Convergence rates of Laplace-transform based estimators

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').