Asymptotics of conditional moments of the summand in Poisson compounds

Methodol Comput Appl Probab

2016

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...

Prediction of Components in Random Sums

Methodol Comput Appl Probab

2016

Prediction in a non-homogeneous Poisson cluster model

Insurance: Mathematics and Economics

2014

A non-homogeneous Poisson cluster model is studied, motivated by insurance applications. The Poisson center process which expresses arrival times of claims, triggers off cluster member processes which correspond to number or amount of payments. The cluster member process is an additive process. Given the past observations of the process we consider expected values of future increments and their mean squared errors, aiming the application in claims reserving problems. Our proposed process can cope with non-homogeneous observations such as the seasonality of claims arrival or the reducing property of payment processes, which are unavailable in the former models where both center and member processes are time homogeneous. Hence results presented in this paper are significant extensions toward applications. We also give numerical examples to show how non-homogeneity appears in predictions.where 0 < T 1 < T 2 < · · · are points of non-homogeneous Poisson (N P for short) processes N (t) and (L j ), j = 1, 2, . . . are an iid sequence of additive processes with L j (t) = 0, a.s. for t ≤ 0, such that (T j ) and (L j ) are independent. In the insurance context T j ≤ 1 would be the arrival of claims within a year, and (L j (t − T j )) j:T j ≤1 are the corresponding payment processes from an insurance company to policyholders. We could also regard the cluster as the counting process of payment number. Hence M (t) would be the total number or amount of payments for the claims arriving in a year and being paid in the interval [0, t], t ≥ 1. Historically such kind of stochastic process modeling goes back to Lundberg (1903) (see comments in [11, p.224]) who introduced the Poisson process for a simple claim counting process. Norberg [12,13] has been considered to give publicity to the point process approach in a non-life insurance context.Our focus in this paper is on the prediction of the future incrementsfor some suitable σ-fields F t i.e. we will calculate E[M (t, t + s] | F t ] and evaluate the mean squared error of the prediction. These kind of problems are known to be claims reserving problems, which

Prediction in a Poisson cluster model with multiple cluster processes

Scandinavian Actuarial Journal

2013

We consider a simple but flexible extension of the Poisson cluster model studied in Matsui & Mikosch (2010). In the former model, only a single cluster process starts at each jump point of the Poisson process, whereas we start a randomly given number of cluster processes at each jump. This simple extension yields additional mathematical problems in prediction of future increments of the process which are based on the past observations. However, by making full use of the Poisson structure of the model, we derive reasonably explicit expressions for predictors, which is of critical importance in the insurance application. Some comparisons of predictors are also made by their mean-squared errors when the cluster process is a compound Poisson process. The result yields a natural conclusion that the finer information we use, the better predictors we obtain.