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In this paper, we study the functional linear multiplicative model based onthe least product relative error criterion. Under some regularizationconditions, we establish the consistency and asymptotic normality of theestimator. Further, we investigate the optimal subsampling for this model withmassive data. Both the consistency and the asymptotic distribution of thesubsampling estimator are first derived. Then, we obtain the optimalsubsampling probabilities based on the A-optimality criterion. Moreover, theuseful alternative subsampling probabilities without computing the inverse ofthe Hessian matrix are also proposed, which are easier to implement inpractise. Finally, numerical studies and real data analysis are done toevaluate the performance of the proposed approaches.

Least product relative error estimation for functional multiplicative model and optimal subsampling