  We propose \textbf{JAWS}, a series of wrapper methods for distribution-free
uncertainty quantification tasks under covariate shift, centered on the core
method \textbf{JAW}, the \textbf{JA}ckknife+ \textbf{W}eighted with
data-dependent likelihood-ratio weights. JAWS also includes computationally
efficient \textbf{A}pproximations of JAW using higher-order influence
functions: \textbf{JAWA}. Theoretically, we show that JAW relaxes the
jackknife+'s assumption of data exchangeability to achieve the same
finite-sample coverage guarantee even under covariate shift. JAWA further
approaches the JAW guarantee in the limit of the sample size or the influence
function order under common regularity assumptions. Moreover, we propose a
general approach to repurposing predictive interval-generating methods and
their guarantees to the reverse task: estimating the probability that a
prediction is erroneous, based on user-specified error criteria such as a safe
or acceptable tolerance threshold around the true label. We then propose
\textbf{JAW-E} and \textbf{JAWA-E} as the repurposed proposed methods for this
\textbf{E}rror assessment task. Practically, JAWS outperform state-of-the-art
predictive inference baselines in a variety of biased real world data sets for
interval-generation and error-assessment predictive uncertainty auditing tasks.
