Is distribution-free inference possible for binary regression? (Q2209818)

This is a study in the domain of distribution-free prediction. The aim is to investigate the extension of the distribution-free framework in the binary regression setting, i.e., to provide distribution-free inference on the conditional label probability \(\pi_P(X)=(\mathbb{P}_P[Y=1|X])\). The author proves that in the binary regression setting, any algorithm providing distribution-free coverage of \(\pi_P\)(\(X\)) must necessarily also cover the binary label \(Y\), for every distribution \(P\) that is nonatomic. Bounds on the length of a distribution-free confidence interval for binary regression are established. Further, the author analyses whether the lower bound on the confidence interval length can be achieved by a distribution-free method. A distribution-free algorithm is developed and the author proves that the constructed confidence interval provides a \((1-\alpha)\)-distribution-free confidence interval for the defined binary regression. An upper bound on its expected length is obtained. Appendix A is devoted to additional proofs connected to lower bounds and Appendix B to additional proofs connected to upper bounds.