Prediction of Components in Random Sums

Abstract

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. PreliminariesMotivated by prediction problems in a Poisson shot noise process, we consider two types of problems for random sums of iid random variables (r.v. or r.v.'s for short). Let N be a nonnegative integer-valued r.v. and denote an iid sequence of non-negative r.v's by (X i denotes the total sum. The distributions of both N and X 1 are assumed to be known. Our problem is how we could obtain the information of the number N or each component X i when we only observe S N . Although there are several methods for these quantities such as linear predictions cS N with c some constant, our methods are those by conditional moments, which are minimizors of the mean square error. More precisely our focus is on the following two types of conditional moments:where (X i ) may take both real and integer values and N denotes the set of natural numbers as usual.This type of random sum S N has been studied for a long time and has applications in a variety of fields. One could find many examples in the book of Feller [5, XII] such as genetics, required service time, cosmic ray showers, and automobile accidents to name just a few. A large number of relevant researches have been conducted, including e.g. calculations for probability of S N (Sundt and Vernic [28]) or various limit theorems (see e.g. Gut [8] and consult a nice summary in Embrechts et al. [4, 2.5]). In recent years tail asymptotics have intensively studied, since accurate calculations of tail probabilities of S N are computationally quite expensive, while they are required in applications. See Jessen and Mikosch [9] for a survey with regularly varying tails and Goldie and Klüpperberg [6] for that with subexponential tails.In this paper we do not go further into asymptotics but investigate precise calculations of quantities (1.1), which have not been studied yet except for some special cases (see Subsection 1.1). We rely on two numerical methods, i.e. the Panjer recursion and the Fourier method, which are useful tools for computing P(S N = n) and which are competitive ([3]). The Panjer recursion scheme 2010 Mathematics Subject Classification. Primary 60G50 ; Secondary 60G25, 6...

Arrival times of Cox process with independent increment with application to prediction problems

“…Such prediction problems are studied lately e.g. in [13,15,25,26,16,14] (see also references therein) motivated by a non-life insurance application. In the model (1.1), T j may describe the arrival of a claim in an insurance portfolio and L j (t − T j ) t≥T j is the corresponding payment process from the insurer to the insured starting at time T j .…”