Estimating Loynes' exponent

TL;DR

This paper investigates the statistical limits of estimating the tail exponent of Loynes' distribution, which describes the stationary solution to Lindley's recursion, proposing a conjecture about non-parametric estimators and providing partial theoretical and simulation support.

Contribution

It introduces a conjecture that consistent non-parametric estimators for Loynes' tail exponent satisfy a large deviation principle, with partial proofs and simulation evidence.

Findings

Support for the conjecture under restrictive assumptions

Simulation evidence suggests broader applicability

Rigorous partial theoretical support provided

Abstract

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.