Convergence rates of Laplace-transform based estimators

This paper considers the problem of estimating probabilities of the form P (Y = w), for a given value of w, in the situation that a sample of i.i.d. observations X_1, …, 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, …, X_n, applying the functional relation to it, and then (if possible) inverting the resulting Laplace transform and evaluating it in y. We show, under mild regularity conditions, that the resulting estimator is weakly consistent and has root mean-square error rate O(n^{-1/2} log n). 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’).