  Loynes' distribution, which characterizes the one dimensional marginal of the
stationary solution to Lindley's recursion, possesses an ultimately exponential
tail for a large class of increment processes. If one can observe increments
but does not know their probabilistic properties, what are the statistical
limits of estimating the tail exponent of Loynes' distribution? We conjecture
that in broad generality a consistent sequence of non-parametric estimators can
be constructed that satisfies a large deviation principle. We present rigorous
support for this conjecture under restrictive assumptions and simulation
evidence indicating why we believe it to be true in greater generality.
