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Basic Statistical Inference

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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

Once data are collected, edited, summarized and otherwise prepared for detailed analysis, basic methods of statistical inference are applied to address stated experimental and management objectives. In this chapter, we look at several key statistical techniques that are used in inference, particularly in the context of reliability and warranty analysis. These include (1) estimation, including maximum likelihood, several other methods of point estimation, and confidence intervals; (2) hypothesis testing, including comparison of two population means; (3) nonparametric methods for comparing populations; (4) tolerance intervals for estimating population fractiles; and (5) rank correlation for measuring data relationships.

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Notes

  1. 1.

    A more complete treatment of incomplete data is given in Chaps. 11ᾢ15.

  2. 2.

    In some cases, it is possible to use the data to check for evidence of bias. A better (but usually expensive) check is to randomly sample the entire population of sold items and compare key measures from that sample with those of the claims data.

  3. 3.

    The chapter is intended to be a review of these topics, and, as such, treats most topics briefly, giving few details and no mathematical derivations. Details can be found in introductory statistical texts such as [13, 17]. Mathematical results and some techniques for their derivation are given in Appendix D; Details may be found in [6, 18].

  4. 4.

    Simulation studies can often be used to investigate alternative methods in this situation, but with obvious limitations.

  5. 5.

    The gamma function may be evaluated by interpolation in the tables found in [1], or by use of the gamma command in Minitab Calculator.

  6. 6.

    For additional results, see [8, 11, 15].

  7. 7.

    Note: In the Minitab parameterization of the exponential, the scale parameter is the inverse of that in the formula used in this book.

  8. 8.

    For further information, see [15]. MLE’s for grouped data from the three-parameter Weibull distribution (See (A.35)) are given by [5].

  9. 9.

    For a thorough discussion, see [19].

  10. 10.

    For further information and additional applications, see [2, 20].

  11. 11.

    There are many other approaches, depending on the data and the assumed distribution. For additional discussion and examples of applications, see [2, 14].

  12. 12.

    For further information and a number of examples, see [19], Chap. 20, [20], Chap. 8, and [2], Chaps. 5 and 8.

  13. 13.

    The estimators in this class are known as Best Asymptotically Normal (BAN) estimators. See [19] for details.

  14. 14.

    For other approaches and many additional results, see [11], Chap. 4.

  15. 15.

    Note: most statistical program packages do not require the specification of a value for α as an input. Instead, the analysis is performed and the output includes a “p-value,” which is the probability of obtaining the observed value of the test statistic, given that H 0 is true. If p ≤ α, H 0 is rejected; if p > α, it is not.

  16. 16.

    Note: The version of the z-statistic used for asymptotic tests may be replaced by the corresponding modification of the t-test for small sample sizes, say n < 100. In either case, the test is approximate in the sense that the level of significance is only approximately achieved. How good the approximation may be depends, among other things, on the true distribution of the test statistic and the sample size.

  17. 17.

    Many additional results may be found in [6, 8, 11, 15, 19, 20], and numerous other texts on theoretical and applied statistics and related field.

  18. 18.

    In general, an F-statistic is the ratio of two Chi-Square variables. df are those associated with the numerator and denominator of this ratio.

  19. 19.

    More extensive tables as well as factors for one-sided nonparametric tolerance intervals are given in [2, 16, 18], who initially published the results.

  20. 20.

    Note: Since Table E.3 gives values only to 100 df, the required fractiles of the Chi-Square distribution were obtained by use of Minitab.

  21. 21.

    A key difficulty is that parametric tests applied to data from distributions very different from that assumed may result in error rates quite different from the nominal.

  22. 22.

    Note: Minitab calculates the exact binomial probability of obtaining a result this or more discrepant from that expected under H 0, which may differ somewhat from that obtained using the normal approximation unless n is quite large.

  23. 23.

    The test is sometimes described as a comparison of the medians of two populations, which is not strictly a correct interpretation.

  24. 24.

    Note: The formula for the standard deviation assumes no ties. If there are only a few, this can safely be ignored. Otherwise, the formula requires an adjustment given in most texts on nonparametrics. Computer packages automatically perform the adjustment.

  25. 25.

    See texts on nonparametric statistics for the tables. Additional details regarding the MannᾢWhitney U may be found in the Wikipedia article by that title and in [7].

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Author information

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). Basic Statistical Inference. In: Warranty Data Collection and Analysis. Springer Series in Reliability Engineering. Springer, London. https://doi.org/10.1007/978-0-85729-647-4_9

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  • DOI: https://doi.org/10.1007/978-0-85729-647-4_9

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