Individual and life-span differences in charitable giving are an important economic force, yet the underlying motives are not well understood. In an adult, life-span sample, we assessed manifestations of prosocial tendencies across three different measurement domains: (a) psychological self-report measures, (b) actual giving choices, and (c) fMRI-derived, neural signals of “pure altruism”. The latter expressed individuals’ activity in valuation areas when charities received money compared to when oneself received money and thus reflected an altruistic concern for others. Results based both on structural equation modeling and unit-weighted aggregate scores revealed a strong higher-order General Benevolence dimension that accounted for variability across all measurement domains. The fact that the neural measures likely reflect pure altruistic tendencies indicates that General Benevolence is based on a genuine concern for others. Furthermore, General Benevolence exhibited a robust increase across the adult life-span, potentially providing one important explanation for why older adults typically contribute more to the public good than young adults.

We consider nonparametric regression in the context of functional data, that is, when a random sample of functions is observed on a fine grid. We obtain a functional asymptotic normality result allowing to build simultaneous confidence bands (SCB) for various estimation and inference tasks. Two applications to a SCB procedure for the regression function and to a goodness-of-fit test for curvilinear regression models are proposed. The first one has improved accuracy upon the other available methods while the second can detect local departures from a parametric shape, as opposed to the usual goodness-of-fit tests which only track global departures. A numerical study of the SCB procedures and an illustration with a speech data set are provided.

Summary Principal component analysis (PCA) is a method of choice for dimension reduction. In the current context of data explosion, online techniques that do not require storing all data in memory are indispensable to perform the PCA of streaming data and/or massive data. Despite the wide availability of recursive algorithms that can efficiently update the PCA when new data are observed, the literature offers little guidance on how to select a suitable algorithm for a given application. This paper reviews the main approaches to online PCA, namely, perturbation techniques, incremental methods and stochastic optimisation, and compares the most widely employed techniques in terms statistical accuracy, computation time and memory requirements using artificial and real data. Extensions of online PCA to missing data and to functional data are detailed. All studied algorithms are available in the  package onlinePCA on CRAN.

The mean function is a central object of inquiry in the analysis of functional data. Typical questions related to the mean function include quantifying estimation uncertainty, testing parametric models, and making comparisons between populations. To make probabilistic statements about the mean function over its entire domain, rather than at a single location, it is necessary to infer all of its values simultaneously. Pointwise inference is not appropriate for this task and indeed produces anticonservative results, i.e., the coverage level of confidence regions is too low and the significance level of hypothesis tests too high. In contrast, simultaneous confidence bands (SCB) provide a flexible framework for conducting simultaneous inference on the mean function and other functional parameters. They also offer powerful visualization tools for communicating analytic results to interdisciplinary audiences. The construction of SCB in the context of functional data requires specific theory and methods. In particular, it is not addressed by the nonparametric regression literature. Although software is available to perform individual steps of an SCB procedure, resources that provide end-to-end computations are scarce. Applications of SCB to one-and two-sample inferences are illustrated here with the R package SCBmeanfd.

When collections of functional data are too large to be exhaustively observed, survey sampling techniques provide an effective way to estimate global quantities such as the population mean function. Assuming functional data are collected from a finite population according to a probabilistic sampling scheme, with the measurements being discrete in time and noisy, we propose to first smooth the sampled trajectories with local polynomials and then estimate the mean function with a Horvitz-Thompson estimator. Under mild conditions on the population size, observation times, regularity of the trajectories, sampling scheme, and smoothing bandwidth, we prove a Central Limit theorem in the space of continuous functions. We also establish the uniform consistency of a covariance function estimator and apply the former results to build confidence bands for the mean function. The bands attain nominal coverage and are obtained through Gaussian process simulations conditional on the estimated covariance function. To select the bandwidth, we propose a cross-validation method that accounts for the sampling weights. A simulation study assesses the performance of our approach and highlights the influence of the sampling scheme and bandwidth choice.