriate in some applications. If all systems are different, then each system can be modeled by its own intensity function with parameter \( \theta_{i} \) [3].

  7. 7.

    Intensity function (12.6) is also known as the exponential law or the Cox-Lewis intensity function.

  8. 8.

    According to [16], these tests appear to be the most frequently used trend tests.

  9. 9.

    Minitab software also provides these tests for trend.

  10. 10.

    In order to analyze the repairable system failure data by Minitab, each system must have a retirement time, which is the largest time for that system.

  11. 11.

    If necessary, the length can be equal to a “week”, “day” and so on.

  12. 12.

    In some cases, manufacturers consider month of production instead of month of sale when investigating engineering changes, product design changes, manufacturing and assembly changes, etc. In these cases, the monthly sales amounts can be estimated using the sales-lag distribution of the claims data.

  13. 13.

    Additional details on this method can be found in [11, 12, 18].

  14. 14.

    Estimation of the variance using (12.27) and (12.28) requires the detailed claims data \( \{ n_{jt} \} \) for all j and t, which are not available for publication but were used to obtain the result \( \hat{\sigma } = 1.358. \)

  15. 15.

    When the probability prediction intervals are computed on the basis of estimates from limited data, they are sometimes called “naive prediction intervals”, and they can serve as a basis for developing more commonly needed statistical prediction intervals ([20], Chap. 12).

  16. 16.

    See [4] for more on calculation of the renewal function.

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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). Parametric Approach to the 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_12

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  • DOI: https://doi.org/10.1007/978-0-85729-647-4_12

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