# Marceau
# Overview
This module provide a fast and efficient way to compute the [Panjer's Algorithm][panjer] 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][git-repo-url] repository or simply use Pypi via pip.
```sh
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:
```sh
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.
```sh
from Marceau import Cossette
```
The command
```sh
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
```sh
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():
```sh
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:
```sh
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():
```sh
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
[//]: # (These are reference links used in the body of this note and get stripped out when the markdown processor does its job)
[git-repo-url]: <https://github.com/despervita/Marceau>
[panjer]: <https://www.casact.org/sites/default/files/database/astin_vol12no1_22.pdf>
[scipy.stats]: <https://docs.scipy.org/doc/scipy/reference/stats.html>
[numpy]: <https://numpy.org/doc/stable/index.html>
[Panjer-Katz]: <https://doi.org/10.1016/j.insmatheco.2010.03.010>
[class of distributions]: <https://www.actuaries.org/ASTIN/Colloquia/Helsinki/Papers/S7_13_Fackler.pdf>
[jQuery]: <http://jquery.com>
[Marceau]: <https://www.act.ulaval.ca/departement-et-professeurs/professeurs-et-personnel/professeurs/fiche-de-professeur/etienne-marceau-138>
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"description": "# Marceau \n# Overview\nThis module provide a fast and efficient way to compute the [Panjer's Algorithm][panjer] in a Python shell.\n\n## Usage\n\nIn the following paragraphs, I am going to describe how you can get and use Marceau for your own projects.\n\n### Getting it\n\nTo download Marceau, either from this [Github][git-repo-url] repository or simply use Pypi via pip.\n```sh\npip install Marceau\n```\n\n### Module\n\nMarceau uses two modules to work properly, you need to make sure to have the following on your computer:\n\n- [Scipy.stats] - Used to generate probability mass function from discrete distributions.\n- [Numpy] - For usefull calculations.\n\nYou are then ready to use it:\n```sh\nimport Marceau\n```\n\n### Using it\n\nThe class Cossette built in the Marceau module calculate the Probability Density Function (PDF) and the Cumulative Distribution Function (CDF) of a Compound Distribution.\n\n```sh\nfrom Marceau import Cossette\n```\n\nThe command\n```sh\nCossette.help()\n```\n\nand\n```sh\nCossette.example()\n```\n\nprovide respectivly an brief help and two example of the following algorithm.\n\n## Panjer's Algorithm\n\nWe are interested in the compound random variable: $$X=\\sum_{i=1}^{N}B_{i}$$\n\nwhere:\n* $M$ 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 $X=0$.\n* $\\underline{B}={B_{k},k\\in\\mathbb{N}^{+}}$ are positive i.i.d random variable defined on $\\mathbb{N}$.\n* $\\underline{B}$ and $M$ are independant.\n\nTherefore, the random variable $X$ has value in $\\mathbb{N}$. And the Panjer's recursive method works as follow:\n* If $B_{i}$ are distributed on a lattice $h\\mathbb{N}$ with latticewidth $h>0$. $B\\in$\\{ $0$, $1h$, $2h$,....\\}\n* We have $X\\in$ $A_{h}$=\\{ $0$, $1h$, $2h$,....\\}\n* With $W_{M}$ beeing the probability generating function of M, we compute $f_{X}(0)=W_{M}(f_{B}(0))$\n* The Panjer's recursive relation states for $k>0$: $$f_{X}(kh)=\\frac{1}{1-af_{B}(0)}\\sum_{i=1}^{k}(a+b\\frac{jh}{kh})f_{B}(jh)\\times f_{X}((k-j)h)$$\n\n\n### Implementation\n\nIn order to compute the Panjer's Algorithm, we need to enter the following feature to our class Cossette.\n\n| Arguments | Data Type| Description| \n| ------ | ------ | ------ |\n| k | a positive integer| the epoch of recursion to find X distribution |\n| h | a strictly positive integer| the latticewidth of $B_{i}$ distribution |\n| parameters | a list of length $1$ (poisson or geometric) or $2$ (binomial or negative binomial)| the parameters for the $X$ compound distribution |\n| method | a string with value 'Binomial', 'NegBinomial', 'Geometric' or 'Poisson'| the law of $X$ compound distribution |\n| fb | a list of length $k+1$ | this correspond to the $f_{B}$ values when those are given, default value is an empty list |\n| 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|\n| 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 |\n\n\n## Example\n#### Example 1\nLet $X\\sim PComp(\\lambda=2,F_{B}),$ with $B\\sim Bin(10,0.4)$.\n\nWe implement the following\n\n```sh\nmodel= Marceau.Cossette(k=10,parameters=[2],method='Poisson',generator_method='Binomial',generator_param=[10,0.4]) \n```\nAnd we get our output with the call model.panjer():\n```sh\nmodel.panjer()\n >>> f(10*1)=0.05434563071580669 \n F(10*1)=0.6980136730471336 \n ```\n#### Example 2 \n\nLet $X\\sim PComp(\\lambda=2,F_{B}),$ with $B \\in$ \\{ $1000$, $2000$, ... , $6000$ \\} and the following values for $f_{B}(hk)$ with $h=1000$:\n\n| $k$ | $0$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ |\n|---|---|-----|-----|-----|------|-----|------|\n| $f_{B}(hk)$ |$0$ | $0.2$ | $0.3$ | $0.2$ | $0.15$ | $0.1$ | $0.05$ |\n\nWe implement the following:\n```sh\nfb=np.zeros(30*1000+1)\nfb[0]=0\nfb[1000]=0.2\nfb[2000]=0.3\nfb[3000]=0.2\nfb[4000]=0.15\nfb[5000]=0.1\nfb[6000]=0.05 \nmodel= Marceau.Cossette(k=10,h=1000,parameters=[1.25],method='Poisson',fb=fb) \n```\n\nAnd we get our output with the call model.panjer():\n```sh\nmodel.panjer()\n >>> f(10*1000)=0.02089842353538644 \nF(10*1000)=0.9536818666811318 \n```\n\n\n## Aknowledgement\nThis module was built with the help of [Marceau] lecture of Risk Theory.\n\n\n## License\n\nMIT\nCopyright (c) 2022 Rayane Vigneron\n\n\n\n\n[//]: # (These are reference links used in the body of this note and get stripped out when the markdown processor does its job)\n\n [git-repo-url]: <https://github.com/despervita/Marceau>\n [panjer]: <https://www.casact.org/sites/default/files/database/astin_vol12no1_22.pdf>\n [scipy.stats]: <https://docs.scipy.org/doc/scipy/reference/stats.html>\n [numpy]: <https://numpy.org/doc/stable/index.html>\n [Panjer-Katz]: <https://doi.org/10.1016/j.insmatheco.2010.03.010>\n [class of distributions]: <https://www.actuaries.org/ASTIN/Colloquia/Helsinki/Papers/S7_13_Fackler.pdf>\n [jQuery]: <http://jquery.com>\n [Marceau]: <https://www.act.ulaval.ca/departement-et-professeurs/professeurs-et-personnel/professeurs/fiche-de-professeur/etienne-marceau-138>",
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