  Conformal regression provides prediction intervals with global coverage
guarantees, but often fails to capture local error distributions, leading to
non-homogeneous coverage. We address this with a new adaptive method based on
rescaling conformal scores with an estimate of local score distribution,
inspired by the Jackknife+ method, which enables the use of calibration data in
conformal scores without breaking calibration-test exchangeability. Our
approach ensures formal global coverage guarantees and is supported by new
theoretical results on local coverage, including an a posteriori bound on any
calibration score. The strength of our approach lies in achieving local
coverage without sacrificing calibration set size, improving the applicability
of conformal prediction intervals in various settings. As a result, our method
provides prediction intervals that outperform previous methods, particularly in
the low-data regime, making it especially relevant for real-world applications
such as healthcare and biomedical domains where uncertainty needs to be
quantified accurately despite low sample data.
