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Regression splines are smooth, flexible, and parsimonious nonparametric function estimators. They are known to be sensitive to knot number and placement, but if assumptions such as monotonicity or convexity may be imposed on the regression…
We consider the problem of approximating smoothing spline estimators in a nonparametric regression model. When applied to a sample of size $n$, the smoothing spline estimator can be expressed as a linear combination of $n$ basis functions,…
In this paper, we propose a new horseshoe-type prior hierarchy for adaptively shrinking spline-based functional effects towards a predefined vector space of parametric functions. Instead of shrinking each spline coefficient towards zero, we…
We propose robust methods for inference on the effect of a treatment variable on a scalar outcome in the presence of very many controls. Our setting is a partially linear model with possibly non-Gaussian and heteroscedastic disturbances.…
In state space models, smoothing refers to the task of estimating a latent stochastic process given noisy measurements related to the process. We propose an unbiased estimator of smoothing expectations. The lack-of-bias property has…
In this paper, we propose an adaptive smoothing spline (AdaSS) estimator for the function-on-function linear regression model where each value of the response, at any domain point, depends on the full trajectory of the predictor. The AdaSS…
For the problem of nonparametric regression of smooth functions, we reconsider and analyze a constrained variational approach, which we call the MultIscale Nemirovski-Dantzig (MIND) estimator. This can be viewed as a multiscale extension of…
Due to developments in instruments and computers, functional observations are increasingly popular. However, effective methodologies for flexibly estimating the underlying trends with valid uncertainty quantification for a sequence of…
We establish sensitivity analysis on the sphere. We present formulas that allow us to decompose a function $f\colon \mathbb S^d\rightarrow \mathbb R$ into a sum of terms $f_{\boldsymbol u,\boldsymbol \xi}$. The index $\boldsymbol u$ is a…
This article studies local and global inference for smoothing spline estimation in a unified asymptotic framework. We first introduce a new technical tool called functional Bahadur representation, which significantly generalizes the…
In this paper we propose an approximation method for high-dimensional $1$-periodic functions based on the multivariate ANOVA decomposition. We provide an analysis on the classical ANOVA decomposition on the torus and prove some important…
We study the semiparametric efficient estimation of a class of linear functionals in settings where a complete multivariate dataset is supplemented by additional datasets recording subsets of the variables of interest. These datasets are…
We consider estimation and inference in a single index regression model with an unknown but smooth link function. In contrast to the standard approach of using kernels or regression splines, we use smoothing splines to estimate the smooth…
Diffusion models achieve remarkable generation quality, yet face a fundamental challenge known as memorization, where generated samples can replicate training samples exactly. We develop a theoretical framework to explain this phenomenon by…
Testing for the significance of a subset of regression coefficients in a linear model, a staple of statistical analysis, goes back at least to the work of Fisher who introduced the analysis of variance (ANOVA). We study this problem under…
The mean dimension of a black box function of $d$ variables is a convenient way to summarize the extent to which it is dominated by high or low order interactions. It is expressed in terms of $2^d-1$ variance components but it can be…
Mediation analysis is widely used for exploring treatment mechanisms; however, it faces challenges when nonignorable missing confounders are present. Efficient inference of mediation effects and the efficiency loss due to nonignorable…
We propose nonparametric estimators for the second-order central moments of possibly anisotropic spherical random fields, within a functional data analysis context. We consider a measurement framework where each random field among an…
In this article, we investigate the problem of estimating a spatially inhomogeneous function and its derivatives in the white noise model using Besov-Laplace priors. We show that smoothness-matching priors attains minimax optimal posterior…
Motivated by recent data analyses in biomedical imaging studies, we consider a class of image-on-scalar regression models for imaging responses and scalar predictors. We propose using flexible multivariate splines over triangulations to…