Measures of influence for the functional linear model with scalar response

This paper studies how to identify influential observations in the functional linear model in which the predictor is functional and the response is scalar. Measurement of the effects of a single observation on estimation and prediction when the model is estimated by the principal components method is undertaken. For that, three statistics are introduced for measuring the influence of each observation on estimation and prediction of the functional linear model with scalar response that are generalizations of the measures proposed for the standard regression model by [D.R. Cook, Detection of influential observations in linear regression, Technometrics 19 (1977) 15–18; D. Peña, A new statistic for influence in linear regression, Technometrics 47 (2005) 1–12] respectively. A smoothed bootstrap method is proposed to estimate the quantiles of the influence measures, which allows us to point out which observations have the larger influence on estimation and prediction. The behavior of the three statistics and the quantile estimation bootstrap based method is analyzed via a simulation study. Finally, the practical use of the proposed statistics is illustrated by the analysis of a real data example, which show that the proposed measures are useful for detecting heterogeneity in the functional linear model with scalar response.