Testing the equality of several gamma means: a parametric bootstrap method with applications

Computational Statistics volume 26, pages55–76(2011)Cite this article

Abstract

We apply a recently developed ‘Computational Approach Test’ (CAT), a variant of the parametric bootstrap method, to test the equality of means of several gamma distributions. All parameters are assumed to be unknown, and we consider two cases-(i) the shape parameters, though unknown, are assumed to be equal; and (ii) the shape parameters are all unknown and possibly unequal. The CAT, as applied to the above two cases, doesn’t require the knowledge of any sampling distribution, depends heavily on numerical computations and Monte-Carlo simulation, and figures out the critical region automatically. The power and/or size of our proposed CAT is quite encouraging compared with the other tests reported in the literature. The proposed method can be used as a logical alternative approach to classical one-way ANOVA when one is not sure about normality, and positively skewed distribution is a possibility for the observed data. Though the proposed CAT has been used recently to compare normal means by the present authors, its usefulness for comparing gamma means hadn’t been studied before. This paper shows that the CAT can be as good as, if not better than, the other proposed methods discussed in the literature to test the equality of several gamma means. Real life datasets have been used to illustrate the applicability of this method. Also, our comprehensive numerical study reveals that some of the frequently cited methods are not as good as they are claimed to be.

This is a preview of subscription content, log in to check access.

References

  1. Aksoy H (2000) Use of gamma distribution in hydrological analysis. Turk J Eng Environ Sci 24(6): 419–428

    MathSciNet  Google Scholar 

  2. Allison JS (2008) Bootstrap-based hypothesis testing. Ph.D. Thesis, North-West University, Potchefstroom, South Africa

  3. Bougeault P (1982) Cloud-ensemble relations based on the ability distribution for the higher order models of the planet layer. J Atmos Sci 39: 2691–2700

    Article  Google Scholar 

  4. Chang CH, Pal N (2008) A computational approach to one-way ANOVA and its model robustness. Far East J Theor Stat 25(1): 111–134

    MATH  MathSciNet  Google Scholar 

  5. Coe R, Stern RD (1982) Fitting models to daily rainfall data. J Appl Meteorol 21: 1024–1031

    Article  Google Scholar 

  6. Coles B, Vrbik P, Giacometti RD, Rothstein SM (2008) Gamma distribution model to provide a direct assessment of the overall quality of quantum monte carlo-generated electron distributions. J Phys Chem 112(10): 2012–2017

    Google Scholar 

  7. Davison AC, Hinkley DV (1997) Bootstrap methods and their application. Cambridge University Press, Cambridge

    Google Scholar 

  8. Duigon MT, Cooper BF (1999) Impact of a public water supply well on availability of ground water to neighboring domestic wells near bel air, Maryland. Interim Technical Report, Department of Natural Resources, Maryland Geological Survey (www.mgs.md.gov/hydro/pub/pdf/impact.pdf)

  9. Efron B (1982) The Jackknife, Bootstrap, and other resampling plans. In: CBMS-NSF regional conference series in applied mathematics, vol 38. Society for Industrial and Applied Mathematics, Philadelphia

  10. Ferth JH, Roberson CE, Polzer WL (1964) Sources of mineral constituents in water from granite rocks, Sierra Nevada, California and Nevada. US Geological Survey Water Supply Paper 1535-I

  11. Greenwood JA, Durand D (1960) Aids for fitting the gamma distribution by maximum likelihood. Technometrics 2: 55–65

    MATH  Article  MathSciNet  Google Scholar 

  12. Grice JV, Bain LJ (1980) Inferences concerning the mean of the gamma distribution. J Am Stat Assoc 75: 929–933

    MATH  Article  MathSciNet  Google Scholar 

  13. Hall P, Wilson SR (1991) Two guidelines for Bootstrap hypothesis testing. Biometrics 47: 757–762

    Article  MathSciNet  Google Scholar 

  14. Hjorth JSU (1994) Computer intensive statistical methods: validation, model selection and Bootstrap. Chapman and Hall, London

    Google Scholar 

  15. Li LW, Kooi PS, Leong MS, Yeo TS (2007) A gamma distribution of raindrop sizes and its application to Singapore’s tropical environment. Microw Opt Technol Lett 7(5): 253–257

    Article  Google Scholar 

  16. Martin MA (2007) Bootstrap hypothesis testing for some common statistical problems: a critical evaluation of size and power properties. Comput Stat Data Anal 51: 6321–6342

    MATH  Article  Google Scholar 

  17. McDonald JB, Jensen BC (1979) An analysis of estimators of alternative measures of income inequality associated with the gamma distribution function. J Am Stat Assoc 74: 856–860

    MATH  Article  Google Scholar 

  18. Ming Tan C, Raghavan N, Roy A (2007) Application of gamma distribution in electromigration for submicron interconnects. J Appl Phys 102(10): 3703–3711

    Article  Google Scholar 

  19. Shiue WK, Bain LJ (1983) A two-sample test of equal gamma distribution scale parameters with unknown common shape parameter. Technometrics 25(4): 377–381

    MATH  Article  MathSciNet  Google Scholar 

  20. Shiue WK, Bain LJ, Engelhardt M (1988) Test of equal gamma-distribution means with unknown and unequal shape parameters. Technometrics 30: 169–174

    MATH  Article  MathSciNet  Google Scholar 

  21. Simpson J (1972) Use of the gamma distribution in single-cloud rainfall analysis. Mon Weather Rev 100(4): 309–312

    Article  MathSciNet  Google Scholar 

  22. Singh VP, Singh K (1985) Derivation of the gamma distribution by using the principle of maximum entropy (POME). Water Resour Bull 21: 941–952

    Google Scholar 

  23. Singh SK (2004) Simplified use of gamma-distribution/nash model for runoff modeling. J Hydrol Eng 9(3): 240–243

    Article  Google Scholar 

  24. Tibshirani R (1991) Bootstrap hypothesis testing (correspondence). Biometrics 47: 969–970

    Google Scholar 

  25. Tripathi RC, Gupta RC, Pair RK (1993) Statistical tests involving several independent gamma distributions. Ann Inst Stat Math 45(4): 773–786

    MATH  Article  MathSciNet  Google Scholar 

  26. Wright WG (1985) Effects of fracturing on well yields in the coal field areas of wise and dickenson counties, southwestern Virginia; US geological survey water resources investigations report, pp 85–4061

Download references

Author information

Affiliations

  1. Department of Applied Statistics and Information Science, Ming Chuan University, Taoyuan County, Taiwan

    Ching-Hui Chang

  2. Department of Statistics, Tamkang University, Tamsui, Taipei, Taiwan

    Jyh-Jiuan Lin

  3. Department of Mathematics, University of Louisiana at Lafayette, Louisiana, LA, 70504, USA

    Nabendu Pal

Authors
  1. Ching-Hui Chang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jyh-Jiuan Lin
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Nabendu Pal
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nabendu Pal.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Chang, C., Lin, J. & Pal, N. Testing the equality of several gamma means: a parametric bootstrap method with applications. Comput Stat 26, 55–76 (2011). https://doi.org/10.1007/s00180-010-0209-1

Download citation

Keywords

  • Hypotheses
  • Parametric bootstrap
  • Power function
  • Size