Functional Linear Regression of Cumulative Distribution Functions
This work addresses risk assessment in predictions and decision-making by providing accurate CDF estimation, though it is incremental as it builds on existing functional regression methods.
The paper tackles the problem of estimating cumulative distribution functions (CDFs) through functional linear regression, achieving minimax optimal estimation error bounds of $\widetilde O(\sqrt{d/n})$ across various design settings and extending results to infinite-dimensional models.
The estimation of cumulative distribution functions (CDF) is an important learning task with a great variety of downstream applications, such as risk assessments in predictions and decision making. In this paper, we study functional regression of contextual CDFs where each data point is sampled from a linear combination of context dependent CDF basis functions. We propose functional ridge-regression-based estimation methods that estimate CDFs accurately everywhere. In particular, given $n$ samples with $d$ basis functions, we show estimation error upper bounds of $\widetilde O(\sqrt{d/n})$ for fixed design, random design, and adversarial context cases. We also derive matching information theoretic lower bounds, establishing minimax optimality for CDF functional regression. Furthermore, we remove the burn-in time in the random design setting using an alternative penalized estimator. Then, we consider agnostic settings where there is a mismatch in the data generation process. We characterize the error of the proposed estimators in terms of the mismatched error, and show that the estimators are well-behaved under model mismatch. Moreover, to complete our study, we formalize infinite dimensional models where the parameter space is an infinite dimensional Hilbert space, and establish a self-normalized estimation error upper bound for this setting. Notably, the upper bound reduces to the $\widetilde O(\sqrt{d/n})$ bound when the parameter space is constrained to be $d$-dimensional. Our comprehensive numerical experiments validate the efficacy of our estimation methods in both synthetic and practical settings.