Prediction of components in random sums

TL;DR

This paper develops numerical methods for predicting the number and size of iid components in a random sum, motivated by Poisson shot noise processes, using conditional techniques, Panjer recursion, and Fourier transforms.

Contribution

It introduces effective numerical procedures for predicting components in random sums, extending previous work limited by numerical difficulties.

Findings

Procedures work reasonably in numerical experiments

Methods are based on conditional techniques, Panjer recursion, Fourier transform

Application to compound mixed Poisson process suggested

Abstract

We consider predictions of the random number and the magnitude of each iid component in a random sum based on its distributional structure, where only a total value of the sum is available and where iid random components are non-negative. The problem is motivated by prediction problems in a Poisson shot noise process. In the context, although conditional moments are best possible predictors under the mean square error, only a few special cases have been investigated because of numerical difficulties. We replace the prediction problem of the process with that of a random sum, which is more general, and establish effective numerical procedures. The methods are based on conditional technique together with the Panjer recursion and the Fourier transform. In view of numerical experiments, procedures work reasonably. An application in the compound mixed Poisson process is also suggested.