Statistics Theory

stat.TH — Statistics Theory

Binary Regression for BV Class Conditional Probability

Contributed by Rajesh Dachiraju

In this article we solve the problem of binary regression when the class conditional probability is a multivariate function of bounded variation. We study the consistency of binary regression under a nonparametric sampling framework. Sufficient conditions are established for the convergence of the binary regression estimator, and quantitative error estimates are obtained. We further strengthen the analysis by introducing the assumption that the feature vectors possess a point density measure of bounded variation. Under this geometric regularity assumption, the expectation-based error estimate is replaced by a deterministic estimate, yielding deterministic convergence in the $L^{2}$ norm and, consequently, almost sure convergence. The analysis demonstrates that the geometric distribution of the sampling points plays a fundamental role in the approximation properties of the estimator. Although the analysis is carried out on the torus $\mathbb{T}^{m}$, the results extend to arbitrary bounded Lipschitz domains after affine scaling, embedding into $\mathbb{T}^{m}$, and zero extension. The point density measure framework provides a deterministic geometric perspective on binary regression and establishes a connection between sampling geometry, bounded variation, and convergence theory.