Tolerance Limits on Order Statistics in Future Samples Coming from the Two-Parameter Exponential Distribution

Nicholas A. Nechval; Konstantin N. Nechval

doi:doi:10.11648/j.ajtas.s.2016050201.11

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Tolerance Limits on Order Statistics in Future Samples Coming from the Two-Parameter Exponential Distribution

Nicholas A. Nechval, Konstantin N. Nechval

Published in American Journal of Theoretical and Applied Statistics (Volume 5, Issue 2-1)

Received: 9 September 2015 Accepted: 10 September 2015 Published: 30 November 2015

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Abstract

This paper presents an innovative approach to constructing lower and upper tolerance limits on order statistics in future samples. Attention is restricted to invariant families of distributions under parametric uncertainty. The approach used here emphasizes pivotal quantities relevant for obtaining tolerance factors and is applicable whenever the statistical problem is invariant under a group of transformations that acts transitively on the parameter space. It does not require the construction of any tables and is applicable whether the past data are complete or Type II censored. The proposed approach requires a quantile of the F distribution and is conceptually simple and easy to use. For illustration, the two-parameter exponential distribution is considered. A practical example is given.

Published in	American Journal of Theoretical and Applied Statistics (Volume 5, Issue 2-1) This article belongs to the Special Issue Novel Ideas for Efficient Optimization of Statistical Decisions and Predictive Inferences under Parametric Uncertainty of Underlying Models with Applications
DOI	10.11648/j.ajtas.s.2016050201.11
Page(s)	1-6
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Order Statistics, F Distribution, Lower Tolerance Limit, Upper Tolerance Limit

References

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[2]	I. Guttman, “On the power of optimum tolerance regions when sampling from normal distributions,” Annals of Mathematical Statistics, vol. XXVIII, pp. 773–778, 1957.
[3]	A. Wald and J. Wolfowitz, “Tolerance limits for a normal distribution,” Annals of Mathematical Statistics, vol. XVII, pp. 208–215, 1946.
[4]	W. A. Wallis, “Tolerance intervals for linear regression,” in Second Berkeley Symposium on Mathematical Statistics and Probability, University of California Press, Berkeley, 1951, pp. 43–51.
[5]	J. K. Patel, “Tolerance limits: a review,” Communications in Statistics: Theory and Methodology, vol. 15, pp. 2719–2762, 1986.
[6]	I. R. Dunsmore, “Some approximations for tolerance factors for the two parameter exponential distribution,” Technometrics, vol. 20, pp. 317–318, 1978.
[7]	W. C. Guenther, S. A. Patil, and V. R. R. Uppuluri, “One-sided -content tolerance factors for the two parameter exponential distribution,” Technometrics, vol. 18, pp. 333–340, 1976.
[8]	M. Engelhardt and L. J. Bain, “Tolerance limits and confidence limits on reliability for the two-parameter exponential distribution,” Technometrics, vol. 20, pp. 37–39, 1978.
[9]	W. C. Guenther, “Tolerance intervals for univariate distributions,” Naval Research Logistics Quarterly, vol. 19, pp. 309–333, 1972.
[10]	G. J. Hahn and W. Q. Meeker, Statistical Intervals: A Guide for Practitioners. New York: John Wiley & Sons, 1991.
[11]	N. A. Nechval, and K. N. Nechval, “Characterization theorems for selecting the type of underlying distribution,” in Proceedings of the 7th Vilnius Conference on Probability Theory and 22nd European Meeting of Statisticians. Vilnius: TEV, 1998, pp. 352– 353.
[12]	P. H. Muller, P. Neumann, and R. Storm, Tables of Mathematical Statistics. Leipzig: VEB Fachbuchverlag, 1979.

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APA Style

Nicholas A. Nechval, Konstantin N. Nechval. (2015). Tolerance Limits on Order Statistics in Future Samples Coming from the Two-Parameter Exponential Distribution. American Journal of Theoretical and Applied Statistics, 5(2-1), 1-6. https://doi.org/10.11648/j.ajtas.s.2016050201.11

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ACS Style

Nicholas A. Nechval; Konstantin N. Nechval. Tolerance Limits on Order Statistics in Future Samples Coming from the Two-Parameter Exponential Distribution. Am. J. Theor. Appl. Stat. 2015, 5(2-1), 1-6. doi: 10.11648/j.ajtas.s.2016050201.11

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AMA Style

Nicholas A. Nechval, Konstantin N. Nechval. Tolerance Limits on Order Statistics in Future Samples Coming from the Two-Parameter Exponential Distribution. Am J Theor Appl Stat. 2015;5(2-1):1-6. doi: 10.11648/j.ajtas.s.2016050201.11

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@article{10.11648/j.ajtas.s.2016050201.11,
  author = {Nicholas A. Nechval and Konstantin N. Nechval},
  title = {Tolerance Limits on Order Statistics in Future Samples Coming from the Two-Parameter Exponential Distribution},
  journal = {American Journal of Theoretical and Applied Statistics},
  volume = {5},
  number = {2-1},
  pages = {1-6},
  doi = {10.11648/j.ajtas.s.2016050201.11},
  url = {https://doi.org/10.11648/j.ajtas.s.2016050201.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.s.2016050201.11},
  abstract = {This paper presents an innovative approach to constructing lower and upper tolerance limits on order statistics in future samples. Attention is restricted to invariant families of distributions under parametric uncertainty. The approach used here emphasizes pivotal quantities relevant for obtaining tolerance factors and is applicable whenever the statistical problem is invariant under a group of transformations that acts transitively on the parameter space. It does not require the construction of any tables and is applicable whether the past data are complete or Type II censored. The proposed approach requires a quantile of the F distribution and is conceptually simple and easy to use. For illustration, the two-parameter exponential distribution is considered. A practical example is given.},
 year = {2015}
}

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TY  - JOUR
T1  - Tolerance Limits on Order Statistics in Future Samples Coming from the Two-Parameter Exponential Distribution
AU  - Nicholas A. Nechval
AU  - Konstantin N. Nechval
Y1  - 2015/11/30
PY  - 2015
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DO  - 10.11648/j.ajtas.s.2016050201.11
T2  - American Journal of Theoretical and Applied Statistics
JF  - American Journal of Theoretical and Applied Statistics
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AB  - This paper presents an innovative approach to constructing lower and upper tolerance limits on order statistics in future samples. Attention is restricted to invariant families of distributions under parametric uncertainty. The approach used here emphasizes pivotal quantities relevant for obtaining tolerance factors and is applicable whenever the statistical problem is invariant under a group of transformations that acts transitively on the parameter space. It does not require the construction of any tables and is applicable whether the past data are complete or Type II censored. The proposed approach requires a quantile of the F distribution and is conceptually simple and easy to use. For illustration, the two-parameter exponential distribution is considered. A practical example is given.
VL  - 5
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Author Information

Nicholas A. Nechval

Department of Mathematics, Baltic International Academy, Riga, Latvia
Konstantin N. Nechval

Department of Applied Mathematics, Transport and Telecommunication Institute, Riga, Latvia

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APA Style

Nicholas A. Nechval, Konstantin N. Nechval. (2015). Tolerance Limits on Order Statistics in Future Samples Coming from the Two-Parameter Exponential Distribution. American Journal of Theoretical and Applied Statistics, 5(2-1), 1-6. https://doi.org/10.11648/j.ajtas.s.2016050201.11

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ACS Style

Nicholas A. Nechval; Konstantin N. Nechval. Tolerance Limits on Order Statistics in Future Samples Coming from the Two-Parameter Exponential Distribution. Am. J. Theor. Appl. Stat. 2015, 5(2-1), 1-6. doi: 10.11648/j.ajtas.s.2016050201.11

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AMA Style

Nicholas A. Nechval, Konstantin N. Nechval. Tolerance Limits on Order Statistics in Future Samples Coming from the Two-Parameter Exponential Distribution. Am J Theor Appl Stat. 2015;5(2-1):1-6. doi: 10.11648/j.ajtas.s.2016050201.11

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@article{10.11648/j.ajtas.s.2016050201.11,
  author = {Nicholas A. Nechval and Konstantin N. Nechval},
  title = {Tolerance Limits on Order Statistics in Future Samples Coming from the Two-Parameter Exponential Distribution},
  journal = {American Journal of Theoretical and Applied Statistics},
  volume = {5},
  number = {2-1},
  pages = {1-6},
  doi = {10.11648/j.ajtas.s.2016050201.11},
  url = {https://doi.org/10.11648/j.ajtas.s.2016050201.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.s.2016050201.11},
  abstract = {This paper presents an innovative approach to constructing lower and upper tolerance limits on order statistics in future samples. Attention is restricted to invariant families of distributions under parametric uncertainty. The approach used here emphasizes pivotal quantities relevant for obtaining tolerance factors and is applicable whenever the statistical problem is invariant under a group of transformations that acts transitively on the parameter space. It does not require the construction of any tables and is applicable whether the past data are complete or Type II censored. The proposed approach requires a quantile of the F distribution and is conceptually simple and easy to use. For illustration, the two-parameter exponential distribution is considered. A practical example is given.},
 year = {2015}
}

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TY  - JOUR
T1  - Tolerance Limits on Order Statistics in Future Samples Coming from the Two-Parameter Exponential Distribution
AU  - Nicholas A. Nechval
AU  - Konstantin N. Nechval
Y1  - 2015/11/30
PY  - 2015
N1  - https://doi.org/10.11648/j.ajtas.s.2016050201.11
DO  - 10.11648/j.ajtas.s.2016050201.11
T2  - American Journal of Theoretical and Applied Statistics
JF  - American Journal of Theoretical and Applied Statistics
JO  - American Journal of Theoretical and Applied Statistics
SP  - 1
EP  - 6
PB  - Science Publishing Group
SN  - 2326-9006
UR  - https://doi.org/10.11648/j.ajtas.s.2016050201.11
AB  - This paper presents an innovative approach to constructing lower and upper tolerance limits on order statistics in future samples. Attention is restricted to invariant families of distributions under parametric uncertainty. The approach used here emphasizes pivotal quantities relevant for obtaining tolerance factors and is applicable whenever the statistical problem is invariant under a group of transformations that acts transitively on the parameter space. It does not require the construction of any tables and is applicable whether the past data are complete or Type II censored. The proposed approach requires a quantile of the F distribution and is conceptually simple and easy to use. For illustration, the two-parameter exponential distribution is considered. A practical example is given.
VL  - 5
IS  - 2-1
ER  -

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