scholarly journals Modified Recursions for a Class of Compound Distributions

1996 ◽  
Vol 26 (2) ◽  
pp. 213-224 ◽  
Author(s):  
Karl-Heinz Waldmann
Keyword(s):  
Distribution Function ◽  
Cumulative Distribution Function ◽  
Mass Function ◽  
Cumulative Distribution ◽  
Probability Mass Function ◽  
Initial Value ◽  
Compound Distributions ◽  
Claim Frequency ◽  
Probability Mass ◽  
Stop Loss

AbstractRecursions are derived for a class of compound distributions having a claim frequency distribution of the well known (a,b)-type. The probability mass function on which the recursions are usually based is replaced by the distribution function in order to obtain increasing iterates. A monotone transformation is suggested to avoid an underflow in the initial stages of the iteration. The faster increase of the transformed iterates is diminished by use of a scaling function. Further, an adaptive weighting depending on the initial value and the increase of the iterates is derived. It enables us to manage an arbitrary large portfolio. Some numerical results are displayed demonstrating the efficiency of the different methods. The computation of the stop-loss premiums using these methods are indicated. Finally, related iteration schemes based on the cumulative distribution function are outlined.

scholarly journals Improved Method for Approximating the Hazard Rate for Convolution Poisson Random Variables

2021 ◽  
Author(s):  
Andrei Volodin ◽  
ALYA AL MUTAIRI
Keyword(s):  
Hazard Rate ◽  
Cumulative Distribution Function ◽  
Random Variables ◽  
Saddlepoint Approximation ◽  
Mass Function ◽  
Rate Function ◽  
Cumulative Distribution ◽  
Probability Mass Function ◽  
Complicated Model ◽  
Probability Mass

In this study, we investigate the performance of the saddlepoint approximation of the probability mass function and the cumulative distribution function for the weighted sum of independent Poisson random variables. The goal is to approximate the hazard rate function for this complicated model. The better performance of this method is shown by numerical simulations and comparison with a performance of other approximation methods.

scholarly journals A New Discrete Distribution Arising from a Generalised Random Game and Its Asymptotic Properties

2021 ◽  
pp. 11-20
Author(s):  
R. Frühwirth ◽  
R. Malina ◽  
W. Mitaroff
Keyword(s):  
Numerical Study ◽  
Asymptotic Properties ◽  
Discrete Distribution ◽  
Rayleigh Distribution ◽  
Mass Function ◽  
Cumulative Distribution ◽  
Probability Mass Function ◽  
Expected Gain ◽  
Probability Mass ◽  
Analytic Expression

The rules of a game of dice are extended to a ``hyper-die'' with \(n\in\mathbb{N}\) equally probable faces, numbered from 1 to \(n\). We derive recursive and explicit expressions for the probability mass function and the cumulative distribution function of the gain \(G_n\) for arbitrary values of \(n\). A numerical study suggests the conjecture that for \(n \to \infty\) the expectation of the scaled gain \(\mathbb{E}[{H_n}]=\mathbb{E} [{G_n/\sqrt{n}\,}]\) converges to \(\sqrt{\pi/\,2}\). The conjecture is proved by deriving an analytic expression of the expected gain \(\mathbb{E} [{G_n}]\). An analytic expression of the variance of the gain \(G_n\) is derived by a similar technique. Finally,  it is proved that \(H_n\) converges weakly to the Rayleigh distribution with scale parameter~1.

ON THE EVALUATION OF MULTIVARIATE COMPOUND DISTRIBUTIONS WITH CONTINUOUS SEVERITY DISTRIBUTIONS AND SARMANOV'S COUNTING DISTRIBUTION

2018 ◽  
Vol 48 (02) ◽  
pp. 841-870 ◽  
Author(s):  
Maissa Tamraz ◽  
Raluca Vernic
Keyword(s):  
Distribution Function ◽  
Hypergeometric Function ◽  
Cumulative Distribution Function ◽  
Cumulative Distribution ◽  
Type Ii ◽  
Similar Function ◽  
Compound Distributions ◽  
Closed Type ◽  
Counting Distribution

AbstractIn this paper, we present closed-type formulas for some multivariate compound distributions with multivariate Sarmanov counting distribution and independent Erlang distributed claim sizes. Further on, we also consider a type-II Pareto dependency between the claim sizes of a certain type. The resulting densities rely on the special hypergeometric function, which has the advantage of being implemented in the usual software. We numerically illustrate the applicability and efficiency of such formulas by evaluating a bivariate cumulative distribution function, which is also compared with the similar function obtained by the classical recursion-discretization approach.

scholarly journals Regional Incomes Structure Analysis in Slovak Republic On the Basis of EU-SILC Data

2017 ◽  
Vol 64 (2) ◽  
pp. 171-185 ◽  
Author(s):  
Milan Terek
Keyword(s):  
Structure Analysis ◽  
Slovak Republic ◽  
Mass Function ◽  
Cumulative Distribution ◽  
Probability Mass Function ◽  
Sampling Weights ◽  
Empirical Cumulative Distribution Function ◽  
Empirical Probability ◽  
Probability Mass ◽  
Household Incomes

Abstract The paper deals with the regional incomes structure analysis in Slovak republic on the basis of European Union statistics on income and living conditions in Slovak republic data. The empirical probability mass function and empirical cumulative distribution function is constructed with aid of given sampling weights. On the basis of these functions the median, medial, standard deviation and population histogram of the whole gross household incomes for the whole Slovak republic and separately for eight Slovak regions are estimated and compared.

scholarly journals Recursions for Compound Distributions

1982 ◽  
Vol 13 (1) ◽  
pp. 1-12 ◽  
Author(s):  
H. H. Panjer ◽  
G. E. Willmot
Keyword(s):  
Distribution Function ◽  
Usual Method ◽  
Claim Amount ◽  
Frequency Distributions ◽  
Compound Distributions ◽  
Compound Distribution ◽  
The Laplace Transform ◽  
Claim Frequency ◽  
Claims Frequency ◽  
Stop Loss

Various methods for developing recursive formulae for compound distributions have been reported recently by Panjer (1980, including discussion), Panjer (1981), Sundt and Jewell (1981) and Gerber (1982) for a class of claim frequency distributions and arbitrary claim amount distributions. The recursions are particularly useful for computational purposes since the number of calculations required to obtain the distribution function of total claims and related values such as net stop-loss premiums may be greatly reduced when compared with the usual method based on convolutions.In this paper a broader class of claims frequency distributions is considered and methods for developing recursions for the corresponding compound distributions are examined. The methods make use of the Laplace transform of the density of the compound distribution.Consider the class of claim frequency distributions which has the property that successive probabilities may be written as the ratio of two polynomials. For convenience we write the polynomials in terms of descending factorial powers. For obvious reasons, only distributions on the non-negative integers are considered.

Customer Arrival Event Processing on Computer Simulation for Discrete Event System

2014 ◽  
Vol 513-517 ◽  
pp. 2133-2136
Author(s):  
Ming Hai Yao ◽  
Xiao Ji Chen ◽  
Lei Zuo
Keyword(s):  
Computer Simulation ◽  
Differential Equations ◽  
Effective Means ◽  
Discrete Event ◽  
Mass Function ◽  
Cumulative Distribution ◽  
Probability Mass Function ◽  
Simulation Techniques ◽  
Probability Mass ◽  
Ticket Machine

Discrete event systems are widely used in the production and life, it is difficult to use conventional differential equations, differential equations, and other models to describe, the theoretical analysis method is difficult to obtain analytical solutions, computer simulation techniques to solve these problems provides an effective means. Arrival event is a typical discrete system event; on arrival event handling is always one of the difficulties of computer simulation, in this paper, banking customer arrival system as an example to study. For banks queuing system, customers arrive to obey the parameter of Poisson distribution is, the probability mass function through the distribution curves and cumulative distribution function curves to study the distribution of customer arrival; construction of single-queue multi-server system of customer arrival event subroutine flow chart, and processing steps will be described. Content of this study, it is suitable for the developed area bank to adopt "number ticket machine" approach to service.

DISSONANCE — A MEASURE OF VARIABILITY FOR ORDINAL RANDOM VARIABLES

2001 ◽  
Vol 09 (01) ◽  
pp. 39-53 ◽  
Author(s):  
RONALD R. YAGER
Keyword(s):  
Distribution Function ◽  
Cumulative Distribution Function ◽  
Probability Distributions ◽  
Random Variables ◽  
Cumulative Distribution ◽  
General Properties

We look at the issue of obtaining a variance like measure associated with probability distributions over ordinal sets. We call these dissonance measures. We specify some general properties desired in these dissonance measures. The centrality of the cumulative distribution function in formulating the concept of dissonance is pointed out. We introduce some specific examples of measures of dissonance.

Estimating the cumulative distribution function for the linear combination of gamma random variables

2017 ◽  
Vol 20 (5) ◽  
pp. 939-951
Author(s):  
Amal Almarwani ◽  
Bashair Aljohani ◽  
Rasha Almutairi ◽  
Nada Albalawi ◽  
Alya O. Al Mutairi
Keyword(s):  
Distribution Function ◽  
Linear Combination ◽  
Cumulative Distribution Function ◽  
Random Variables ◽  
Cumulative Distribution
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