  We develop a general framework for distribution-free predictive inference in
regression, using conformal inference. The proposed methodology allows for the
construction of a prediction band for the response variable using any estimator
of the regression function. The resulting prediction band preserves the
consistency properties of the original estimator under standard assumptions,
while guaranteeing finite-sample marginal coverage even when these assumptions
do not hold. We analyze and compare, both empirically and theoretically, the
two major variants of our conformal framework: full conformal inference and
split conformal inference, along with a related jackknife method. These methods
offer different tradeoffs between statistical accuracy (length of resulting
prediction intervals) and computational efficiency. As extensions, we develop a
method for constructing valid in-sample prediction intervals called {\it
rank-one-out} conformal inference, which has essentially the same computational
efficiency as split conformal inference. We also describe an extension of our
procedures for producing prediction bands with locally varying length, in order
to adapt to heteroskedascity in the data. Finally, we propose a model-free
notion of variable importance, called {\it leave-one-covariate-out} or LOCO
inference. Accompanying this paper is an R package {\tt conformalInference}
that implements all of the proposals we have introduced. In the spirit of
reproducibility, all of our empirical results can also be easily (re)generated
using this package.
