Convergence rates of Laplace-transform based estimators

TL;DR

This paper introduces a Laplace-transform based estimator for probabilities involving non-negative random variables, demonstrating its consistency and convergence rates, with applications in queueing theory and process decompounding.

Contribution

It develops a novel plug-in estimator leveraging Laplace transform relations, providing theoretical guarantees and practical examples in queueing and stochastic process analysis.

Findings

Estimator is weakly consistent under mild conditions.

Expected absolute error decreases at rate O(n^{-1/2} log(n+1)).

Applications include workload estimation in queues and decompounding of Poisson processes.

Abstract

This paper considers the problem of estimating probabilities of the form $\mathbb{P}(Y \leq w)$, for a given value of $w$, in the situation that a sample of i.i.d.\ observations $X_1, \ldots, X_n$ of $X$ is available, and where we explicitly know a functional relation between the Laplace transforms of the non-negative random variables $X$ and $Y$. A plug-in estimator is constructed by calculating the Laplace transform of the empirical distribution of the sample $X_1, \ldots, X_n$, applying the functional relation to it, and then (if possible) inverting the resulting Laplace transform and evaluating it in $w$. We show, under mild regularity conditions, that the resulting estimator is weakly consistent and has expected absolute estimation error $O(n^{-1/2} \log(n+1))$. We illustrate our results by two examples: in the first we estimate the distribution of the workload in an M/G/1 queue from observations of the input in fixed time intervals, and in the second we identify the distribution of the increments when observing a compound Poisson process at equidistant points in time (usually referred to as `decompounding').