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Unbiased condition of the dispersion effects estimator in unreplicated two-level factorial experiments

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Abstract

This article presents a general form of the estimator for identifying dispersion effects from unreplicated two-level factorial experiments, and shows that the widely used estimators such as the BH, MH, and AMH estimators are all special cases of the proposed one, designated as the G estimator. The unbiased condition of the G estimator is proved, and a lower bound of variance of the G estimator is provided. A simulation based on a realistic design illustrates the variation of the variance and MSE (mean square error) of the G estimator on different coefficients. This estimator may be more flexible and has better performance than other methods such as the BH and MH estimators by appropriately selecting the coefficients.

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References

  1. Brenneman WA and Nair V N, Methods for identifying dispersion effects in unreplicated factorial experiments: A critical analysis and proposed strategies, Technometrics, 2001, 43: 388–404.

    Article  MathSciNet  Google Scholar 

  2. Wu Jeff C F and Hamada M, Experiments: Planning, Analysis, and Parameter Design Optimization, John Wiley & Sons, Inc, New York, 2000.

    MATH  Google Scholar 

  3. Bartlett M S and Kendall D G, The statistical analysis of variance-heterogeneity and the logarithmic transformation, Journal of the Royal Statistical Society, Ser. B, 1946, 8: 128–138.

    MathSciNet  MATH  Google Scholar 

  4. Nair V N and Pregibon D, Analyzing dispersion effects from replicated factorial experiments, Technometrics, 1988, 30: 247–257.

    Article  MathSciNet  MATH  Google Scholar 

  5. Box G E P and Meyer R D, Dispersion effects from fractional designs, Technometrics, 1986, 28: 19–27.

    Article  MathSciNet  MATH  Google Scholar 

  6. Wang P C, Tests for dispersion effects from orthogonal arrays, Computational Statistics and Data Analysis, 1989, 8: 109–117.

    Article  Google Scholar 

  7. Engel J and Huele A F, A generalized linear modeling approach to robust design, Technometrics, 1996, 38: 365–373.

    Article  MATH  Google Scholar 

  8. Bergman B and Hynén A, Dispersion effects from unreplicated designs in the 2k-p series, Technometrics, 1997, 30: 1–40.

    MATH  Google Scholar 

  9. Liao C T and Iyer H K, Optimal 2n-p fractional factorial designs for dispersion effects under a location-dispersion model, Communication in Statistics: Theory and Methods, 2000, 29: 823–835.

    Article  MathSciNet  MATH  Google Scholar 

  10. McGrath R N and Lin K J, Confounding of location and dispersion effects in unreplicated fractional factorials, Journal of Quality Technology, 2001, 33: 129–139.

    Google Scholar 

  11. Wiklander K and Holm S, Dispersion effects in unreplicated factorial designs, Applied Stochastic Models in Business and Industry, 2003, 19: 13–30.

    Article  MathSciNet  MATH  Google Scholar 

  12. Ven de van P M, On the equivalence of three estimators for dispersion effects in unreplicated two-level factorial designs, Journal of Statistical Planning and Inference, 2008, 138: 18–29.

    Article  MathSciNet  MATH  Google Scholar 

  13. Li J H, Wang Y, and Zhang X, Estimation of dispersion effect in two-level unreplicated factorial experiments, Acta Mathematicae Applicatae Sinica, 2010, 33: 710–721.

    MATH  Google Scholar 

  14. Li J H, Ren G X, and Wang Y, Properties of BH estimator of dispersion effect in unreplicated two-Level factorial experiments, Journal of Applied Probability and Statistics, 2010, 26: 179–189.

    MathSciNet  MATH  Google Scholar 

  15. Wang Y, Li J H, and Feng X, The ML estimator of dispersion effects in unreplicated factorial experiments, Journal of Systems Science and Mathematical Science, 2011, 31(7): 804–816 (in Chinese).

    MathSciNet  MATH  Google Scholar 

  16. Chen Y, A significance test in multi-level orthogonal designs with only one replicate, Journal of Applied Probability and Statistics, 2011, 27: 497–510.

    MathSciNet  MATH  Google Scholar 

  17. Bergquist B, Vanhatalo E, and Nordenvaad M L, A Bayesian analysis of unreplicated two-Level factorials using effects sparsity, hierarchy, and heredity, Quality Engineering, 2011, 23(2): 152–166.

    Article  Google Scholar 

  18. Yang L, A method for estimating dispersion effects in unreplicated two-level factorial experiments, Journal of Mathematical Research with Applications, 2015, 35: 581–590.

    MathSciNet  MATH  Google Scholar 

  19. Harvey A C, Estimating regression models with multiplicative heteroscedasticity, Econometrica, 1976, 44: 461–465.

    Article  MathSciNet  MATH  Google Scholar 

  20. Satterthwaite F E, Synthesis of variance, Psychometrika, 1941, 6: 309–316.

    Article  MathSciNet  MATH  Google Scholar 

  21. Satterthwaite F E, An approximate distribution of estimates of variance components, Biometrics Bulletin, 1946, 2: 110–114.

    Article  Google Scholar 

  22. Abramowitz M and Stegun I A, Handbook of Mathematical Functions, Dover Publications, Inc, New York, 1965.

    MATH  Google Scholar 

  23. Mao S S, Wang J L, and Pu X L, Advanced Mathematical Statistics, Higher Education Press, Beijing, 2006.

    Google Scholar 

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Authors and Affiliations

  1. Department of Applied Mathematics, Shanxi University of Finance and Economics, Taiyuan, 030006, China

    Liu Yang

  2. Computer Center, Shanxi University, Taiyuan, 030006, China

    Jihong Li & Yu Wang

Authors
  1. Liu Yang
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  2. Jihong Li
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  3. Yu Wang
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Corresponding author

Correspondence to Liu Yang.

Additional information

This research was supported by the National Natural Science Fund of China under Grant No. 61503228, Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi under Grant No. 2015106.

This paper was recommended for publication by Editor SHI Jianjun.

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Yang, L., Li, J. & Wang, Y. Unbiased condition of the dispersion effects estimator in unreplicated two-level factorial experiments. J Syst Sci Complex 29, 1716–1736 (2016). https://doi.org/10.1007/s11424-016-5194-1

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  • DOI: https://doi.org/10.1007/s11424-016-5194-1

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