scholarly journals Statistical Methodology for Large Claims

1977 ◽  
Vol 9 (1-2) ◽  
pp. 1-9 ◽  
Author(s):  
J. Tiago de Oliveira
Keyword(s):  
Distribution Function ◽  
Poisson Process ◽  
Critical Level ◽  
Compound Poisson Process ◽  
Distribution Functions ◽  
Asymptotic Distributions ◽  
Jump Function ◽  
Image Position ◽  
Asymptotic Forms ◽  
Independent And Identically Distributed

The question of large claims in insurance is, evidently, a very important one, chiefly if we consider it in relation with reinsurance. To a statistician it seems that it can be approached, essentially, in two different ways.The first one can be the study of overpassing of a large bound, considered to be a critical one. If N(t) is the Poisson process of events (claims) of intensity v, each claim having amounts Yi, independent and identically distributed with distribution function F(x), the compound Poisson processwhere a denotes the critical level, can describe the behaviour of some problems connected with the overpassing of the critical level. For instance, if h(Y, a) = H(Y − a), where H(x) denotes the Heavside jump function (H(x) = o if x < o, H(x) = 1 if x ≥ o), M(t) is then the number of claims overpassing a; if h(Y, a) = Y H(Y − a), M(t) denotes the total amount of claims exceeding the critical level; if h(Y, a) = (Y − a) H(Y − a), M(t) denotes the total claims reinsured for some reinsurance policy, etc.Taking the year as unit of time, the random variables M(1), M(2) − M(1), … are evidently independent and identically distributed; its distribution function is easy to obtain through the computation of the characteristic function of M(1). For details see Parzen (1964) and the papers on The ASTIN Bulletin on compound processes; for the use of distribution functions F(x), it seems that the ones developed recently by Pickands III (1975) can be useful, as they are, in some way, pre-asymptotic forms associated with tails, leading easily to the asymptotic distributions of extremes.

scholarly journals A Generalized Model for the Risk Process and its Application to a Tentative Evaluation of Outstanding Liabilities

1965 ◽  
Vol 3 (3) ◽  
pp. 215-238 ◽  
Author(s):  
Carl Philipson
Keyword(s):  
Poisson Process ◽  
Compound Poisson Process ◽  
Random Variable ◽  
Weak Sense ◽  
Risk Process ◽  
Distribution Functions ◽  
Wide Sense ◽  
Risk Distribution ◽  
Image Position ◽  
The Right

A compound Poisson process, in this context abbreviated to cPp, is defined by a probability distribution of the number m of events in the interval (o, τ) of the original scale of the process parameter, assumed to be one-dimensional, in the following form.where du shall be inserted for t, λτ being the intensity function of a Poisson process with the expected number t of events in the interval (O, τ) and U(ν, τ) is the distribution function of ν for every fixed value of τ, here called the risk distribution. If the inverse of is substituted for τ, in the right membrum of (1), the function obtained is a function of t.If the risk distribution is defined by the general form U(ν, τ) the process defined by (1) is called a cPp in the wide sense (i.w.s.). In the sequel two particular cases for U(ν, τ) shall be considered, namely when it has the form of distribution functions, which define a primary process being stationary (in the weak sense) or non-stationary, and when it is equal to U1(ν) independently of τ. The process defined by (1) is in these cases called a stationary or non-stationary (s. or n.s.)cPp and a cPpin the narrow sense (i.n.s.) respectively. If a process is non-elementary i.e. the size of one change in the random function constituting the process is a random variable, the distribution of this variable conditioned by the hypothesis that such a change has occurred at τ is here called the change distribution and denoted by V(x, τ), or, if it is independent of τ, by V1(x). In an elementary process the size of one change is a constant, so that, in this case, the change distribution reduces to the unity distribution E(x — k), where E(ξ) is equal to I, o, if ξ is non-negative, negative respectively, and k is a given constant.

scholarly journals On the Numerical Calculation of the Distribution Functions Defining Some Compound Poisson Processes

1963 ◽  
Vol 3 (1) ◽  
pp. 20-42 ◽  
Author(s):  
Carl Philipson
Keyword(s):  
Probability Distribution ◽  
Poisson Process ◽  
Random Function ◽  
Compound Poisson Process ◽  
Distribution Functions ◽  
Wide Sense ◽  
Stieltjes Integral ◽  
Compound Poisson ◽  
Risk Distribution ◽  
Image Position

1. The comfound Poisson process in the wide sense is defined as a process for which the probability distribution of the number i of changes in the random function attached to the process, while the parameter passes from o to a fixed value τ of the parameter measured on a suitable scale, is given by the Laplace-Stieltjes integral where U(ν, τ) for a fixed value of τ defines the distribution of ν. U(ν, τ) is called the risk distribution and is either τ-independent or, dependent on ν, τ.2. The compound Poisson process in the narrow sense is defined as a process for which the probability distribution of the number of changes can be written in the form of (I) with a τ-independent risk distribution.In their general form these processes have been analyzed by Ove Lundberg (1940). For such processes the following relation holds for the probability of i changes in the interval ο to τ, P̅i (τ) say this relation does not hold for processes with τ-dependent risk distribution. Hofmann (1955) has introduced a sub-set of the processes concerned in this section for which the probability for non-occurrence of a change in the interval o to τ is defined as a solution of the differential equation and ϰ ≥ o; the solutions may be written in the form where η is independent of and of two alternative forms one for ϰ = I and one for other values of ϰ. The probabilities for i changes in the interval o to τ in the processes defined by the solutions of Hofmann's equation are derived by Leibniz's formula, and are designated by and, in this paper, called Hofmann probabilities.

scholarly journals On the rate of convergence of waiting times

1965 ◽  
Vol 5 (3) ◽  
pp. 365-373 ◽  
Author(s):  
C. K. Cheong ◽  
C. R. Heathcote
Keyword(s):  
Distribution Function ◽  
Rate Of Convergence ◽  
Waiting Times ◽  
Distribution Functions ◽  
Initial Condition ◽  
Hopf Equation ◽  
Unknown Distribution ◽  
Image Position

Let K(y) be a known distribution function on (−∞, ∞) and let {Fn(y), n = 0, 1,…} be a sequence of unknown distribution functions related by subject to the initial condition If the sequence {Fn(y)} converges to a distribution function F(y) then F(y) satisfies the Wiener-Hopf equation

Asymptotic distributions of weighted compound Poisson bridges

1992 ◽  
Vol 112 (3) ◽  
pp. 613-629 ◽  
Author(s):  
Barbara Szyszkowicz
Keyword(s):  
Poisson Process ◽  
Random Variables ◽  
Asymptotic Distributions ◽  
Independent Random Variables ◽  
Intensity Parameter ◽  
Compound Poisson ◽  
Image Position

Let S(N(t)) be defined bywhere {N(t), t ≥ 0} is a Poisson process with intensity parameter 1/μ > 0 and {Xi i ≥ 1} is a family of independent random variables which are also independent of {N(t), t ≥ 0}.

High density shot noise and gaussianity

1971 ◽  
Vol 8 (1) ◽  
pp. 118-127 ◽  
Author(s):  
A. Papoulis
Keyword(s):  
Distribution Function ◽  
Poisson Process ◽  
Shot Noise ◽  
Average Density ◽  
Normal Process ◽  
Noise Process ◽  
Shot Noise Process ◽  
Mean And Variance ◽  
Image Position ◽  
Random Times

The distance from Gaussianity of the shot noise process is considered, where ti are the random times of a Poisson process with average density λ(t). With F(x) the distribution function of x(t) and G(x) that of a normal process with the same mean and variance as x(t) it is shown that where If the process x(t) is stationary with λ(t) =λ and h(t, τ) = h(t – τ) and the function h(t) is bandlimited by ωc, then the above yields

scholarly journals On a transformation of the weighted compound Poisson process

1971 ◽  
Vol 6 (1) ◽  
pp. 42-46 ◽  
Author(s):  
Hans Bühlmann ◽  
Roberto Buzzi
Keyword(s):  
Poisson Process ◽  
Compound Poisson Process ◽  
Random Variable ◽  
Characteristic Functions ◽  
Compound Poisson ◽  
Common Distribution ◽  
Common Distribution Function ◽  
Poisson Variable ◽  
Image Position ◽  
Independent Identically Distributed

We are using the following terminology—essentially following Feller:a) Compound Poisson VariableThis is a random variable where X1, X2, … Xn, … independent, identically distributed (X0 = o) and N a Poisson counting variablehence(common) distribution function of the Xj with j ≠ 0 or in the language of characteristic functionsb) Weighted Compound Poisson VariableThis is a random variable Z obtained from a class of Compound Poisson Variables by weighting over λ with a weight function S(λ)henceor in the language of characteristic functionsLet [Z(t); t ≥ o] be a homogeneous Weighted Compound Poisson Process. The characteristic function at the time epoch t reads thenIt is most remarkable that in many instances φt(u) can be represented as a (non weighted) Compound Poisson Variable. Our main result is given as a theorem.

Stability of extremes with random sample size

1989 ◽  
Vol 26 (04) ◽  
pp. 734-743 ◽  
Author(s):  
W. J. Voorn
Keyword(s):  
Distribution Function ◽  
Sample Size ◽  
Random Sample ◽  
Distribution Functions ◽  
Random Sample Size ◽  
Maximum Value ◽  
Arbitrary Real Constant ◽  
Independent Observations ◽  
Degenerate Distribution ◽  
Image Position

A non-degenerate distribution function F is called maximum stable with random sample size if there exist positive integer random variables Nn, n = 1, 2, ···, with P(Nn = 1) less than 1 and tending to 1 as n → ∞ and such that F and the distribution function of the maximum value of Nn independent observations from F (and independent of Nn ) are of the same type for every index n. By proving the converse of an earlier result of the author, it is shown that the set of all maximum stable distribution functions with random sample size consists of all distribution functions F satisfying where c 2, c 3, · ·· are arbitrary non-negative constants with 0 &lt; c2 + c3 + · ·· &lt;∞, and all distribution functions G and H defined by F(x)= G(c + exp(x)) and F(x) = H(c – exp(–x)), –∞ &lt; x &lt;∞, where c is an arbitrary real constant.

scholarly journals The Effect of Reinsurance on the Degree of Risk Associated with an Insurer's Portfolio

1980 ◽  
Vol 11 (2) ◽  
pp. 119-135 ◽  
Author(s):  
M. Andreadakis ◽  
H. R. Waters
Keyword(s):  
Distribution Function ◽  
Poisson Process ◽  
Poisson Distribution ◽  
Compound Poisson Distribution ◽  
Proportional Reinsurance ◽  
Insurance Portfolio ◽  
Degree Of Risk ◽  
Definition Of ◽  
Image Position ◽  
Annual Premium

There are many reasons why an insurer may choose to reinsure a part of his portfolio (see, for example, Carter (1979, p. 5 ff.)) and many ways in which he can assess the effectiveness of the reinsurance arrangements he makes. In this paper we assume the insurer wishes to reinsure a part of his portfolio in order to reduce its “riskiness”. We take as given a portfolio consisting of n independent risks together with the total premium charged to insure these risks and we investigate the effect on the degree of risk associated with the portfolio (see §3 for a definition) of varying the excess of loss or proportional reinsurance limits for each risk.We are given an insurance portfolio consisting of n independent risks. A risk may consist of a single policy or a group of policies: the essential points being that a reinsurance limit, either excess of loss or proportional, is the same for all claims arising from a particular risk, although reinsurance limits may vary from one risk to another. We assume the claims arising from each risk have a compound Poisson distribution. To be more precise, we assume the number of claims arising from the i-th risk is a Poisson process with mean Pi claims each year and the size of each claim has distribution function Fi. As usual, the size of a claim is independent of the time at which it occurs and of all other claims. We also assume that Fi(O) = O for each i, so that we consider only positive claims amounts. We take as given the total annual premium, P, charged by the insurer in respect of these risks. We make no assumption about the way in which P is calculated but we do assume that

scholarly journals Calculation of Ruin Probabilities when the Claim Distribution is Lognormal

1977 ◽  
Vol 9 (1-2) ◽  
pp. 231-246 ◽  
Author(s):  
Olof Thorin ◽  
Nils Wikstad
Keyword(s):  
Distribution Function ◽  
Poisson Process ◽  
Linear Combination ◽  
Lognormal Distribution ◽  
Ruin Probabilities ◽  
Exponential Distributions ◽  
Absolutely Continuous ◽  
Image Position

In this paper some ruin probabilities are calculated for an example of a lognormal claim distribution. For that purpose it is shown that the lognormal distribution function, Λ(y), may be written in the formwhere V(x) is absolutely continuous and without being a distribution function preserves some useful properties of such a function.An attempt is also made to give an approximant Λα(y) to Λ(y) such that Λα(y) is a linear combination of a low number of exponential distributions. For comparison, ruin probabilities are also calculated for two examples of Λα(y).In the considered numerical cases it is assumed that the occurrence of claims follows a Poisson process.

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