  We consider Wald's sequential probability ratio test for deciding whether a
sequence of independent and identically distributed observations comes from a
specified phase-type distribution or from an exponentially tilted alternative
distribution. In this setting, we derive exact decision boundaries for given
Type I and Type II errors by establishing a link with ruin theory. Information
on the mean sample size of the test can be retrieved as well. The approach
relies on the use of matrix-valued scale functions associated to a certain
one-sided Markov additive process. By suitable transformations the results also
apply to other types of distributions including some distributions with
regularly varying tail.
