scholarly journals Further Results on Recursive Evaluation of Compound Distributions

1981 ◽  
Vol 12 (1) ◽  
pp. 27-39 ◽  
Author(s):  
Bjørn Sundt ◽  
William S. Jewell
Keyword(s):  
Recent Result ◽  
Recursive Algorithm ◽  
Compound Distributions ◽  
Compound Distribution ◽  
Aggregate Claims ◽  
Recursive Evaluation

A recent result by Panjer provides a recursive algorithm for the compound distribution of aggregate claims when the counting law belongs to a special recursive family. In the present paper we first give a characterization of this recursive family, then describe some generalizations of Panjer's result.

scholarly journals Recursions for certain bivariate counting distributions and their compound distributions

1996 ◽  
Vol 26 (1) ◽  
pp. 35-52 ◽  
Author(s):  
Ole Hesselager
Keyword(s):  
Recursive Algorithm ◽  
Compound Distributions ◽  
Compound Distribution

AbstractWe consider three classes of bivariate counting distributions and the corresponding compound distributions. For each class we derive a recursive algorithm for calculating the bivariate compound distribution.

scholarly journals Asymptotic Behaviour of Compound Distributions and Stop-loss Premiums

1982 ◽  
Vol 13 (2) ◽  
pp. 89-98 ◽  
Author(s):  
Bjørn Sundt
Keyword(s):  
Asymptotic Behaviour ◽  
Negative Binomial ◽  
Asymptotic Results ◽  
Compound Distributions ◽  
Number Distribution ◽  
Claim Number ◽  
Compound Distribution ◽  
Aggregate Claims ◽  
Stop Loss

AbstractThe paper gives some asymptotic results for the compound distribution of aggregate claims when the claim number distribution is negative binomial. The case when the claim numbers are geometrically distributed, is treated separately.

scholarly journals A Recursive Procedure for Calculation of some Compound Distributions

1994 ◽  
Vol 24 (1) ◽  
pp. 19-32 ◽  
Author(s):  
Ole Hesselager
Keyword(s):  
Poisson Process ◽  
Recursive Algorithm ◽  
Recursive Procedure ◽  
Probability Of Ruin ◽  
Compound Distributions ◽  
Compound Distribution ◽  
Counting Distribution ◽  
Mixed Poisson Process ◽  
The One

AbstractWe consider compound distributions where the counting distribution has the property that the ratio between successive probabilities may be written as the ratio of two polynomials. We derive a recursive algorithm for the compound distribution, which is more efficient than the one suggested by Panjer & Willmot (1982) and Willmot & Panjer (1987). We also derive a recursive algorithm for the moments of the compound distribution. Finally, we present an application of the recursion to the problem of calculating the probability of ruin in a particular mixed Poisson process.

On a generalization of the Rényi–Srivastava characterization of the Poisson law

2021 ◽  
Vol 58 (1) ◽  
pp. 68-82
Author(s):  
Jean-Renaud Pycke
Keyword(s):  
Life Insurance ◽  
Collective Model ◽  
New Method ◽  
Bell Polynomials ◽  
Explicit Formulas ◽  
Compound Distributions ◽  
Poisson Law ◽  
Pierre Loti

AbstractWe give a new method of proof for a result of D. Pierre-Loti-Viaud and P. Boulongne which can be seen as a generalization of a characterization of Poisson law due to Rényi and Srivastava. We also provide explicit formulas, in terms of Bell polynomials, for the moments of the compound distributions occurring in the extended collective model in non-life insurance.

scholarly journals On Multivariate Vernic Recursions

2000 ◽  
Vol 30 (1) ◽  
pp. 111-122 ◽  
Author(s):  
Bjørn Sundt
Keyword(s):  
Recursive Algorithm ◽  
Compound Distributions ◽  
Counting Distribution

AbstractIn the present paper we extend a recursive algorithm developed by Vernic (1999) for compound distributions with bivariate counting distribution and univariate severity distributions to more general multivariate counting distributions.

scholarly journals Recursive Formulae for Some Bivariate Counting Distributions Obtained by the Trivariate Reduction Method

2000 ◽  
Vol 30 (1) ◽  
pp. 141-155 ◽  
Author(s):  
J.F. Walhin ◽  
J. Paris
Keyword(s):  
Reduction Method ◽  
Bivariate Distribution ◽  
Recursive Algorithms ◽  
Data Set ◽  
Compound Distributions ◽  
Aggregate Claims

AbstractIn this paper we study some bivariate counting distributions that are obtained by the trivariate reduction method. We work with Poisson compound distributions and we use their good properties in order to derive recursive algorithms for the bivariate distribution and bivariate aggregate claims distribution. A data set is also fitted.

A REPRESENTATION THEORY FOR POLYNOMIAL COFRACTIONALITY IN VECTOR AUTOREGRESSIVE MODELS

2009 ◽  
Vol 26 (4) ◽  
pp. 1201-1217 ◽  
Author(s):  
Massimo Franchi
Keyword(s):  
Representation Theory ◽  
Autoregressive Model ◽  
Moving Average ◽  
Recursive Algorithm ◽  
Autoregressive Models ◽  
Vector Autoregressive Models ◽  
Vector Autoregressive ◽  
Moving Average Representation ◽  
Average Representation

We extend the representation theory of the autoregressive model in the fractional lag operator of Johansen (2008, Econometric Theory 24, 651–676). A recursive algorithm for the characterization of cofractional relations and the corresponding adjustment coefficients is given, and it is shown under which condition the solution of the model is fractional of order d and displays cofractional relations of order d − b and polynomial cofractional relations of order d − 2b,…, d − cb ≥ 0 for integer c; the cofractional relations and the corresponding moving average representation are characterized in terms of the autoregressive coefficients by the same algorithm. For c = 1 and c = 2 we find the results of Johansen (2008).

scholarly journals A Characterization of the Esscher-Transformation

1982 ◽  
Vol 13 (1) ◽  
pp. 57-59 ◽  
Author(s):  
Erhard Kremer
Keyword(s):  
Distribution Function ◽  
Iterative Methods ◽  
Edgeworth Expansion ◽  
Mean Value ◽  
Risk Theory ◽  
Approximation Methods ◽  
Classical Approximation ◽  
Aggregate Claims ◽  
The Mean

One of the central problems in risk theory is the calculation of the distribution function F of aggregate claims of a portfolio. Whereas formerly mainly approximation methods could be used, nowadays the increased speed of the computers allows application of iterative methods of numerical mathematics (see Bertram (1981), Küpper (1971) and Strauss (1976)). Nevertheless some of the classical approximation methods are still of some interest, especially a method developed by Esscher (1932).The idea of this so called Esscher-approximation (see Esscher (1932), Grandell and Widaeus (1969) and Gerber (1980)) is rather simple:In order to calculate 1 –F(x) for large x one transforms F into a distribution function such that the mean value of is equal to x and applies the Edgeworth expansion to the density of The reason for applying the transformation is the fact that the Edgeworth expansion produces good results for x near the mean value, but poor results in the tail (compare also Daniels (1954)).

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