# Determining Distribution for the Product of Random Variables by Using Copulas

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## Abstract

**:**

## 1. Introduction

## 2. Background Theory

## 3. Copulas

**Definition**

**1.**

**(Copula)**A two-dimensional copula is a function C: ${\mathbb{I}}^{2}\to \mathbb{I}$ satisfying the following conditions:

- (i)
- $C(u,0)=C(0,v)=0$ for any u and $v\in \mathbb{I}$.
- (ii)
- $C(u,1)=u$ and $C(1,v)=v$ for any u and $v\in \mathbb{I}$.
- (iii)
- for any ${u}_{1},{u}_{2},{v}_{1}$, and ${v}_{2}\in \mathbb{I}$ such that ${u}_{1}\le {u}_{2}$ and ${v}_{1}\le {v}_{2},$. We have$$C({u}_{2},{v}_{2})+C({u}_{1},{v}_{1})-C({u}_{2},{v}_{1})-C({u}_{1},{v}_{2})\ge 0.$$

**Definition**

**2.**

- (i)
- C is grounded; that is, $C({u}_{1},{u}_{2},\dots ,{u}_{n})=0$ where $({u}_{1},{u}_{2},\dots ,{u}_{n})\in {\mathbb{I}}^{n}$ such that at least one ${u}_{i}=0$ for $i=1,2,\dots ,n$.
- (ii)
- $C(1,\dots ,1,{u}_{i},1,\dots ,1)={u}_{i}$, $\forall {u}_{i}\in \mathbb{I},\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{4pt}{0ex}}i=1,2,\dots ,n$.
- (iii)
- C is an n-increasing function; that is, $\forall B={B}_{1}\times {B}_{2}\dots \times {B}_{n}$ where ${B}_{i}=[{a}_{i},{b}_{i}]\subset [0,1]$ for $i=1,2,\dots ,n$. Then, we have:$$\begin{array}{c}\hfill {V}_{C}\left(B\right)={\int}_{B}dC({u}_{1},\dots ,{u}_{n})=\sum _{v\in B}\mathrm{sign}\left(v\right)C\left(v\right)\ge 0,\end{array}$$$$\begin{array}{c}\hfill sign\left(v\right)=\left\{\begin{array}{c}1,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{4pt}{0ex}}if\phantom{\rule{1.em}{0ex}}{v}_{k}={a}_{k},\phantom{\rule{1.em}{0ex}}for\phantom{\rule{4.pt}{0ex}}an\phantom{\rule{4.pt}{0ex}}even\phantom{\rule{4.pt}{0ex}}number\phantom{\rule{4.pt}{0ex}}of\phantom{\rule{4.pt}{0ex}}k\u2019s;and\hfill \\ -1,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}if\phantom{\rule{1.em}{0ex}}{v}_{k}={a}_{k},\phantom{\rule{1.em}{0ex}}for\phantom{\rule{4.pt}{0ex}}an\phantom{\rule{4.pt}{0ex}}odd\phantom{\rule{4.pt}{0ex}}number\phantom{\rule{4.pt}{0ex}}of\phantom{\rule{4.pt}{0ex}}k\u2019s.\hfill \end{array}\right.\end{array}$$

## 4. Theory

#### 4.1. Bivariate Model

**Theorem**

**1.**

**Proof.**

**Corollary**

**1.**

#### 4.2. Multivariate Model

**Theorem**

**2.**

**Proof.**

**Corollary**

**2.**

**Corollary**

**3.**

**Corollary**

**4.**

## 5. Simulation Study

- (i)
- For each y belonging to the sequence $\{0,0.01,0.02,0.03,\dots ,4\}$, generate the uniform random variable U on the unit interval; that is, $U\sim Uniform(0,1)$ with the sample size N, say N =10,000.
- (ii)
- Estimate the values for ${f}_{Y}\left(y\right)$ and ${F}_{Y}\left(y\right)$ by using$$\begin{array}{ccc}\hfill {\widehat{f}}_{Y}\left(y\right)& \approx & \frac{1}{N}\sum _{i=1}^{N}\frac{1}{|{F}_{1}^{-1}\left({u}_{i}\right)|}{c}_{\theta}\left({u}_{i},{F}_{2}\left(\frac{y}{{F}_{1}^{-1}\left({u}_{i}\right)}\right)\right){f}_{2}\left(\frac{y}{{F}_{1}^{-1}\left({u}_{i}\right)}\right),\hfill \end{array}$$$$\begin{array}{ccc}\hfill {\widehat{F}}_{Y}\left(y\right)& \approx & {F}_{1}\left(0\right)+\frac{1}{N}\sum _{i=1}^{N}\mathrm{sgn}\left({F}_{1}^{-1}\left({u}_{i}\right)\right)\frac{\partial}{\partial u}{C}_{\theta}\left({u}_{i},{F}_{2}\left(\frac{y}{{F}_{1}^{-1}\left({u}_{i}\right)}\right)\right),\hfill \end{array}$$
- (iii)
- Plot ${\widehat{f}}_{Y}\left(y\right)$ and ${\widehat{F}}_{Y}\left(y\right)$, with $y\in \{0,0.01,0.02,0.03,\dots ,4\}$.

- (1)
- For each repetition $k=1,2,\dots ,5000$:
- (i)
- Generate $({X}_{1},{X}_{2})$ from ${H}_{\theta}({x}_{1},{x}_{2})$ of sample size ${10}^{4}$ by using the package copula in R language and define$${y}_{i}^{\left(k\right)}={x}_{1i}^{\left(k\right)}{x}_{2i}^{\left(k\right)},\phantom{\rule{1.em}{0ex}}i=1,2,\dots ,{10}^{4}.$$
- (ii)
- Estimate the mean ${\overline{y}}^{\left(k\right)}$, median ${\tilde{y}}^{\left(k\right)}$, standard deviation ${s}^{\left(k\right)}$, skewness $skew{\left(y\right)}^{\left(k\right)}$ and kurtosis $kur{\left(y\right)}^{\left(k\right)}$ of Y by using the following formula$$\begin{array}{ccc}\hfill {\overline{y}}^{\left(k\right)}& =& \frac{1}{{10}^{4}}\sum _{i=1}^{{10}^{4}}{y}_{i}^{\left(k\right)},\hfill \\ \hfill {\tilde{y}}^{\left(k\right)}& =& \frac{{y}_{\left(5000\right)}^{\left(k\right)}+{y}_{\left(5001\right)}^{\left(k\right)}}{2},\mathrm{where}{y}_{\left(j\right)}^{\left(k\right)}\mathrm{denotes}\mathrm{order}\mathrm{statistic}\mathrm{of}{y}^{\left(k\right)},\hfill \\ \hfill {s}^{\left(k\right)}& =& \sqrt{\frac{1}{{10}^{4}-1}\sum _{i=1}^{{10}^{4}}{\left({y}_{i}^{\left(k\right)}-{\overline{y}}^{\left(k\right)}\right)}^{2}},\hfill \\ \hfill skew{\left(y\right)}^{\left(k\right)}& =& \frac{\frac{1}{{10}^{4}}{\displaystyle \sum _{i=1}^{{10}^{4}}}{\left({y}_{i}^{\left(k\right)}-{\overline{y}}^{\left(k\right)}\right)}^{3}}{{\left[\frac{1}{{10}^{4}}{\displaystyle \sum _{i=1}^{{10}^{4}}}{\left({y}_{i}^{\left(k\right)}-{\overline{y}}^{\left(k\right)}\right)}^{2}\right]}^{3/2}},\hfill \\ \hfill kur{\left(y\right)}^{\left(k\right)}& =& \frac{\frac{1}{{10}^{4}}{\displaystyle \sum _{i=1}^{{10}^{4}}}{\left({y}_{i}^{\left(k\right)}-{\overline{y}}^{\left(k\right)}\right)}^{4}}{{\left[\frac{1}{{10}^{4}}{\displaystyle \sum _{i=1}^{{10}^{4}}}{\left({y}_{i}^{\left(k\right)}-{\overline{y}}^{\left(k\right)}\right)}^{2}\right]}^{2}}.\hfill \end{array}$$

- (2)
- Finally, take the mean for each of the above quantities by using the following formula:$$\begin{array}{ccc}\hfill \overline{y}& =& \frac{1}{5000}\sum _{k=1}^{5000}{\overline{y}}^{\left(k\right)},\hfill \\ \hfill \tilde{y}& =& \frac{1}{5000}\sum _{k=1}^{5000}{\tilde{y}}^{\left(k\right)},\hfill \\ \hfill s& =& \frac{1}{5000}\sum _{k=1}^{5000}{s}^{\left(k\right)},\hfill \\ \hfill skew\left(y\right)& =& \frac{1}{5000}\sum _{k=1}^{5000}skew{\left(y\right)}^{\left(k\right)},\hfill \\ \hfill kur\left(y\right)& =& \frac{1}{5000}\sum _{k=1}^{5000}kur{\left(y\right)}^{\left(k\right)},\hfill \end{array}$$

#### 5.1. Gaussian Copula

#### 5.2. Student-t Copula

#### 5.3. Clayton Copula

#### 5.4. Gumbel Copula

#### 5.5. Frank Copula

#### 5.6. Joe Copula

#### 5.7. Comparison of Copulas for the Same Measure of Dependence

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**PDFs and CDFs of the product of two log-normal distributed random variables having Gaussian Copulas.

**Figure 2.**PDFs and CDFs of the product of two log-normal distributed random variables having Student t-Copulas $\nu =3$.

**Figure 3.**PDFs and CDFs of the product of two log-normal distributed random variables having Clayton Copulas.

**Figure 4.**PDFs and CDFs of the product of two log-normal distributed random variables having Gumbel Copulas.

**Figure 5.**PDFs and CDFs of the product of two log-normal distributed random variables having Frank Copulas.

**Figure 6.**PDFs and CDFs of the product of two log-normal distributed random variables having Joe Copulas.

**Figure 7.**PDFs and CDFs of the product of two log-normal distributed random variables with six copulas having the same Kendall coefficient.

**Table 1.**Descriptive Statistics for $Y={X}_{1}{X}_{2}$ when $({X}_{1},{X}_{2})$ follows Gaussian copulas.

r | $\mathit{\tau}\left({\mathit{C}}_{\mathit{r}}\right)$ | Mean | Median | sd | Skewness | Kurtosis |
---|---|---|---|---|---|---|

$-0.9$ | $-0.71$ | 1.11 | 1 | 0.52 | 1.51 | 7.30 |

$-0.5$ | $-0.33$ | 1.65 | 1 | 2.16 | 5.84 | 85.43 |

0 | 0 | 2.72 | 1 | 6.77 | 14.27 | 466.41 |

0.5 | 0.33 | 4.48 | 1 | 18.21 | 23.23 | 1045.26 |

0.9 | 0.71 | 6.68 | 1 | 38.59 | 30.15 | 1585.27 |

**Table 2.**Descriptive Statistics for $Y={X}_{1}{X}_{2}$ when $({X}_{1},{X}_{2})$ follows Student-t copulas, $\nu =3$.

r | $\mathit{\tau}\left({\mathit{C}}_{\mathit{r}}\right)$ | Mean | Median | sd | Skewness | Kurtosis |
---|---|---|---|---|---|---|

$-0.9$ | $-0.71$ | 1.13 | 1 | 1.42 | 3.62 | 2221.42 |

$-0.5$ | $-0.33$ | 1.92 | 1 | 9.26 | 40.75 | 2562.20 |

0 | 0 | 3.30 | 1 | 20.44 | 37.01 | 2182.05 |

0.5 | 0.33 | 5.51 | 1 | 32.58 | 34.03 | 1903.21 |

0.9 | 0.71 | 6.89 | 1 | 44.09 | 32.31 | 1766.27 |

**Table 3.**Descriptive Statistics for $Y={X}_{1}{X}_{2}$ when $({X}_{1},{X}_{2})$ follows Clayton copulas.

$\mathit{\theta}$ | $\mathit{\tau}\left({\mathit{C}}_{\mathit{\theta}}\right)$ | Mean | Median | sd | Skewness | Kurtosis |
---|---|---|---|---|---|---|

1 | 0.33 | 3.53 | 1.12 | 8.93 | 13.12 | 403.90 |

2 | 0.5 | 4.01 | 1.13 | 10.43 | 12.81 | 389.76 |

3 | 0.60 | 4.34 | 1.11 | 11.55 | 12.65 | 379.99 |

4 | 0.67 | 4.59 | 1.08 | 12.41 | 12.36 | 359.40 |

**Table 4.**Descriptive Statistics for $Y={X}_{1}{X}_{2}$ when $({X}_{1},{X}_{2})$ follows Gumbel copulas.

$\mathit{\theta}$ | $\mathit{\tau}\left({\mathit{C}}_{\mathit{\theta}}\right)$ | Mean | Median | sd | Skewness | Kurtosis |
---|---|---|---|---|---|---|

1 | 0 | 2.72 | 1 | 6.78 | 14.41 | 477.10 |

2 | 0.5 | 6.47 | 0.95 | 41.52 | 32.34 | 1756.88 |

3 | 0.67 | 7.01 | 0.97 | 44.46 | 31.88 | 1719.80 |

4 | 0.75 | 7.19 | 0.98 | 44.83 | 31.66 | 1696.71 |

**Table 5.**Descriptive Statistics for $Y={X}_{1}{X}_{2}$ when $({X}_{1},{X}_{2})$ follows Frank copulas.

$\mathit{\theta}$ | $\mathit{\tau}\left({\mathit{C}}_{\mathit{\theta}}\right)$ | Mean | Median | sd | Skewness | Kurtosis |
---|---|---|---|---|---|---|

1 | 0.11 | 3.10 | 1 | 8.06 | 13.84 | 440.25 |

2 | 0.21 | 3.47 | 1 | 9.26 | 13.49 | 419.00 |

3 | 0.31 | 3.81 | 1 | 10.35 | 13.07 | 394.79 |

4 | 0.39 | 4.11 | 1 | 11.34 | 12.88 | 384.20 |

**Table 6.**Descriptive Statistics for $Y={X}_{1}{X}_{2}$ when $({X}_{1},{X}_{2})$ follows Joe copulas.

$\mathit{\theta}$ | $\mathit{\tau}\left({\mathit{C}}_{\mathit{\theta}}\right)$ | Mean | Median | sd | Skewness | Kurtosis |
---|---|---|---|---|---|---|

1 | 0 | 2.72 | 1.00 | 6.76 | 14.28 | 466.67 |

2 | 0.36 | 6.30 | 0.87 | 42.01 | 32.67 | 1789.96 |

3 | 0.52 | 6.91 | 0.88 | 44.31 | 31.65 | 1692.84 |

4 | 0.61 | 7.11 | 0.90 | 45.01 | 31.79 | 1720.81 |

**Table 7.**Descriptive Statistics for $Y={X}_{1}{X}_{2}$ in which $({X}_{1},{X}_{2})$ is modeled with six copulas having the same Kendall coefficient $\tau =0.49$.

Copulas | Parameters | $\mathit{\tau}\left(\mathit{C}\right)$ | Mean | Median | sd | Skewness | Kurtosis |
---|---|---|---|---|---|---|---|

Gaussian | 0.7 | 0.49 | 5.47 | 1.00 | 26.98 | 27.42 | 1375.98 |

Student-t | 0.7, $\nu =3$ | 0.49 | 5.95 | 1.00 | 38.33 | 33.58 | 1876.99 |

Clayton | 1.90 | 0.49 | 3.97 | 1.13 | 10.27 | 12.74 | 382.27 |

Gumbel | 1.95 | 0.49 | 6.42 | 0.95 | 41.54 | 32.41 | 1764.93 |

Frank | 5.5 | 0.49 | 4.47 | 1.00 | 12.58 | 12.59 | 362.17 |

Joe | 2.8 | 0.49 | 6.84 | 0.87 | 44.67 | 32.37 | 1766.43 |

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**MDPI and ACS Style**

Ly, S.; Pho, K.-H.; Ly, S.; Wong, W.-K.
Determining Distribution for the Product of Random Variables by Using Copulas. *Risks* **2019**, *7*, 23.
https://doi.org/10.3390/risks7010023

**AMA Style**

Ly S, Pho K-H, Ly S, Wong W-K.
Determining Distribution for the Product of Random Variables by Using Copulas. *Risks*. 2019; 7(1):23.
https://doi.org/10.3390/risks7010023

**Chicago/Turabian Style**

Ly, Sel, Kim-Hung Pho, Sal Ly, and Wing-Keung Wong.
2019. "Determining Distribution for the Product of Random Variables by Using Copulas" *Risks* 7, no. 1: 23.
https://doi.org/10.3390/risks7010023