Core Concepts
The author explores functional linear regression methods for estimating cumulative distribution functions, providing theoretical bounds and optimality.
Abstract
The content delves into the study of functional linear regression for estimating cumulative distribution functions (CDFs), proposing ridge-regression-based methods with upper bounds on estimation errors. The analysis covers various design settings, including fixed, random, and adversarial contexts. The paper also discusses agnostic settings and infinite-dimensional models, showcasing the efficacy of the proposed estimators through numerical experiments. Key points include the importance of CDF estimation in risk assessment and decision-making applications, the development of least-squares and ridge regression estimators for contextual CDF bases, and the establishment of minimax optimality for CDF functional regression.
Stats
Given n samples with d basis functions, estimation error upper bounds scale like rOp{a}d{nq.
For any positive definite matrix A in Rdˆd, the weighted ℓ2-norm is defined as }x}A = xJAx.
The KS distance between two CDFs F1 and F2 is denoted by KSpF1, F2q.
The estimator pθλ in (2) minimizes the squared L2-distance between estimated and empirical CDFs.
In Scheme I (Adversarial), samples are generated from convex combinations of context-dependent CDF bases.