# Parameter Estimation Methods

## Abstract

This chapter covers topics related to estimation of model parameters and to model identification of univariate and multivariate problems not covered in earlier chapters. First, the fundamental notion of estimability of a model is introduced which includes both structural and numerical identifiability concepts. Recall that in Chap. 5 issues addressed were relevant to parameter estimation of linear models and to variable selection using step-wise regression in multivariate analysis. This chapter extends these basic notions by presenting the general statistical parameter estimation problem, and then presenting a few important estimation methods. Multivariate estimation methods (such as principle component analysis, ridge regression and stagewise regression) are discussed along with case study examples. Next, the error in variable (EIV) situation is treated when the errors in the regressor variables are large. Subsequently, another powerful and widely used estimation method, namely maximum likelihood estimation (MLE) is described, and its application to parameter estimation of probability functions and logistic models is presented. Also covered is parameter estimation of models non-linear in the parameters which can be separated into those that are transformable into linear ones, and those which are intrinsically non-linear. Finally, computer intensive numerical methods are discussed since such methods are being increasingly used nowadays because of the flexibility and robustness they provide. Different robust regression methods, whereby the influence of outliers on parameter estimation can be deemphasized, are discussed. This is, followed by the bootstrap resampling approach which is applicable for parameter estimation, and for ascertaining confidence limits of estimated model parameters and of model predictions.

### Keywords

Migration Toxicity Covariance Nicotine Marketing### References

- Andersen, K.K., and Reddy, T.A., 2002. “The error in variable (EIV) regression approach as a means of identifying unbiased physical parameter estimates: Application to chiller performance data”,
*HVAC&R Research Journal*, vol.8, no.3, pp. 295-309, July.CrossRefGoogle Scholar - Bard, Y., 1974. Nonlinear Parameter Estimation, Academic Press, New York.MATHGoogle Scholar
- Beck, J.V. and K.J., Arnold, 1977. Parameter Estimation in Engineering and Science, John Wiley and Sons, New YorkMATHGoogle Scholar
- Belsley, D.A., Kuh, E. and Welsch, R.E., 1980, Regression Diagnostics, John Wiley & Sons, New York.CrossRefMATHGoogle Scholar
- Chapra, S.C. and Canale, R.P., 1988. Numerical Methods for Engineers, 2nd Edition, McGraw-Hill, New York.Google Scholar
- Chatfield, C., 1995. Problem Solving: A Statistician’s Guide, 2nd Ed., Chapman and Gall, London, U.K.MATHGoogle Scholar
- Chatterjee, S. and B. Price, 1991. Regression Analysis by Example, 2nd Edition, John Wiley & Sons, New YorkGoogle Scholar
- Devore J., and N. Farnum, 2005. Applied Statistics for Engineers and Scientists, 2nd Ed., Thomson Brooks/Cole, Australia.Google Scholar
- Davison, A.C. and D.V.Hinkley, 1997. Bootstrap Methods and their Applications, Cambridge University Press, CambridgeCrossRefGoogle Scholar
- Draper, N.R. and H. Smith, 1981. Applied Regression Analysis, 2nd Ed., John Wiley and Sons, New York.MATHGoogle Scholar
- Edwards, C.H. and D.E. Penney, 1996. Differential Equations and Boundary Value Problems, Prentice Hall, Englewood Cliffs, NJMATHGoogle Scholar
- Efron, B. and J. Tibshirani, 1982. An Introduction to the The Bootstrap, Chapman and Hall, New YorkGoogle Scholar
- Freedman, D.A., and S.C. Peters, 1984. Bootstrapping a regression equation: Some empirical results,
*J. of the American Statistical Association,*79(385), pp.97-106, March.CrossRefMathSciNetGoogle Scholar - Fuller, W. A., 1987. Measurement Error Models, John Wiley & Sons, NY.CrossRefMATHGoogle Scholar
- Godfrey, K., 1983. Compartmental Models and Their Application, Academic Press, New York.Google Scholar
- Kachigan, S.K., 1991. Multivariate Statistical Analysis, 2nd Ed., Radius Press, New York.Google Scholar
- Kowalski, W.J., 2002. Immune Building Systems Technology, McGraw-Hill, New YorkGoogle Scholar
- Lipschutz, S., 1966. Finite Mathematics, Schaum’s Outline Series, McGraw-Hill, New YorkGoogle Scholar
- Mandel, J., 1964. The Statistical Analysis of Experimental Data, Dover Publications, New York.Google Scholar
- Manly, B.J.F., 2005. Multivariate Statistical Methods: A Primer, 3rd Ed., Chapman & Hall/CRC, Boca Raton, FLGoogle Scholar
- Masters, G.M. and W.P. Ela, 2008. Introduction to Environmental Engineering and Science,3rd Ed. Prentice Hall, Englewood Cliffs, NJGoogle Scholar
- Mullet, G.M., 1976. Why regression coefficients have the wrong sign,
*J. Quality Technol.*, 8(3).Google Scholar - Pindyck, R.S. and D.L. Rubinfeld, 1981. Econometric Models and Economic Forecasts, 2nd Edition, McGraw-Hill, New York, NY.Google Scholar
- Reddy, T.A., and D.E. Claridge, 1994 “Using synthetic data to evaluate multiple regression and principal component analysis for statistical models of daily energy consumption”,
*Energy and Buildings*, Vol.21, pp. 35-44.Google Scholar - Reddy, T.A., S. Deng, and D.E. Claridge, 1999. “Development of an inverse method to estimate overall building and ventilation parameters of large commercial buildings”,
*ASME Journal of Solar Energy Engineering*, vol.121, pp.40-46, February.Google Scholar - Reddy, T.A. and K.K. Andersen, 2002. “An evaluation of classical steady-state off-line linear parameter estimation methods applied to chiller performance data
*”, HVAC&R Research Journal*, vol.8, no.1, pp.101-124.Google Scholar - Reddy, T.A., 2007. Application of a generic evaluation methodology to assess four different chiller FDD Methods (RP1275),
*HVAC&R Research Journal*, vol.13, no.5, pp 711-729, September.Google Scholar - Saunders, D.H., J.J. Duffy, F.S. Luciani, W. Clarke, D. Sherman, L. Kinney, and N.H. Karins, 1994. “Measured performance rating method for weatherized homes”
*, ASHRAE Trans.,*V.100, Pt.1, paper no. 94-25-3.Google Scholar - Shannon, R.E., 1975. System Simulation: The Art and Science, Prentice-Hall, Englewood Cliffs, NJ.Google Scholar
- Sinha, N.K. and B. Kuszta, 1983. Modeling and Identification of Dynamic Systems, Van Nostrand Reinhold Co., New York.Google Scholar
- Subbarao, K., 1988. “PSTAR-Primary and Secondary Terms Analysis and Renormalization: A Unified Approach to Building Energy Simulations and Short-Term Monitoring”, SERI/TR - 253- 3175, Solar Energy Research Institute, Golden CO.Google Scholar
- Walpole, R.E., R.H. Myers, S.L. Myers, and K. Ye, 2007, Probability and Statistics for Engineers and Scientists, 8th Ed., Prentice-Hall, Upper Saddle River, NJ.Google Scholar