Least absolute relative error estimation for functional quadratic multiplicative model

ABSTRACT

As an alternative to the functional quadratic model due to Yao and Müller (2010 Yao, F., Müller, H.-G. (2010). Functional quadratic regression. Biometrika 97:49–64.Crossref], Web of Science ®] , Google Scholar]), we consider a functional quadratic multiplicative model. This multiplicative model provides a useful alternative when the relative error is considered for analyzing data with positive responses. The existing work for functional models are mainly based on absolute errors. The commonly used least squares criterion is just such an example. In many practical applications, however, people concern on the size of relative error rather than that of error itself. Therefore, the estimation procedure based on least absolute relative errors, which is proposed by Chen et al. (2010 Chen, K., Guo, S., Lin, Y., Ying, Z. (2010). Least absolute relative error estimation. J. Am. Stat. Assoc. 105:1104–1112.Taylor & Francis Online], Web of Science ®] , Google Scholar]) for the linear multiplicative model, is developed for functional quadratic multiplicative model. The asymptotic behaviors of the proposed estimators are established. Some simulation studies show that the estimation procedure has good prediction performance. Moreover, a real data set is analyzed for illustrating the proposed methods.