The proposed Goodness-of-Fit (GoF) test for checking the linear autocorrelation model in a functional time series is based on an empirical process, whose residual marks and covariate index set are in a separable Hilbert space H. A functional central limit theorem is derived providing the convergence of the empirical process to a time-changed Wiener process evaluated in a separable Hilbert space H, with subordinator given by the marginal probability of the involved strictly stationary Autoregressive Hilbertian process (ARH(1) process). The large sample behavior of the test statistics is obtained under simple and composite null hypotheses. The consistency of the test is addressed under simple null hypothesis. The finite-sample performance of the testing procedure, under different families of alternatives, and random projection schemes, is illustrated in the Supplementary Material.
Expected shortfall (ES) is defined as the average over the tail below (or above) a certain quantile of a probability distribution. Functional linear expected shortfall regression provides powerful tools for learning the relationship between a scalar response variable and functional covariates while exploring homogeneous and heterogeneous effects of the covariates. In this paper, we consider two types of ES regression approaches that model both quantile and ES simultaneously. The first approach employs a joint estimation procedure by minimizing a non-differentiable and non-convex joint loss function and obtains the estimates of the quantile and ES slope functions simultaneously. The second method adopts a two-step estimation procedure to mitigate computational complexity. In the first step, the quantile slope function is estimated using conventional quantile regression techniques. Then we derive the ES slope function via the least squares regression method in the second step. Under certain regularity conditions, we establish the convergence rates of slope functions and mean squared prediction error. Furthermore, extensions to the model with multiple curves are also considered. The superiority of two estimation methods is demonstrated by simulations and empirical studies based on intraday S&P 500 index data.
In this paper, we study multivariate linear regression models with both multivariate responses and covariates. Beyond parameter estimation, our primary objectives include identifying informative covariates and uncovering the underlying graphical structure among the responses, which naturally arise in multivariate data settings. A key practical challenge is the presence of measurement error in both responses and covariates, which, if ignored, can lead to biased estimators and misleading conclusions. To address this, we propose two valid estimation procedures that simultaneously correct for measurement error, perform covariate selection, and recover the graphical structure among responses. In addition, we extend two proposed estimation methods by incorporating repeated measurements and establish the theoretical properties of the proposed estimators. Numerical studies demonstrate the effectiveness of the proposed methods and highlight the importance of accounting for measurement error.
Interpretable functional coefficients are essential in functional linear regression models. In this paper, we introduce a method that yields functions estimates that are identically zero on domain subsets with positive Lebesgue measure, thereby explicitly identifying regions where the functional predictors have no effect on the response. The approach leverages a B-spline expansion combined with an Overlap Group Lasso penalty. To enhance accuracy and stability, the framework integrates adaptive group weights, an additional smoothness penalty, and supports the inclusion of scalar covariates. Estimation is formulated as a convex optimization problem and efficiently solved using an ADMM scheme. We characterize key properties of the approach, including a closed-form upper expression for the penalization parameter that guarantees a non-trivial solution, and an analytic expression for the effective degrees of freedom, enabling direct use of BIC-type model assessment. Empirical performance is investigated via simulation and illustrated on mid-infrared spectroscopy measurements from milk samples.
This work introduces a novel smoothing technique for functional data observed over space and time, based on a regularization term defined through a time-dependent nonlinear Partial Differential Equation (PDE). The proposed approach extends physics-informed statistical models, which have so far primarily relied on linear PDE-based penalties. By incorporating nonlinear PDEs into the regularization term, the method substantially broadens the applicability of smoothing models, while introducing new theoretical and computational challenges. Simulation studies illustrate the benefits of integrating physical knowledge into statistical modeling, with promising implications for real-world applications. The methodology is further demonstrated in a neuroimaging application, where nonlinear PDE regularization is used to model the progression of Alzheimer’s disease from longitudinal Positron Emission Tomography data.
