Parametric Inference on the Mean of Functional Data Applied to Lifetime Income Curves

We propose a framework for estimation of the conditional mean function in a parametric model with function space covariates. The approach employs a functional mean squared error objective criterion and allows for possible model misspecification. Under regulatory conditions, consistency and asymptotic normality are established. The analysis extends to situations where the asymptotic properties are influenced by estimation errors arising from the presence of nuisance parameters. Wald, Lagrange multiplier, and quasi-likelihood ratio statistics are studied and asymptotic theory is provided. These procedures enable inference about curve shapes in the observed functional data. Several model specifications where our results are useful are analyzed, including random coefficient models, distributional mixtures, and copula mixture models. Simulations exploring the finite sample properties of our methods are provided. An empirical application conducts lifetime income path comparisons across different demographic groups according to years of work experience. Gender and education levels produce differences in mean income paths corroborating earlier research. However, the mean income paths are found to be proportional so that, upon rescaling, the paths match over gender and across education levels.