Instrumental variable approach to estimating the scalar-on-function regression model with measurement error with application to energy expenditure assessment in childhood obesity.

Wearable device technology allows continuous monitoring of biological markers and thereby enables study of time-dependent relationships. For example, in this paper, we are interested in the impact of daily energy expenditure over a period of time on subsequent progression toward obesity among children. Data from these devices appear as either sparsely or densely observed functional data and methods of functional regression are often used for their statistical analyses. We study the scalar-on-function regression model with imprecisely measured values of the predictor function. In this setting, we have a scalar-valued response and a function-valued covariate that are both collected at a single time period. We propose a generalized method of moments-based approach for estimation, while an instrumental variable belonging in the same time space as the imprecisely measured covariate is used for model identification. Additionally, no distributional assumptions regarding the measurement errors are assumed, while complex covariance structures are allowed for the measurement errors in the implementation of our proposed methods. We demonstrate that our proposed estimator is L2 consistent and enjoys the optimal rate of convergence for univariate nonparametric functions. In a simulation study, we illustrate that ignoring measurement error leads to biased estimations of the functional coefficient. The simulation studies also confirm our ability to consistently estimate the function-valued coefficient when compared to approaches that ignore potential measurement errors. Our proposed methods are applied to our motivating example to assess the impact of baseline levels of energy expenditure on body mass index among elementary school-aged children.