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Data-driven individualized decision making has recently received increasing research interest. However, most existing methods rely on the assumption of no unmeasured confounding, which cannot be ensured in practice especially in observational studies. Motivated by the recently proposed proximal causal inference, we develop several proximal learning methods to estimate optimal individualized treatment regimes (ITRs) in the presence of unmeasured confounding. Explicitly, in terms of two types of proxy variables, we are able to establish several identification results for different classes of ITRs respectively, exhibiting the tradeoff between the risk of making untestable assumptions and the potential improvement of the value function in decision making. Based on these identification results, we propose several classification-based approaches to finding a variety of restricted in-class optimal ITRs and establish their theoretical properties. The appealing numerical performance of our proposed methods is demonstrated via extensive simulation experiments and a real data application. for this article are available online. [ABSTRACT FROM AUTHOR]

Measuring and testing the dependency between multiple random functions is often an important task in functional data analysis. In the literature, a model-based method relies on a model which is subject to the risk of model misspecification, while a model-free method only provides a correlation measure which is inadequate to test independence. In this paper, we adopt the Hilbert–Schmidt Independence Criterion (HSIC) to measure the dependency between two random functions. We develop a two-step procedure by first pre-smoothing each function based on its discrete and noisy measurements and then applying the HSIC to recovered functions. To ensure the compatibility between the two steps such that the effect of the pre-smoothing error on the subsequent HSIC is asymptotically negligible when the data are densely measured, we propose a new wavelet thresholding method for pre-smoothing and to use Besov-norm-induced kernels for HSIC. We also provide the corresponding asymptotic analysis. The superior numerical performance of the proposed method over existing ones is demonstrated in a simulation study. Moreover, in a magnetoencephalography (MEG) data application, the functional connectivity patterns identified by the proposed method are more anatomically interpretable than those by existing methods. [ABSTRACT FROM AUTHOR]

Multidimensional function data arise from many fields nowadays. The covariance function plays an important role in the analysis of such increasingly common data. In this article, we propose a novel nonparametric covariance function estimation approach under the framework of reproducing kernel Hilbert spaces (RKHS) that can handle both sparse and dense functional data. We extend multilinear rank structures for (finite-dimensional) tensors to functions, which allow for flexible modeling of both covariance operators and marginal structures. The proposed framework can guarantee that the resulting estimator is automatically semipositive definite, and can incorporate various spectral regularizations. The trace-norm regularization in particular can promote low ranks for both covariance operator and marginal structures. Despite the lack of a closed form, under mild assumptions, the proposed estimator can achieve unified theoretical results that hold for any relative magnitudes between the sample size and the number of observations per sample field, and the rate of convergence reveals the phase-transition phenomenon from sparse to dense functional data. Based on a new representer theorem, an ADMM algorithm is developed for the trace-norm regularization. The appealing numerical performance of the proposed estimator is demonstrated by a simulation study and the analysis of a dataset from the Argo project. for this article are available online. [ABSTRACT FROM AUTHOR]