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Normalization of the origin-shifted exponential distribution for control chart construction

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  • Shih-Chou Kao

Abstract

This study demonstrates that a location parameter of an exponential distribution significantly influences normalization of the exponential. The Kullback-Leibler information number is shown to be an appropriate index for measuring data normality using a location parameter. Control charts based on probability limits and transformation are compared for known and estimated location parameters. The probabilities of type II error (β-risks) and average run length (ARL) without a location parameter indicate an ability to detect an out-of-control signal of an individual chart using a power transformation similar to using probability limits. The β-risks and ARL of control charts with an estimated location parameter deviate significantly from their theoretical values when a small sample size of n≤50 is used. Therefore, without taking into account of the existence of a location parameter, the control charts result in inaccurate detection of an out-of-control signal regardless of whether a power or natural logarithmic transformation is used. The effects of a location parameter should be eliminated before transformation. Two examples are presented to illustrate these findings.

Suggested Citation

  • Shih-Chou Kao, 2010. "Normalization of the origin-shifted exponential distribution for control chart construction," Journal of Applied Statistics, Taylor & Francis Journals, vol. 37(7), pages 1067-1087.
    • Handle: RePEc:taf:japsta:v:37:y:2010:i:7:p:1067-1087
    DOI: 10.1080/02664760802571333
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    References listed on IDEAS

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    1. Sat Ya D. Dubey, 1967. "Normal and weibull distributions," Naval Research Logistics Quarterly, John Wiley & Sons, vol. 14(1), pages 69-79.
    2. Zhenlin Yang & Min Xie, 2000. "Process monitoring of exponentially distributed characteristics through an optimal normalizing transformation," Journal of Applied Statistics, Taylor & Francis Journals, vol. 27(8), pages 1051-1063.
    3. Shih-Chou Kao & Chuan-Ching Ho & Ying-Chin Ho, 2006. "Transforming the exponential by minimizing the sum of the absolute differences," Journal of Applied Statistics, Taylor & Francis Journals, vol. 33(7), pages 691-702.
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    Cited by:

    1. Ignacio Cascos, 2021. "Simultaneous monitoring of origin and scale in left-bounded processes via depth," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 105(4), pages 649-673, December.

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