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Tolerance Limits Under Gamma Mixtures: Application in Hydrology

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Abstract

In this study, the authors proposed upper tolerance limits for the gamma mixture distribution based on generalized fiducial inference, and an MCMC simulation is performed to sample from the generalized fiducial distributions. The simulation results and a real hydrological data example show that the proposed tolerance limits are more efficient.

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References

  1. Hauck W W and Shaikh R, Modified two-sided normal tolerance intervals for batch acceptance of dose uniformity, Pharmaceutical Statistics, 2004, 3(2): 89–97.

    Article  Google Scholar 

  2. Ryan T P, Modern Engineering Statistics, John Wiley & Sons, Inc, New York, NY, 2007.

    Book  MATH  Google Scholar 

  3. Shirke D T, Kumbhar R R, and Kundu D, Tolerance intervals for exponentiated scale family of distributions, Journal of Applied Statistics, 2005, 32(10): 1067–1074.

    Article  MathSciNet  MATH  Google Scholar 

  4. Du J and Fang X, Tolerance interval for exponential distribution, Frontiers of Mathematics in China, 2011, 6(6): 1059–1066.

    Article  MathSciNet  MATH  Google Scholar 

  5. Chen C and Wang H, Tolerance interval for the mixture normal distribution, Journal of Quality Technology, 2020, 52(2): 145–154.

    Article  Google Scholar 

  6. Bhaumik D K and Gibbons R D, One-sided approximate prediction intervals for at least p of m observations from a gamma population at each of r locations, Technometrics, 2006, 48(1): 112–119.

    Article  MathSciNet  Google Scholar 

  7. Aryal S, Bhaumik D K, Mathew T, et al., Approximate tolerance limits and prediction limits for the gamma distribution, Journal of Applied Statistical Science, 2008, 16(2): 253–261.

    MathSciNet  Google Scholar 

  8. Fernandez A J, Tolerance limits for k-out-of-n systems with exponentially distributed component lifetimes, IEEE Transactions on Reliability, 2010, 59(2): 331–337.

    Article  Google Scholar 

  9. Wang H and Tsung F, Tolerance intervals with improved coverage probabilities for binomial and Poisson variables, Technometrics, 2009, 51(1): 25–33.

    Article  MathSciNet  Google Scholar 

  10. Zimmer Z, Park D, and Mathew T, Tolerance limits under normal mixtures: Application to the evaluation of nuclear power plant safety and to the assessment of circular error probable, Computational Statistics and Data Analysis, 2016, 103: 304–315.

    Article  MathSciNet  MATH  Google Scholar 

  11. Krishnamoorthy K and Wang X, Fiducial confidence limits and prediction limits for a Gamma distribution: Censored and uncensored cases, Environmetrics, 2016, 27(8): 479–493.

    Article  MathSciNet  Google Scholar 

  12. Chen P and Ye Z S, Approximate statistical limits for a Gamma distribution, Journal of Quality Technology, 2017, 49(1): 64–77.

    Article  Google Scholar 

  13. Krishnamoorthy K and Mathew T, Statistical Tolerance Regions: Theory, Applications, and Computation, John Wiley & Sons, New York, NY, 2009.

    Book  MATH  Google Scholar 

  14. Meeker W Q, Hahn G J, and Escobar L A, Statistical Intervals: A Guide for Practitioners and Researchers, John Wiley & Sons, New York, NY, 2017.

    Book  MATH  Google Scholar 

  15. Wilks S S, Determination of sample sizes for setting tolerance limits, The Annals of Mathematical Statistics, 1941, 12(1): 91–96.

    Article  MathSciNet  MATH  Google Scholar 

  16. Wilks S S, Statistical prediction with special reference to the problem of tolerance limits, The Annals of Mathematical Statistics, 1942, 13(4): 400–409.

    Article  MathSciNet  MATH  Google Scholar 

  17. Liao C T and Iyer H K, A Tolerance interval for the normal distribution with several variance components, Statistica Sinica, 2004, 14: 217–229.

    MathSciNet  MATH  Google Scholar 

  18. Krishnamoorthy K, Mathew T, and Mukherjee S, Normal-based methods for a gamma distribution: Prediction and tolerance intervals and stress-strength reliability, Technometrics, 2008, 50(1): 69–78.

    Article  MathSciNet  Google Scholar 

  19. Cai T and Wang H, Tolerance intervals for discrete distributions in exponential families, Statistical Sinica, 2009, 19: 905–923.

    MathSciNet  MATH  Google Scholar 

  20. Yuan M, Hong Y, Escobar L A, et al., Two-sided tolerance intervals for members of the (log)-location-scale family of distributions, Quality Technology and Quantitative Management, 2018, 15(3): 374–392.

    Article  Google Scholar 

  21. Guo B, Zhu N, Wang W, et al., Constructing exact tolerance intervals for the exponential distribution based on record values, Quality and Reliability Engineering International, 2020, 36(7): 2398–2410.

    Article  Google Scholar 

  22. Lei Q and Qin Y, A modified likelihood ratio test for homogeneity in bivariate normal mixtures of two samples, Journal of Systems Science & Complexity, 2009, 22(3): 460–468.

    Article  MathSciNet  MATH  Google Scholar 

  23. Wiper M, Insua D R, and Ruggeri F, Mixtures of gamma distributions with applications, Journal of Computational and Graphical Statistics, 2001, 10(3): 440–454.

    Article  MathSciNet  Google Scholar 

  24. Yoo C, Jung K S, and Kim T W, Rainfall frequency analysis using a mixed Gamma distribution: Evaluation of the global warming effect on daily rainfall, Hydrological Processes, 2005, 19: 3851–3861.

    Article  Google Scholar 

  25. Yoo C, Kim K, Kim H S, et al., Estimation of areal reduction factors using a mixed Gamma distribution, Journal of Hydrology, 2007, 335: 271–284.

    Article  Google Scholar 

  26. Kong L and Kaddoum G, Secrecy characteristics with assistance of mixture Gamma distribution, IEEE Wireless Communications Letters, 2019, 8(4): 1086–1089.

    Article  Google Scholar 

  27. Young D S, Chen X, Hewage D C, et al., Finite mixture-of-Gamma distributions: Estimation, inference, and model-based clustering, Advances in Data Analysis and Classification, 2019, 13: 1053–1082.

    Article  MathSciNet  MATH  Google Scholar 

  28. Dilip D, Freris N, and Jabari S E, Sparse estimation of travel time distributions using Gamma kernels, Proceedings of 96th Annual Meeting of the Transportation Research Board, Washington, DC: Transportation Research of the National Academies of Sciences, Engineering, and Medicine, 2017.

  29. Jabari S E, Freris N, and Dilip D, Sparse travel time estimation from streaming data, Transportation Science, 2020, 54(1): 1–20.

    Article  Google Scholar 

  30. Xu Z, Jabari S E, and Prassas E, Applying finite mixture models to new york city travel times, Journal of Transportation Engineering, Part A: Systems, 2020, 146(5): 05020001.

    Google Scholar 

  31. Hannig J, On generalized fiducial inference, Statistica Sinica, 2009, 19: 491–544.

    MathSciNet  MATH  Google Scholar 

  32. Hannig J, Iyer H, and Patterson P, Fiducial generalized confidence intervals, Journal of the American Statistical Association, 2006, 101(473): 254–269.

    Article  MathSciNet  MATH  Google Scholar 

  33. Wang C, Hannig J, and Iyer H K, Fiducial prediction intervals, Journal of Statistical Planning and Inference, 2012, 142(7): 1980–1990.

    Article  MathSciNet  MATH  Google Scholar 

  34. Fisher R A, Inverse probability, Proceedings of the Cambridge Philosophical Society XXVi, London, UK, 1930, 528–535.

  35. Zabell S L, R. A. Fisher and the fiducial argument, Statistic Science, 1992, 7: 369–387.

    MathSciNet  MATH  Google Scholar 

  36. Abdel-Karim A, Applications of generalized inference, Ph.D. Thesis, Colorado State University, Fort Collins, CO., 2005.

    MATH  Google Scholar 

  37. Bain L and Engelhardt M, A two-moment chi-square approximation for the statistic \(\log (\bar x/\tilde x)\), Journal of the American Statistical Association, 1975, 70(352): 948–950.

    MATH  Google Scholar 

  38. Ye Z S and Chen N, Closed-form estimators for the gamma distribution derived from likelihood equations, The American Statistician, 2017, 71: 177–181.

    Article  MathSciNet  MATH  Google Scholar 

  39. R Development Core Team, R: A Language and Environment for Statistical Computing, Vienna, Austria, URL http://www.r-project.org/, 2014.

  40. Gelman A, Carlin J B, Stern H S, et al., Bayesian Data Analysis, 3rd Edition, Chapman & Hall, London, 2013.

    Book  MATH  Google Scholar 

  41. Tsai S F, Comparing coefficients across subpopulations in gaussian mixture regression models, Journal of Agricultural, Biological, and Environmental Statistics, 2019, 24(4): 610–633.

    Article  MathSciNet  MATH  Google Scholar 

  42. McLachlan G J and Peel D, Finite Mixture Models, Wiley, New York, 2000.

    Book  MATH  Google Scholar 

  43. Yin G and Ma Y, Pearson-type goodness-of-fit test with bootstrap maximum likelihood estimation, Electronic Journal of Statistics, 2013, 7: 412–427.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Faculty of Science, Beijing University of Technology, Beijing, 100124, China

    Junjun Jiao & Weihu Cheng

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  1. Junjun Jiao
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  2. Weihu Cheng
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Correspondence to Junjun Jiao or Weihu Cheng.

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Jiao, J., Cheng, W. Tolerance Limits Under Gamma Mixtures: Application in Hydrology. J Syst Sci Complex 36, 1285–1301 (2023). https://doi.org/10.1007/s11424-023-1156-6

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  • DOI: https://doi.org/10.1007/s11424-023-1156-6

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