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November 2017 Convergence rates of Laplace-transform based estimators
Arnoud V. den Boer, Michel Mandjes
Bernoulli 23(4A): 2533-2557 (November 2017). DOI: 10.3150/16-BEJ818

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”).

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Arnoud V. den Boer. Michel Mandjes. "Convergence rates of Laplace-transform based estimators." Bernoulli 23 (4A) 2533 - 2557, November 2017. https://doi.org/10.3150/16-BEJ818

Information

Received: 1 August 2015; Revised: 1 January 2016; Published: November 2017
First available in Project Euclid: 9 May 2017

zbMATH: 06778248
MathSciNet: MR3648037
Digital Object Identifier: 10.3150/16-BEJ818

Rights: Copyright © 2017 Bernoulli Society for Mathematical Statistics and Probability

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Vol.23 • No. 4A • November 2017
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