Next-generation functional data arise from many fields nowadays, such as imaging data or spatial data in medical research. The covariance function takes an important role in analysis, serving as the first step for principal component decomposition and dimension reduction. In our paper, we propose a bivariate spline estimator for the covariance function based on discretely observed, error prone raw next-generation functional data. Flexible bivariate splines over triangulations are employed to accommodate the irregular domains that frequently appear in such data. The proposed covariance estimator enjoys oracle property in the sense that it is as efficient as the infeasible estimator when all individual images are entirely observed without errors. Both local and uniform weak convergence are investigated under some mild conditions, with the added bonus of asymptotic simultaneous confidence regions (SCRs). Extensive simulation experiments yield appealing results that strongly corroborate the asymptotic theory. The proposed method is finally illustrated by analyzing the brain positron emission tomography data from the Alzheimer’s Disease Neuroimaging Initiative (ADNI) database.

The Median-of-Mean (MoM) estimation is an efficient statistical method for handling data with contamination. In this paper, we propose a variance-dependent MoM estimation method using the tail probability of a binomial distribution. The bound of this method is better than the classical Hoeffding method under mild conditions. This method is then used to study the concentration of variance-dependent MoM empirical processes and sub-Gaussian intrinsic moment norm. Finally, we give the bound of the variance-dependent MoM estimator with distribution-free contaminated data.

We propose a new non-parametric conditional independence test for a scalar response and a functional covariate over a continuum of quantile levels. We build a Cramer–von Mises type test statistic based on an empirical process indexed by random projections of the functional covariate, effectively avoiding the “curse of dimensionality” under the projected hypothesis, which is almost surely equivalent to the null hypothesis. The asymptotic null distribution of the proposed test statistic is obtained under some mild assumptions. The asymptotic global and local power properties of our test statistic are then investigated. We specifically demonstrate that the statistic is able to detect a broad class of local alternatives converging to the null at the parametric rate. Additionally, we recommend a simple multiplier bootstrap approach for estimating the critical values. The finite-sample performance of our statistic is examined through several Monte Carlo simulation experiments. Finally, an analysis of an EEG data set is used to show the utility and versatility of our proposed test statistic.