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Factorial designs are widely used due to their ability to accommodate multiple factors simultaneously. The factor-based regression with main effects and some interactions is the dominant strategy for downstream data analysis, delivering point estimators and standard errors via one single regression. Justification of these convenient estimators from the design-based perspective requires quantifying their sampling properties under the assignment mechanism conditioning on the potential outcomes. To this end, we derive the sampling properties of the factor-based regression estimators from both saturated and unsaturated models, and demonstrate the appropriateness of the robust standard errors for the Wald-type inference. We then quantify the bias-variance trade-off between the saturated and unsaturated models from the design-based perspective, and establish a novel design-based Gauss--Markov theorem that ensures the latter's gain in efficiency when the nuisance effects omitted indeed do not exist. As a byproduct of the process, we unify the definitions of factorial effects in various literatures and propose a location-shift strategy for their direct estimation from factor-based regressions. Our theory and simulation suggest using factor-based inference for general factorial effects, preferably with parsimonious specifications in accordance with the prior knowledge of zero nuisance effects.

Physical activity (PA) is an important risk factor for many health outcomes. Wearable-devices such as accelerometers are increasingly used in biomedical studies to understand the associations between PA and health outcomes. Statistical analyses involving accelerometer data are challenging due to the following three characteristics: (i) high-dimensionality, (ii) temporal dependence, and (iii) measurement error. To address these challenges we treat accelerometer-based measures of physical activity as a single function-valued covariate prone to measurement error. Specifically, in order to determine the relationship between PA and a health outcome of interest, we propose a regression model with a functional covariate that accounts for measurement error. Using regression calibration, we develop a two-step estimation method for the model parameters and establish their consistency. A test is also proposed to test the significance of the estimated model parameters. Simulation studies are conducted to compare the proposed methods with existing alternative approaches under varying scenarios. Finally, the developed methods are used to assess the relationship between PA intensity and BMI obtained from the National Health and Nutrition Examination Survey data.

We introduce ProtoPool, an interpretable image classification model with a pool of prototypes shared by the classes. The training is more straightforward than in the existing methods because it does not require the pruning stage. It is obtained by introducing a fully differentiable assignment of prototypes to particular classes. Moreover, we introduce a novel focal similarity function to focus the model on the rare foreground features. We show that ProtoPool obtains state-of-the-art accuracy on the CUB-200-2011 and the Stanford Cars datasets, substantially reducing the number of prototypes. We provide a theoretical analysis of the method and a user study to show that our prototypes are more distinctive than those obtained with competitive methods.

In the study of causal inference, statisticians show growing interest in estimating and analyzing heterogeneity in causal effects in observational studies. However, there usually exists a trade-off between accuracy and interpretability when developing a desirable estimator for treatment effects. To make efforts to address the issue, we propose a non-parametric framework for estimating the Conditional Average Treatment Effect (CATE) function in this paper. The framework integrates two components: (i) leverage the joint use of propensity and prognostic scores in a matching algorithm to obtain a proxy of the heterogeneous treatment effects for each observation, (ii) utilize non-parametric regression trees to construct an estimator for the CATE function conditioning on the two scores. The method naturally stratifies treatment effects into subgroups over a 2d grid whose axis are the propensity and prognostic scores. We conduct benchmark experiments on multiple simulated data and demonstrate clear advantages of the proposed estimator over state of the art methods. We also evaluate empirical performance in real-life settings, using two observational data from a clinical trial and a complex social survey, and interpret policy implications following the numerical results

Many physical and mathematical models involve random fields in their input data. Examples are ordinary differential equations, partial differential equations and integro--differential equations with uncertainties in the coefficient functions described by random fields. They also play a dominant role in problems in machine learning. In this article, we do not assume to have knowledge of the moments or expansion terms of the random fields but we instead have only given discretized samples for them. We thus model some measurement process for this discrete information and then approximate the covariance operator of the original random field. Of course, the true covariance operator is of infinite rank and hence we can not assume to get an accurate approximation from a finite number of spatially discretized observations. On the other hand, smoothness of the true (unknown) covariance function results in effective low rank approximations to the true covariance operator. We derive explicit error estimates that involve the finite rank approximation error of the covariance operator, the Monte-Carlo-type errors for sampling in the stochastic domain and the numerical discretization error in the physical domain. This permits to give sufficient conditions on the three discretization parameters to guarantee that an error below a prescribed accuracy $\varepsilon$ is achieved.

This paper introduces an Ordinary Differential Equation (ODE) notion for survival analysis. The ODE notion not only provides a unified modeling framework, but more importantly, also enables the development of a widely applicable, scalable, and easy-to-implement procedure for estimation and inference. Specifically, the ODE modeling framework unifies many existing survival models, such as the proportional hazards model, the linear transformation model, the accelerated failure time model, and the time-varying coefficient model as special cases. The generality of the proposed framework serves as the foundation of a widely applicable estimation procedure. As an illustrative example, we develop a sieve maximum likelihood estimator for a general semi-parametric class of ODE models. In comparison to existing estimation methods, the proposed procedure has advantages in terms of computational scalability and numerical stability. Moreover, to address unique theoretical challenges induced by the ODE notion, we establish a new general sieve M-theorem for bundled parameters and show that the proposed sieve estimator is consistent and asymptotically normal, and achieves the semi-parametric efficiency bound. The finite sample performance of the proposed estimator is examined in simulation studies and a real-world data example.

We propose a theoretical study of two realistic estimators of conditional distribution functions and conditional quantiles using random forests. The estimation process uses the bootstrap samples generated from the original dataset when constructing the forest. Bootstrap samples are reused to define the first estimator, while the second requires only the original sample, once the forest has been built. We prove that both proposed estimators of the conditional distribution functions are consistent uniformly a.s. To the best of our knowledge, it is the first proof of consistency including the bootstrap part. We also illustrate the estimation procedures on a numerical example.

Unbiased and consistent variance estimators generally do not exist for design-based treatment effect estimators because experimenters never observe more than one potential outcome for any unit. The problem is exacerbated by interference and complex experimental designs. In this paper, we consider variance estimation for linear treatment effect estimators under interference and arbitrary experimental designs. Experimenters must accept conservative estimators in this setting, but they can strive to minimize the conservativeness. We show that this task can be interpreted as an optimization problem in which one aims to find the lowest estimable upper bound of the true variance given one's risk preference and knowledge of the potential outcomes. We characterize the set of admissible bounds in the class of quadratic forms, and we demonstrate that the optimization problem is a convex program for many natural objectives. This allows experimenters to construct less conservative variance estimators, making inferences about treatment effects more informative. The resulting estimators are guaranteed to be conservative regardless of whether the background knowledge used to construct the bound is correct, but the estimators are less conservative if the knowledge is reasonably accurate.

We study a semidiscrete analogue of the Unified Transform Method introduced by A. S. Fokas, to solve initial-boundary-value problems for linear evolution partial differential equations with constant coefficients on the finite interval $x \in (0,L)$. The semidiscrete method is applied to various spatial discretizations of several first and second-order linear equations, producing the exact solution for the semidiscrete problem, given appropriate initial and boundary data. From these solutions, we derive alternative series representations that are better suited for numerical computations. In addition, we show how the Unified Transform Method treats derivative boundary conditions and ghost points introduced by the choice of discretization stencil and we propose the notion of "natural" discretizations. We consider the continuum limit of the semidiscrete solutions and compare with standard finite-difference schemes.

In this paper, we study the optimal convergence rate for distributed convex optimization problems in networks. We model the communication restrictions imposed by the network as a set of affine constraints and provide optimal complexity bounds for four different setups, namely: the function $F(\xb) \triangleq \sum_{i=1}^{m}f_i(\xb)$ is strongly convex and smooth, either strongly convex or smooth or just convex. Our results show that Nesterov's accelerated gradient descent on the dual problem can be executed in a distributed manner and obtains the same optimal rates as in the centralized version of the problem (up to constant or logarithmic factors) with an additional cost related to the spectral gap of the interaction matrix. Finally, we discuss some extensions to the proposed setup such as proximal friendly functions, time-varying graphs, improvement of the condition numbers.