Heteroscedastic additive models - Estimating the fixed effects and covariance matrix parameters

Year 2021, Volume: 50 Issue: 2, 579 - 593, 11.04.2021

Adilson Silva , Miguel Fonseca

https://doi.org/10.15672/hujms.647481

Abstract

This work aims to deduce estimators for the unknown parameters of fixed effects and covariance matrix structure in heteroscedastic additive design. In order to do that, the design will be projected onto the orthogonal complement of the subspace spanned by columns of design matrix for the fixed effects, and the Kronecker product will be used to produced unbiased estimators for the parameters of covariance matrix, and then such estimators used to produce an estimator for the fixed effect vector. Moreover, the coefficient of determination for both fixed effects and covariance structure will be derived. A simulation study will be conducted, and a numerical example will be explored. 

Keywords

orthogonal matrix, Kronecker product, additive design, heteroscedasticity

Supporting Institution

New University of Lisbon

Project Number

PEst-OE/MAT/UI0297/2011

Thanks

To Prof. Dr. João T. Mexia for the guidance.

References

• [1] A.C. Atkinson and R.D. Cook, D-optimum designs for heteroscedastic linear models, Amer.Statist. Assoc. 90 (429), 204-212, 1995.
• [2] G.E. Battese and B.P. Bonyhady, Estimation of household expenditure functions: An application of a class of heteroscedastic regression models, Economic Record 57 (1), 1-22, 1981.
• [3] C. Brenton, Linear Models. The Theory and Application of Analysis of Variance, John Wiley & Sons, Inc., 2008.
• [4] M. Carapeto and W. Holt, Testing for heteroscedasticity in regression models, J. Appl. Stat. 30 (1), 13-20, 2003.
• [5] K.S. Gordon, An efficient algorithm for REML in heteroscedastic regression, J. Comput. Graph. Statist. 11 (4), 836-847, 2002.
• [6] A.C. Harvey, Estimating regression models with multiplicative heteroscedasticity, Econometrica 44 (3), 461-465, 1976.
• [7] S.D. Horn, R.A. Horn and D.B. Duncan, Estimating heteroscedastic variances in linear models, J. Amer. Statist. Assoc. 70 (350), 380-385, 1975.
• [8] D.J. Nott, M. Tran and C. Leng, Variational approximation for heteroscedastic linear models and matching pursuit algorithms, Stat. Comput. 22 (2), 497-512, 2012.
• [9] J. Schott, Matrix Analysis for Statistics, John Wiley & Sons, Inc., 1997.
• [10] A. Silva, Variance Components Estimation in Mixed Linear Models, Ph.D Thesis, New University of Lisbon, 2017.
• [11] A.H. Welsh, R.J. Carroll and D. Ruppert, Fitting heteroscedastic regression models, J. Amer. Statist. Assoc. 89 (425), 100-116, 1994.