Parameter design or robust parameter design (RPD) is a statistical methodology used mostly in engineering fields as a cost-effective approach for improving the quality of products and processes. The primary goal of parameter design is to choose the levels of the control variables, which optimizes a defined quality characteristic. Modeling both the mean and variance is commonly referred to as dual modeling. In parametric dual modeling, estimations of the mean and variance parameters are interrelated. When one or both of the models (the mean or variance model) are mis-specified, parametric dual modeling can lead to faulty inferences. An alternative to parametric dual modeling is nonparametric dual modeling. However, nonparametric techniques often result in estimates characterized by high variability, which leads us to ignore important knowledge. We develop a dual modeling approach called dual model robust regression (DMRR), which is robust against user misspecification of the mean and/or variance models. Numerical and asymptotic results illustrate the advantages of DMRR over several other dual model procedures. The proposed method will be illustrated with simulations.

Navaee, M., Mobin, M., Vardani, M & Ahmadi, N. (2012). A Semi parametric approach to dual modeling.Management Science Letters, 2(2), 665-672.

Aitkin, M. (1987). Modeling variance heterogeneity in Normal regression using GLIM. Applied Statistics, 36, 332-339.

Anderson-Cook, C.M., & Prewitt, K. (2005). Some guidelines for using nonparametric methods for modeling data from response surface designs. Journal of Modern Applied Statistical Methods, 4, 106-119.

Bartlett, M.S., & Kendall, D.G. (1946). The statistical analysis of variance heterogeneity and the logarithmic transformation. Journal of Royal Statistical Society Series B, 8, 128- 138.Box, G., & Draper B. (1987). Empirical Model Building and Response Surface, Wiley, New York.Box, G.E.P., & Meyer, R.D. (1986). Dispersion effects from fractional designs. Technimetrics, 28, 19-27.

Box, G.E.P., & Draper, N.R., (1987). Empirical Model Building and Response Surface. Wiley, New York.Burman, P., & Chaudhuri, P. (1992). A Hybrid Approach to Parametric and Nonparametric Regression, Technical Report No. 243, Division of Statistics, University of California Davis, CA, USA.

Einsporn, R., & Birch, J.B. (1993). Model robust regression: using nonparametric regression to improve parametric regression analyses. Technical Report 93-5.Department of Statistics, Virginia Polytechnic Institute & State University, Blacksburg, VA.Fan, J., & Gijbels, I. (1996). Local Polynomial Modeling and its Applications. Chapman & Hall, London.Fan, J., Heckman, N.E., Wand, M.P., (1995). Local polynomial kernel regression for generalized linear models and quasi-likelihood functions. Journal of American Statistical Association, 90, 141-150.

Lin, X., & Carroll, R.J. (2000). Nonparametric function estimation for clustered data when the predictor is measured without/with error. Journal of the American Statistical Association.95, 520-534.

Mays, J. E., Birch, J. B., & R. L. Einsporn. (2000). An overview of model robust regression. Journal of Statistical Computation and Simulation. 66, 79-100.

Mays, J., Brich, J., & Starnes, B. (2001). Model robust regression: combining parametric, nonparametric and semi parametric methods. Journal of Nonparametric Statistics, 13, 245-270.

Pickle, S. M., Robinson, T.J., Birch, J.B., Anderson-Cook, C.M. (2008). A semi-parametric approach to robust parameter design. Journal of Statistical Planning and inference, 138, 114-131.

Robinson, T. J., Birch, J. and Alden Starnes, B. (2010). A semi-parametric approach to dual modeling when no replication exists. Journal of Statistical Planning and Inference, 140, 2860-2869.

Rahman, M., Gokhale, D.V., & Ullah, A. (1997). A Note on Combining Parametric and Nonparametric Regression. Communications in Statistics-Simulation and Computation, 26, 519-529.

Robinson T.J., & Birch, J.B. (2000). Model misspecification in parametric dual modeling. Journal of Statistical Computation and Simulation, 66, 113-126.

Ruppert, D., Wand, M.R., & Carroll, R. J. (2003). Semiparametric Regression. Cambridge University Press.Starnes, B.A. (1999). Asymptotic results for model robust regression. Unpublished dissertation, Virginia Polytechnic Institute and State University, Blacksburg, VA.