Research Article
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Year 2018, Volume: 14 Issue: 3, 1020 - 1037, 25.12.2018

Orkun Coşkuntuncel

https://doi.org/10.17860/mersinefd.443584

Abstract

References

  • ARSLAN, O., BILLOR, N. (1996). Robust Ridge Regression Estimation Based on the GM-Estimators. Jour. of Math. & Comp. Sci. (Math. Ser.), Vol.9-1, 1-9.
  • ARSLAN, O., BILLOR, N. (2000). Robust Liu Estimator for Regression Based on an M-Estimators. Journal of Applied Statistics, Vol.27-1, 39-47.
  • BELSLEY, D.A., KUH, E., WELSCH, R.E. (1980). Regression Diagnostics: Identifying Influential Data and Sources of Colinearity, Wiley, New York.
  • CORNELL, J. A. (1990). Experiments with Mixtures - Designs, Models and the Analysis of Mixture Data. 2nd edition, John Wiley& Sons, Inc., New York, USA.
  • COŞKUNTUNCEL, O. (2005). Robust Estimators for the Regression Parameters of Experiment with Mixtures Models. International Jour. of Pure and App. Math., Vol.24, No.4, 459-469.
  • DU, Z., WIENS, D.P. (2000). Jackknifing, Weighting, Diagnostics and Variance Estimation in Generalized M-Estimation. Statistics and Probability Letters, Vol.46, 287-299.
  • GORMAN, J. W. (1970). Fitting equations to mixture data with restraints on compositions. Journal of Quality Technology, Vol. 2, pp. 186-194.
  • HAMPEL, F.R., RONCHETTI, E.M., ROUSSEEUW, P.J., STAHEL, W.A. (1986). Robust Statistics: The Approach Based on Influential Functions, Wiley, New York.
  • HOERL, A.E., KENNARD, R.W. (1970a). Ridge Regression: Biased Estimation for Nonorthogonal Problems. Technometrics, Vol. 12-1, 55-67.
  • HOERL, A.E., KENNARD, R.W. (1970b). Ridge Regression: Applications to Nonorthogonal Problems. Technometrics, Vol. 12-1, 69-82.
  • HUBER, P.J. (1964). Robust Estimation of a Location Parameters. The Annals of Mathematical Statistics, Vol 35, 73-101.
  • HUBER, P.J. (2003). Robust Statistics, Wiley, New York.
  • LIU, K. (1993). A new class of biased estimate in linear regression. Communication in Statistics A, Vol. 22, 105-123.
  • KRASKER, W.S., WELSCH, R.E. (1982). Efficent bounded-influence regression estimation. Journal of the American Statistical Association, Vol. 77-379, 595-604.
  • MARONNA, R.A., MARTIN, R.D., YOHAI, V.J. (2006). Robust Statistics: Theory and Methods, Wiley, New York.
  • MARONNA, R.A., YOHAI, V.J. (1981). Asymptotic Behaviour of General M-Estimates for Regression and Scale with Random Carriers. Z. Wahrsch. Verb. Geb., 58, 7-20
  • MARQUARDT, D.W. (1970). Generalized inverses, Ridge regression, biased linear estimation, and nonlinear estimation. Technometrics, 12, 591-612.
  • MONTGOMERY, D.C., PECK, E.A., VINING, G.G. (2001). Introduction to Linear Regression Analysis. Wiley, New York.
  • MONTGOMERY, D.C., VOTH, S.R. (1994). Multicolinearity and Leverage in Mixture experiments. Journal of Quality Technology, Vol. 26, pp. 96-108.
  • MYERS, R.H., MONTGOMERY, D.C. (2002). Response Surface Methodology. Wiley, New York.
  • ROUSSEEUW, P.J. (1985). A regression diagnostic for multiple outliers and leverage points. Abstract in IMS Bull., 14, p. 399.
  • ROUSSEEUW, P.J., LEROY, A.M. (1987). Robust Regression and Outlier Detection. Wiley, New York.
  • ROUSSEEUW, P.J., ZOMEREN, B.C. (1990). Unmasking Multivariate Outliers and Leverage Points. Journal of American Statistical Association, Vol. 85-411, 633-651.
  • SCHEFFÉ, H. (1958). Experiments with mixtures. Journal of the Royal Statistical Society - B, Vol. 20, pp. 344-360.
  • SCHEFFÉ, H. (1963). The simplex-centroid design for experiments with mixtures. Journal of the Royal Statistical Society - B, Vol. 25, pp. 235-263.
  • SILVAPULLE, M.J. (1991). Robust Ridge Regression Based on M-Estimator. Australian and New Zealand Journal of Statistics, Vol.33, 319-333.
  • SIMPSON, D.G., RUPPERT, D., CARROLL, R.J. (1992). On One-Step GM Estimates and Stability of Inferences in Linear Regression. Journal of American Statistical Association, Vol. 87-418, 439-450.
  • SIMPSON, D.G., CHANG, Y.C.I. (1997). Reweighting Approximate GM Estimators: Asymptotics and Residual-based Graphics. Journal of Statistical Planning and Inference, 57, 273-293.
  • SNEE, R. D. (1975). Experimental designs for quadratic models in constrained mixture spaces. Technometrics, Vol. 17, pp. 149-159.
  • ST. JOHN, R. C. (1984). Experiments with mixtures, ill conditionning, and Ridge regression. Journal of Quality Technology, Vol. 16, pp. 81-96.
  • STAT-EASE (2004). DESIGN-EXPERT Software for Response Surface Methodology and Mixture Experiments. Version 6, 45 days Trial. Stat-Ease, Inc. Minneapolis, MN.

Bounded-Influence Regression Estimation for Mixture Experiments

Year 2018, Volume: 14 Issue: 3, 1020 - 1037, 25.12.2018

Orkun Coşkuntuncel

https://doi.org/10.17860/mersinefd.443584

Abstract

Ordinary
Least Squares (OLS) estimator is widely used technique for estimating the
regression coefficient in mixture experiments. But this estimator is very
sensitive to outliers and/or multicollinearity problems. The aim of this paper
is to propose estimators for the regression parameters of a mixture model that
can combat with the above problems. For this purpose, Generalized M (GM)
estimation, which is more resistant to outliers in the y and / or x directions
and regression estimators such as ridge and Liu, which is effective against the
multicollinearity, were used together. The Mean Square Error (MSE) properties
of proposed estimator has been examined and shown to be smaller than biased and
GM estimates. Also performance of the combined estimator is illustrated by
examples.

Keywords

Regression Ridge regression Liu estimator Robustness GM estimator

References

  • ARSLAN, O., BILLOR, N. (1996). Robust Ridge Regression Estimation Based on the GM-Estimators. Jour. of Math. & Comp. Sci. (Math. Ser.), Vol.9-1, 1-9.
  • ARSLAN, O., BILLOR, N. (2000). Robust Liu Estimator for Regression Based on an M-Estimators. Journal of Applied Statistics, Vol.27-1, 39-47.
  • BELSLEY, D.A., KUH, E., WELSCH, R.E. (1980). Regression Diagnostics: Identifying Influential Data and Sources of Colinearity, Wiley, New York.
  • CORNELL, J. A. (1990). Experiments with Mixtures - Designs, Models and the Analysis of Mixture Data. 2nd edition, John Wiley& Sons, Inc., New York, USA.
  • COŞKUNTUNCEL, O. (2005). Robust Estimators for the Regression Parameters of Experiment with Mixtures Models. International Jour. of Pure and App. Math., Vol.24, No.4, 459-469.
  • DU, Z., WIENS, D.P. (2000). Jackknifing, Weighting, Diagnostics and Variance Estimation in Generalized M-Estimation. Statistics and Probability Letters, Vol.46, 287-299.
  • GORMAN, J. W. (1970). Fitting equations to mixture data with restraints on compositions. Journal of Quality Technology, Vol. 2, pp. 186-194.
  • HAMPEL, F.R., RONCHETTI, E.M., ROUSSEEUW, P.J., STAHEL, W.A. (1986). Robust Statistics: The Approach Based on Influential Functions, Wiley, New York.
  • HOERL, A.E., KENNARD, R.W. (1970a). Ridge Regression: Biased Estimation for Nonorthogonal Problems. Technometrics, Vol. 12-1, 55-67.
  • HOERL, A.E., KENNARD, R.W. (1970b). Ridge Regression: Applications to Nonorthogonal Problems. Technometrics, Vol. 12-1, 69-82.
  • HUBER, P.J. (1964). Robust Estimation of a Location Parameters. The Annals of Mathematical Statistics, Vol 35, 73-101.
  • HUBER, P.J. (2003). Robust Statistics, Wiley, New York.
  • LIU, K. (1993). A new class of biased estimate in linear regression. Communication in Statistics A, Vol. 22, 105-123.
  • KRASKER, W.S., WELSCH, R.E. (1982). Efficent bounded-influence regression estimation. Journal of the American Statistical Association, Vol. 77-379, 595-604.
  • MARONNA, R.A., MARTIN, R.D., YOHAI, V.J. (2006). Robust Statistics: Theory and Methods, Wiley, New York.
  • MARONNA, R.A., YOHAI, V.J. (1981). Asymptotic Behaviour of General M-Estimates for Regression and Scale with Random Carriers. Z. Wahrsch. Verb. Geb., 58, 7-20
  • MARQUARDT, D.W. (1970). Generalized inverses, Ridge regression, biased linear estimation, and nonlinear estimation. Technometrics, 12, 591-612.
  • MONTGOMERY, D.C., PECK, E.A., VINING, G.G. (2001). Introduction to Linear Regression Analysis. Wiley, New York.
  • MONTGOMERY, D.C., VOTH, S.R. (1994). Multicolinearity and Leverage in Mixture experiments. Journal of Quality Technology, Vol. 26, pp. 96-108.
  • MYERS, R.H., MONTGOMERY, D.C. (2002). Response Surface Methodology. Wiley, New York.
  • ROUSSEEUW, P.J. (1985). A regression diagnostic for multiple outliers and leverage points. Abstract in IMS Bull., 14, p. 399.
  • ROUSSEEUW, P.J., LEROY, A.M. (1987). Robust Regression and Outlier Detection. Wiley, New York.
  • ROUSSEEUW, P.J., ZOMEREN, B.C. (1990). Unmasking Multivariate Outliers and Leverage Points. Journal of American Statistical Association, Vol. 85-411, 633-651.
  • SCHEFFÉ, H. (1958). Experiments with mixtures. Journal of the Royal Statistical Society - B, Vol. 20, pp. 344-360.
  • SCHEFFÉ, H. (1963). The simplex-centroid design for experiments with mixtures. Journal of the Royal Statistical Society - B, Vol. 25, pp. 235-263.
  • SILVAPULLE, M.J. (1991). Robust Ridge Regression Based on M-Estimator. Australian and New Zealand Journal of Statistics, Vol.33, 319-333.
  • SIMPSON, D.G., RUPPERT, D., CARROLL, R.J. (1992). On One-Step GM Estimates and Stability of Inferences in Linear Regression. Journal of American Statistical Association, Vol. 87-418, 439-450.
  • SIMPSON, D.G., CHANG, Y.C.I. (1997). Reweighting Approximate GM Estimators: Asymptotics and Residual-based Graphics. Journal of Statistical Planning and Inference, 57, 273-293.
  • SNEE, R. D. (1975). Experimental designs for quadratic models in constrained mixture spaces. Technometrics, Vol. 17, pp. 149-159.
  • ST. JOHN, R. C. (1984). Experiments with mixtures, ill conditionning, and Ridge regression. Journal of Quality Technology, Vol. 16, pp. 81-96.
  • STAT-EASE (2004). DESIGN-EXPERT Software for Response Surface Methodology and Mixture Experiments. Version 6, 45 days Trial. Stat-Ease, Inc. Minneapolis, MN.
There are 31 citations in total.

Details

Primary Language English
Subjects Studies on Education
Journal Section Makaleler
Authors

Orkun Coşkuntuncel 0000-0002-0599-1881

Publication Date December 25, 2018
Published in Issue Year 2018 Volume: 14 Issue: 3

Cite

APA Coşkuntuncel, O. (2018). Bounded-Influence Regression Estimation for Mixture Experiments. Mersin Üniversitesi Eğitim Fakültesi Dergisi, 14(3), 1020-1037. https://doi.org/10.17860/mersinefd.443584

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