A new regression model that combines the LLR estimates of both the raw response and its residuals

The modeling phase of response surface methodology (RSM) involves the use of regression models to estimate the functional relationship between the response and the explanatory variables using data obtained from a suitable experimental design. In RSM, the Ordinary Least Squares (OLS) is traditionally used to model the data via user-specified low-order polynomials. The OLS model is found to perform poorly if for instance the constant variance assumption is violated. Recently, nonparametric regression models, such as the Local Linear Regression (LLR), have been proposed to address the model inadequacy issue associated with the use of the OLS model. The LLR model is flexible, hence, can capture local trend and structure in the data that are missed by an inadequate OLS model. The successful application of the LLR model has been limited to studies with three unique features, namely: a single explanatory variable, fairly large sample sizes and space-filling designs. Therefore, the LLR model is scantily used in RSM which general underpinning include economy of data points (small sample size), typically sparse data, and oftentimes, more than one explanatory variables. In this paper, we propose a new nonparametric regression model that incorporates the smoothing of residuals to provide a second opportunity of fitting part of the data that is not captured by the LLR model. Using an example from the literature, it is observed that the goodness-of-fits of the proposed model are considerably better when compared with those of the OLS and the LLR models.

Aydin, D., Memmedli, M. and Omay, R. E. (2013). Smoothing parameter selection for nonparametric regression using smoothing spline. European Journal of Pure and Applied Mathematics, 6(2), 222 - 238.

Castillo, D. E. (2007). Process Optimization: A Statistical Method, Springer International Series in Operations Research and Management Science, New York.

Edionwe, E., Mbegbu, J. I. and Chinwe, R. (2016). A New Function for Generating Local Bandwidths for Semi-parametric MRR2 Model in Response Surface Methodology. Journal of Quality Technology, 48(4), pp. 388 – 404.

Fathi, H. K., Moghadam, M. B., Ahmadi, N., Fayaz, M. and Navaee, M. (2011). Application of Nonparametric Optimization Methods for Dyeing of Wool. Middle-East Journal of Scientific Research, 9(2), pp. 270 - 278.

Hastie, T., Tibshirani, R. and Friedman, T. A. (2009). The Elements of Statistical Learning, Data Mining, Inference and Prediction. Springer Series in Statistics, Second Edition, Springer, New York.

He, Z., Wang, J., Oh, J. and Park, S. H. (2009). Robust optimization for multiple responses using response surface methodology. Applied Stochastic Models in Business and Industry, 26, 157 – 171.

He, Z., Zhu, P. F. and Park, S. H. (2012). A robust desirability function for multi-response surface optimization. European Journal of Operational Research, 221, 241 - 247.

Kai, B., Li, R. and Zou, H. (2010). Local Composite Quantile Regression Smoothing: an efficient and safe alternative to the local polynomial regression. Journal of the Royal Statistical Society, 72, 49 – 69.

Mays, J., Birch, J. B. and Starnes, B. A., (2001). Model robust regression: combining parametric, nonparametric and semiparametric methods. Journals of Nonparametric Statistics, 13, 245 - 277.

Pickle, S. M. (2006). Semiparametric Techniques for Response Surface Methodology. Unpublished Ph.D. Dissertation, Department of Statistics, Virginia Polytechnic Institute & State University, Blacksburg, VA.

Prewitt, K. and Lohr, S. (2006). Bandwidth selection in local polynomial regression using eigenvalues. Journal of Royal Statistical Society, Series B, 68(1), 135 – 154.

Ramakrishnan, R. and Arumugam, R. (2012). Application to Response Surface Methodology (RSM) for Optimization of Operating Parameters and Performance Evaluation of Cooling Tower Cold Water Temperature. An International Journal of Optimization and Control, Theories and Applications, 1, 39 - 50.

Rivers, D. L. (2009). A Graphical Analysis of Simultaneously Choosing the Bandwidth and Mixing Parameter for Semiparametric Regression Techniques. Unpublished M.Sc. Dissertation, Virginia Commonwealth University, Richmond, VA 23284, USA.

Swamy, P. A. V. B., Tavlas, G. S., Hall, S. T. and Hondroyiannis, G. (2008). Estimation of Parameters in the Presence of Model Misspecification and Measurement Error, University of Leicester, No. 8-27.

Wan, W. (2007). Semi-parametric techniques for multi-response optimization. Unpublished Ph.D. Dissertation submitted to the Department of Statistics, Virginia Polytechnic Institute, and State University, Blacksburg, Virginia, USA.

Wan, W. and Birch, J. B. (2011). A semi-parametric technique for multi-response optimization. Quality and Reliability Engineering International, 27(1), 47 - 59.