20171101
Convergence rates of Laplacetransform based estimators
Publication
Publication
Bernoulli , Volume 23  Issue 4A p. 2533 2557
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 X1,..., Xn of X is available, and where we explicitly know a functional relation between the Laplace transforms of the nonnegative random variables X and Y. A plugin estimator is constructed by calculating the Laplace transform of the empirical distribution of the sample X1,..., Xn, 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(n1/2 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").
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doi.org/10.3150/16BEJ818  
Bernoulli  
Organisation  Centrum Wiskunde & Informatica, Amsterdam, The Netherlands 
den Boer, A.V, & Mandjes, M.R.H. (2017). Convergence rates of Laplacetransform based estimators. Bernoulli, 23(4A), 2533–2557. doi:10.3150/16BEJ818
