# On the Evaluation of the Distribution of a General Multivariate Collective Model: Recursions versus Fast Fourier Transform

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## Abstract

**:**

## 1. Introduction

- The m-variate claim size random vectors ${\left(\right)}_{{\mathbf{X}}_{i}^{\left(\right)}}i\ge 1$ are i.i.d. as the generic m-variate random vector ${\mathbf{X}}^{\left(\right)}$ whose jth univariate component ${X}_{j}^{\left(\right)}$ if $j\notin \left(\right)open="\{"\; close="\}">{i}_{1},\dots ,{i}_{k}$ meaning that ${\mathbf{X}}^{\left(\right)}$ results from those claim events simultaneously affecting solely the lines $\left(\right)$; these events are counted by the r.v. ${N}_{{i}_{1}\dots {i}_{k}}$. Moreover, the ${\mathbf{X}}_{i}^{\left(\right)}$s are also independent of the other claim size random vectors (i.e., of each ${\left(\right)}_{{\mathbf{X}}_{i}^{\left(\right)}}i\ge 1$, where $\left(\right)open="\{"\; close="\}">{i}_{1},\dots ,{i}_{k}$) and of the claim numbers. We let ${X}_{i,j}^{\left(\right)}$ denote the jth univariate component of ${\mathbf{X}}_{i}^{\left(\right)}$ ${f}_{{i}_{1}\dots {i}_{k}}$ the probability function (p.f.) of ${\mathbf{X}}^{\left(\right)}$ (in the discrete case) and, by convention, ${\mathbf{X}}_{0}^{\left(\right)}$.
- The components of the random vector number of claims $\mathbf{N}={\left(\right)}_{{N}_{{i}_{1}\dots {i}_{k}}}1\le k\le m;1\le {i}_{1}\dots {i}_{k}\le m$ are dependent r.v.s, in total (maximum) $\nu ={2}^{m}-1.$

## 2. Evaluation of the Compound Distribution

**Proposition**

**1.**

**Proof.**

**Proposition**

**2.**

**Proof.**

#### 2.1. Recursive Evaluation

#### 2.1.1. Case 1 Assumptions

**Proposition**

**3.**

**Proof.**

#### 2.1.2. Case 2 Assumptions

**A1**- The p.f. of the total number of claims ${N}_{tot}={\sum}_{1\le k\le m;1\le {i}_{1}<\dots <{i}_{k}\le m}{N}_{{i}_{1}\dots {i}_{k}}$ satisfies Panjer’s recursion for $a,b\in \mathbb{R}$.
**A2**- Given ${N}_{tot}=n,$ the conditional distribution of the random vector number of claims $\mathbf{N}$ is assumed to be multinomial $Mnom(n;\mathbf{p})$ with parameters $n\in \mathbb{N}$ and $\mathbf{p}={\left({p}_{{i}_{1}\dots {i}_{k}}\right)}_{1\le k\le m;1\le {i}_{1}<\dots <{i}_{k}\le m},$ where ${p}_{{i}_{1}\dots {i}_{k}}\in (0,1)$ such that ${\sum}_{1\le k\le m;1\le {i}_{1}<\dots <{i}_{k}\le m}{p}_{{i}_{1}\dots {i}_{k}}=1.$ Therefore, with $\mathbf{n}={\left({n}_{{i}_{1}\dots {i}_{k}}\right)}_{1\le k\le m;1\le {i}_{1}<\dots <{i}_{k}\le m}$ and $n={\sum}_{1\le k\le m;1\le {i}_{1}<\dots <{i}_{k}\le m}{n}_{{i}_{1}\dots {i}_{k}},$$$Pr(\mathbf{N}=\mathbf{n}|{N}_{tot}=n)={\displaystyle \frac{n!}{{\prod}_{1\le k\le m;1\le {i}_{1}<\dots <{i}_{k}\le m}{n}_{{i}_{1}\dots {i}_{k}}!}}\prod _{1\le k\le m;1\le {i}_{1}<\dots <{i}_{k}\le m}{p}_{{i}_{1}\dots {i}_{k}}^{{n}_{{i}_{1}\dots {i}_{k}}}.$$

**Proposition**

**4.**

**Proof.**

**Proposition**

**5.**

**Proof.**

**Particular case**: $m=3$. Let us now have a look at a recursive formula in the trivariate case, where the general Model (2) is $\mathbf{S}=({S}_{1},{S}_{2},{S}_{3})$ with

#### 2.1.3. Case 3 Assumptions

**Remark**

**1.**

**Particular case**: Simpler recursions are obtained when $\Theta $ is gamma $Ga\left(\right)open="("\; close=")">\delta ,\beta $ distributed, with $\delta ,\beta >0$. In this case, the univariate mixed Poisson $Po\left(\right)open="("\; close=")">\theta {\lambda}_{+}$ distribution becomes a Negative Binomial distribution $NB\left(\right)open="("\; close=")">\delta ,\frac{\beta}{\beta \phantom{\rule{3.33333pt}{0ex}}+\phantom{\rule{3.33333pt}{0ex}}{\lambda}_{+}}$ which satisfies Panjer’s recursion with $a=\frac{{\lambda}_{+}}{\beta \phantom{\rule{3.33333pt}{0ex}}+\phantom{\rule{3.33333pt}{0ex}}{\lambda}_{+}}$ and $b=\left(\right)open="("\; close=")">\delta -1$. Since

#### 2.2. Fast Fourier Transform Evaluation

**FFT Algorithm for model**(2)

**Particular cases**: Under the particular assumptions considered in the previous section to allow for a recursive evaluation, one should use the following formulas at Step 3 of the above algorithm:

- -
- When $\mathbf{N}\sim MPo(\lambda ;\tilde{\lambda}),{\tilde{\phi}}_{\mathbf{S}}$ is given by Equation (6);
- -
- Under the Case 2 assumptions (A1 and A2), ${\tilde{\phi}}_{\mathbf{S}}$ is given by Equation (13);
- -
- Under the Case 3 mixed Poisson assumption, ${\tilde{\phi}}_{\mathbf{S}}$ is given by Equation (18).

#### 2.3. Numerical Illustration

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

## 3. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Table 1.**Example 1: Comparing recursive and FFT methods for $h=1,\theta =7/r$ and various $r,{x}_{M}$.

${r}={{x}}_{{M}}=16$ | ${r}=32,{{x}}_{{M}}=20$ | ${r}=64,{{x}}_{{M}}=20$ | ${r}=128,{{x}}_{{M}}=20$ | |
---|---|---|---|---|

Rec. ${F}_{\mathbf{S}}\left({\mathbf{x}}_{M}\right)$ | 0.219737 | 0.312845 | 0.312845 | 0.312845 |

FFT ${F}_{\mathbf{S}}^{FFT}\left({\mathbf{x}}_{M}\right)$ | 0.219884 | 0.312909 | 0.312855 | 0.312847 |

FFT time up to $\mathbf{r}$ | 0.016 s | 0.124 s | 0.952 s | 9.484 s |

$AE$ | 1.4743 × ${10}^{-4}$ | 6.3771 × ${10}^{-5}$ | 1.0251 × ${10}^{-5}$ | 1.7488 × ${10}^{-6}$ |

$Max.err$ | 8.8571 × ${10}^{-8}$ | 2.0810 × ${10}^{-8}$ | 3.5244 × ${10}^{-9}$ | 6.9685 × ${10}^{-10}$ |

${r}=16$ | ${r}=32$ | ${r}=64$ | |
---|---|---|---|

Rec/FFT in time | 12 | 130 | 781 |

**Table 3.**Example 2: Comparing recursive and FFT methods for $h=1,\theta =7/r$ and various $r,{x}_{M}$.

${r}={{x}}_{{M}}=16$ | ${r}={{x}}_{{M}}=32$ | ${r}={{x}}_{{M}}=64$ | ${r}=128,{{x}}_{{M}}=70$ | |
---|---|---|---|---|

Rec. ${F}_{\mathbf{S}}\left({\mathbf{x}}_{M}\right)$ | 0.80035 | 0.91543 | 0.96436 | 0.96804 |

FFT ${F}_{\mathbf{S}}^{FFT}\left({\mathbf{x}}_{M}\right)$ | 0.80039 | 0.91544 | 0.96436 | 0.96804 |

$AE$ | 4.3580 × ${10}^{-5}$ | 1.4294 × ${10}^{-5}$ | 3.8798 × ${10}^{-6}$ | 9.0937 × ${10}^{-7}$ |

$Max.err$ | 7.9393 × ${10}^{-7}$ | 1.8276 × ${10}^{-7}$ | 4.1642 × ${10}^{-8}$ | 9.6893 × ${10}^{-9}$ |

Recursion | FFT no tilt. | $\begin{array}{c}\mathbf{FFT}\mathbf{tilt}.\\ \mathit{\theta}=5/\mathit{r}\end{array}$ | $\begin{array}{c}\mathbf{FFT}\mathbf{tilt}.\\ \mathit{\theta}=7/\mathit{r}\end{array}$ | $\begin{array}{c}\mathbf{FFT}\mathbf{tilt}.\\ \mathit{\theta}=9/\mathit{r}\end{array}$ | |
---|---|---|---|---|---|

${F}_{\mathbf{S}}\left({\mathbf{x}}_{M}\right)$ | 0.96436 | 0.96863 | 0.96439 | 0.96436 | 0.96436 |

$AE$ | 4.2693 × ${10}^{-3}$ | 2.8485 × ${10}^{-5}$ | 3.8798 × ${10}^{-6}$ | 5.4867 × ${10}^{-6}$ | |

$Max.err$ | 4.5823 × ${10}^{-5}$ | 3.0770 × ${10}^{-7}$ | 4.1642 × ${10}^{-8}$ | 2.7813 × ${10}^{-8}$ |

**Table 5.**Example 3: Comparing recursive and FFT methods for $h=1,\theta =7/r$ and various $r,{x}_{M}$.

${r}={{x}}_{{M}}=16$ | ${r}={{x}}_{{M}}=32$ | ${r}={{x}}_{{M}}=64$ | ${r}=128,{{x}}_{{M}}=70$ | |
---|---|---|---|---|

Rec. ${F}_{\mathbf{S}}\left({\mathbf{x}}_{M}\right)$ | 0.49044 | 0.72191 | 0.88701 | 0.90087 |

FFT ${F}_{\mathbf{S}}^{FFT}\left({\mathbf{x}}_{M}\right)$ | 0.49055 | 0.72200 | 0.88705 | 0.90088 |

$AE$ | 1.0650 × ${10}^{-4}$ | 8.3639 × ${10}^{-5}$ | 3.5553 × ${10}^{-5}$ | 6.4368 × ${10}^{-6}$ |

$Max.err$ | 1.6674 × ${10}^{-7}$ | 3.2871 × ${10}^{-8}$ | 6.8457 × ${10}^{-9}$ | 1.5338 × ${10}^{-9}$ |

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Vernic, R.
On the Evaluation of the Distribution of a General Multivariate Collective Model: Recursions versus Fast Fourier Transform. *Risks* **2018**, *6*, 87.
https://doi.org/10.3390/risks6030087

**AMA Style**

Vernic R.
On the Evaluation of the Distribution of a General Multivariate Collective Model: Recursions versus Fast Fourier Transform. *Risks*. 2018; 6(3):87.
https://doi.org/10.3390/risks6030087

**Chicago/Turabian Style**

Vernic, Raluca.
2018. "On the Evaluation of the Distribution of a General Multivariate Collective Model: Recursions versus Fast Fourier Transform" *Risks* 6, no. 3: 87.
https://doi.org/10.3390/risks6030087