scholarly journals Between Individual and Collective Model for the Total Claims

1988 ◽  
Vol 18 (2) ◽  
pp. 169-174 ◽  
Author(s):  
R. Kaas ◽  
A. E. van Heerwaarden ◽  
M. J. Goovaerts
Keyword(s):  
Life Insurance ◽  
Random Variables ◽  
Random Variable ◽  
Poisson Approximation ◽  
Collective Model ◽  
Individual Model ◽  
Compound Poisson ◽  
Compound Poisson Approximation ◽  
The Real ◽  
Stop Loss

AbstractThis article studies random variables whose stop-loss rank falls between a certain risk (assumed to be integer-valued and non-negative, but not necessarily of life-insurance type) and the compound Poisson approximation to this risk. They consist of a compound Poisson part to which some independent Bernoulli-type variables are added.Replacing each term in an individual model with such a random variable leads to an approximating model for the total claims on a portfolio of contracts that is computationally almost as attractive as the compound Poisson approximation used in the standard collective model. The resulting stop-loss premiums are much closer to the real values.

scholarly journals A Compound Poisson Approximation Inequality

2006 ◽  
Vol 43 (01) ◽  
pp. 282-288
Author(s):  
Erol A. Peköz
Keyword(s):  
System Reliability ◽  
Random Variable ◽  
Poisson Approximation ◽  
Poisson Random Variable ◽  
Sequence Pattern ◽  
Compound Poisson ◽  
Compound Poisson Approximation ◽  
Numerical Example

We give conditions under which the number of events which occur in a sequence ofm-dependent events is stochastically smaller than a suitably defined compound Poisson random variable. The results are applied to counts of sequence pattern appearances and to system reliability. We also provide a numerical example.

scholarly journals A Compound Poisson Approximation Inequality

2006 ◽  
Vol 43 (1) ◽  
pp. 282-288 ◽  
Author(s):  
Erol A. Peköz
Keyword(s):  
System Reliability ◽  
Random Variable ◽  
Poisson Approximation ◽  
Poisson Random Variable ◽  
Sequence Pattern ◽  
Compound Poisson ◽  
Compound Poisson Approximation ◽  
Numerical Example

We give conditions under which the number of events which occur in a sequence of m-dependent events is stochastically smaller than a suitably defined compound Poisson random variable. The results are applied to counts of sequence pattern appearances and to system reliability. We also provide a numerical example.

scholarly journals Compound Poisson Approximation for Dissociated Random Variables via Stein's Method

1999 ◽  
Vol 8 (4) ◽  
pp. 335-346 ◽  
Author(s):  
PETER EICHELSBACHER ◽  
MAŁGORZATA ROOS
Keyword(s):  
System Reliability ◽  
Random Variables ◽  
Poisson Approximation ◽  
Stein’S Method ◽  
Stein's Method ◽  
Compound Poisson ◽  
Compound Poisson Approximation ◽  
Stein Equation

In the present paper we consider compound Poisson approximation by Stein's method for dissociated random variables. We present some applications to problems in system reliability. In particular, our examples have the structure of an incomplete U-statistics. We mainly apply techniques from Barbour and Utev, who gave new bounds for the solutions of the Stein equation in compound Poisson approximation in two recent papers.

Approximation of sums by compound Poisson distributions with respect to stop-loss distances

1990 ◽  
Vol 22 (2) ◽  
pp. 350-374 ◽  
Author(s):  
S. T. Rachev ◽  
L. Rüschendorf
Keyword(s):  
Random Variables ◽  
Independent Random Variables ◽  
Compound Poisson ◽  
Poisson Distributions ◽  
Stop Loss ◽  
Scale Transformations

The approximation of sums of independent random variables by compound Poisson distributions with respect to stop-loss distances is investigated. These distances are motivated by risk-theoretic considerations. In contrast to the usual construction of approximating compound Poisson distributions, the method suggested in this paper is to fit several moments. For two moments, this can be achieved by scale transformations. It is shown that the new approximations are more stable and improve the usual approximations by accompanying laws in examples where the probability 1 – pi that the ith summand is zero is not too large.

Poisson approximation for a sum of dependent indicators: an alternative approach

2002 ◽  
Vol 34 (03) ◽  
pp. 609-625 ◽  
Author(s):  
N. Papadatos ◽  
V. Papathanasiou
Keyword(s):  
Total Variation ◽  
Upper Bound ◽  
Random Variables ◽  
Random Variable ◽  
Poisson Approximation ◽  
Total Variation Distance ◽  
Poisson Random Variable ◽  
Birthday Problem ◽  
Alternative Approach ◽  
Variation Distance

The random variablesX1,X2, …,Xnare said to be totally negatively dependent (TND) if and only if the random variablesXiand ∑j≠iXjare negatively quadrant dependent for alli. Our main result provides, for TND 0-1 indicatorsX1,x2, …,Xnwith P[Xi= 1] =pi= 1 - P[Xi= 0], an upper bound for the total variation distance between ∑ni=1Xiand a Poisson random variable with mean λ ≥ ∑ni=1pi. An application to a generalized birthday problem is considered and, moreover, some related results concerning the existence of monotone couplings are discussed.

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