Marceau 0.45

Panjer's Algorithm in Python

Marceau

Overview

This module provide a fast and efficient way to compute the Panjer's Algorithm in a Python shell.

Usage

In the following paragraphs, I am going to describe how you can get and use Marceau for your own projects.

Getting it

To download Marceau, either from this Github repository or simply use Pypi via pip.

pip install Marceau

Module

Marceau uses two modules to work properly, you need to make sure to have the following on your computer:

• Scipy.stats - Used to generate probability mass function from discrete distributions.
• Numpy - For usefull calculations.

You are then ready to use it:

import Marceau

Using it

The class Cossette built in the Marceau module calculate the Probability Density Function (PDF) and the Cumulative Distribution Function (CDF) of a Compound Distribution.

from Marceau import Cossette

The command

Cossette.help()
```and
```sh
Cossette.example()

provide respectivly an brief help and two example of the following algorithm.

Panjer's Algorithm

We are interested in the compound random variable: $$X=\sum_{i=1}^{N}B_{i}$$

where:

• $M$ is a frequence random variable from Panjer-Katz probability distribution family, otherwise known as (a,b, $0$)class of distributions. For $M=0$ we have $X=0$.
• $\underline{B}={B_{k},k\in\mathbb{N}^{+}}$ are positive i.i.d random variable defined on $\mathbb{N}$.
• $\underline{B}$ and $M$ are independant.

Therefore, the random variable $X$ has value in $\mathbb{N}$. And the Panjer's recursive method works as follow:

• If $B_{i}$ are distributed on a lattice $h\mathbb{N}$ with latticewidth $h>0$. $B\in${ $0$, $1h$, $2h$,....}
• We have $X\in$ $A_{h}$={ $0$, $1h$, $2h$,....}
• With $W_{M}$ beeing the probability generating function of M, we compute $f_{X}(0)=W_{M}(f_{B}(0))$
• The Panjer's recursive relation states for $k>0$: $$f_{X}(kh)=\frac{1}{1-af_{B}(0)}\sum_{i=1}^{k}(a+b\frac{jh}{kh})f_{B}(jh)\times f_{X}((k-j)h)$$

Implementation

In order to compute the Panjer's Algorithm, we need to enter the following feature to our class Cossette.

Arguments	Data Type	Description
k	a positive integer	the epoch of recursion to find X distribution
h	a strictly positive integer	the latticewidth of $B_{i}$ distribution
parameters	a list of length $1$ (poisson or geometric) or $2$ (binomial or negative binomial)	the parameters for the $X$ compound distribution
method	a string with value 'Binomial', 'NegBinomial', 'Geometric' or 'Poisson'	the law of $X$ compound distribution
fb	a list of length $k+1$	this correspond to the $f_{B}$ values when those are given, default value is an empty list
generat or_param	a list of length $1$ (poisson or geometric) or $2$ (binomial or negative binomial)	the parameters of the $B$ distribution, only needed if $f_{B}$ is empty, default value is empty
generator_method	a string with value 'Binomial', 'NegBinomial', 'Geometric' or 'Poisson'	the law of $B$ distribution, only needed if $f_{B}$ is empty, default value is empty

Example

Example 1

Let $X\sim PComp(\lambda=2,F_{B}),$ with $B\sim Bin(10,0.4)$.

We implement the following

model= Marceau.Cossette(k=10,parameters=[2],method='Poisson',generator_method='Binomial',generator_param=[10,0.4]) 

And we get our output with the call model.panjer():

model.panjer()
 >>> f(10*1)=0.05434563071580669 
 F(10*1)=0.6980136730471336 

Example 2

Let $X\sim PComp(\lambda=2,F_{B}),$ with $B \in$ { $1000$, $2000$, ... , $6000$ } and the following values for $f_{B}(hk)$ with $h=1000$:

$k$	$0$	$1$	$2$	$3$	$4$	$5$	$6$
$f_{B}(hk)$	$0$	$0.2$	$0.3$	$0.2$	$0.15$	$0.1$	$0.05$

We implement the following:

fb=np.zeros(30*1000+1)
fb[0]=0
fb[1000]=0.2
fb[2000]=0.3
fb[3000]=0.2
fb[4000]=0.15
fb[5000]=0.1
fb[6000]=0.05 
model= Marceau.Cossette(k=10,h=1000,parameters=[1.25],method='Poisson',fb=fb) 

And we get our output with the call model.panjer():

model.panjer()
 >>> f(10*1000)=0.02089842353538644 
F(10*1000)=0.9536818666811318  

Aknowledgement

This module was built with the help of Marceau lecture of Risk Theory.

License

MIT Copyright (c) 2022 Rayane Vigneron