Panjer's Algorithm in Python
Project description
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
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
Project details
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