  This article proposes an alternative to the Hosmer-Lemeshow (HL) test for
evaluating the calibration of probability forecasts for binary events. The
approach is based on e-values, a new tool for hypothesis testing. An e-value is
a random variable with expected value less or equal to one under a null
hypothesis. Large e-values give evidence against the null hypothesis, and the
multiplicative inverse of an e-value is a p-value. Our test uses online
isotonic regression to estimate the calibration curve as a `betting strategy'
against the null hypothesis. We show that the test has power against
essentially all alternatives, which makes it theoretically superior to the HL
test and at the same time resolves the well-known instability problem of the
latter. A simulation study shows that a feasible version of the proposed eHL
test can detect slight miscalibrations in practically relevant sample sizes,
but trades its universal validity and power guarantees against a reduced
empirical power compared to the HL test in a classical simulation setup.We
illustrate our test on recalibrated predictions for credit card defaults during
the Taiwan credit card crisis, where the classical HL test delivers equivocal
results.
