# Adaptive Bernstein Copulas and Risk Management

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Some Important Facts about Multivariate Bernstein Polynomials

**Definition**

**1.**

**Proposition**

**1.**

**Proof.**

**Example**

**1.**

**Definition**

**2.**

**Proposition**

**2.**

**Proof.**

## 3. From Bernstein Polynomials to Bernstein Copulas

**Remark**

**1.**

**Example**

**2.**

**Proposition**

**3.**

**Proof.**

**Corollary**

**1.**

**Proof.**

**Remark**

**2.**

## 4. Empirical Bernstein Copulas

- Step 1: Select an index N randomly and uniformly among 1, …, n.
- Step 2: Generate d independent beta distributed random variables ${V}_{1}$, …, ${V}_{d}$ (also independent of N), where ${V}_{i}$ follows a beta distribution with parameters ${r}_{iN}$ and $n+1-{r}_{iN}$, $i=1,\dots ,d$.

**Example**

**3.**

## 5. Adaptive Bernstein Copula Estimation

- Step 1: AugmentationSelect an integer M such that all ${n}_{i}$, $i=1,\dots ,d$, are divisors of M, for instance, their least common multiple. We construct pseudo-ranks ${r}_{ij}^{+}$ in the following way:$${r}_{ij}^{+}:={r}_{i,\u2308\frac{j}{M}\u2309}M+\left(\u2308\frac{j}{M}\u2309-1\right)M+1-j,\phantom{\rule{1.em}{0ex}}i=1,\dots ,d,\phantom{\rule{1.em}{0ex}}j=1,\dots ,Mn.$$Here $\lceil x\rceil :=min\{m\in \mathbb{Z}\phantom{\rule{0.166667em}{0ex}}|\phantom{\rule{0.166667em}{0ex}}x\le m\}$, $x\in \mathbb{R}$, stands for “rounding up”. Let ${\mathbf{U}}^{+}=({U}_{1}^{+},\dots ,{U}_{d}^{+})$ be the uniformly discretly distributed random vector over ${\{0,1,\dots ,Mn-1\}}^{d}$ with support points ${\mathbf{s}}_{1}$, …, ${\mathbf{s}}_{Mn}$, where ${\mathbf{s}}_{j}=({r}_{1j}^{+}-1,\dots ,{r}_{dj}^{+}-1)$, $j=1,\dots ,nM$. Please note that the probability mass is $\frac{1}{Mn}$ for each support point, and that ${\mathbf{U}}^{+}$ is an admissible discrete skeleton.
- Step 2: ReductionConstruct the final ranks ${r}_{ij}^{\ast}$ in the following way:$${r}_{ij}^{\ast}:=\u2308\frac{{r}_{ij}^{+}{n}_{i}}{nM}\u2309,\phantom{\rule{1.em}{0ex}}i=1,\dots ,d,\phantom{\rule{1.em}{0ex}}j=1,\dots ,Mn.$$It follows from the above definition that there will be replicates in the final ranks and that ${r}_{ij}^{\ast}$ takes values in the set ${T}_{i}^{\ast}=\{0,1,\dots ,{n}_{i}-1\}$. A point $\mathbf{s}=({s}_{1},\dots ,{s}_{d})$ will be a support point of the final admissible skeleton ${\mathbf{U}}^{\ast}$ if there exist final ranks such that $\mathbf{s}=({r}_{1,{j}_{1}},\dots ,{r}_{d,{j}_{d}})$ for some ${j}_{1},\dots ,{j}_{d}\in \{1,\dots ,Mn\}$. The probability mass attached to $\mathbf{s}$ is given by the number $\frac{K}{Mn}$, where K is the number of rank combinations $({r}_{1,{j}_{1}},\dots ,{r}_{d,{j}_{d}})$ that lead to the same $\mathbf{s}$. This also enables very simple Monte Carlo realizations of the corresponding Bernstein copula as described in Section 4 by first selecting an index N randomly and uniformly among 1, …, $Mn$ and then by proceeding as in Step 2 there with all of the ${r}_{ij}^{\ast}$.

## 6. Applications to Risk Management

**Example**

**5.**

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Bernstein, S. Démonstration du théorème de Weierstrass fondée sur le calcul des probabilités. Commun. Kharkov Math. Soc.
**1912**, 13, 1–2. [Google Scholar] - Lorentz, G.G. Bernstein Polynomials, 2nd ed.; Chelsea Publishing Company: New York, NY, USA, 1986. [Google Scholar]
- Babu, G.J.; Canty, A.J.; Chaubey, Y.P. Application of Bernstein polynomials for smooth estimation of a distribution and density function. J. Statist. Plann. Inference
**2002**, 105, 377–392. [Google Scholar] [CrossRef] - Cheng, C. The Bernstein polynomial estimator of a smooth quantile function. Statist. Probab. Lett.
**1995**, 24, 321–330. [Google Scholar] [CrossRef] - Cottin, C.; Pfeifer, D. From Bernstein polynomials to Bernstein copulas. J. Appl. Funct. Anal.
**2014**, 9, 277–288. [Google Scholar] - Guan, Z. Efficient and robust density estimation using Bernstein type polynomials. J. Nonparametr. Stat.
**2016**, 28, 250–271. [Google Scholar] [CrossRef] [Green Version] - Guan, Z. Bernstein polynomial model for grouped continuous data. J. Nonparametr. Stat.
**2017**, 29, 831–848. [Google Scholar] [CrossRef] [Green Version] - Leblanc, A. On estimating distribution functions using Bernstein polynomials. Ann. Inst. Statist. Math.
**2012**, 64, 919–943. [Google Scholar] [CrossRef] - Pfeifer, D.; Nešlehová, J. Modeling dependence in finance and insurance: The copula approach. Bl. DGVFM
**2003**, 26, 177–191. [Google Scholar] [CrossRef] - Pfeifer, D.; Straßburger, D.; Philipps, J. Modelling and simulation of dependence structures in nonlife insurance with Bernstein copulas. In Proceedings of the Paper presented on the occasion of the International ASTIN Colloquium, Helsinki, Finland, 1–4 June 2009. [Google Scholar]
- Sancetta, A.; Satchell, S. The Bernstein copula and its applications to modeling and approximations of multivariate distributions. Econom. Theory
**2004**, 20, 535–562. [Google Scholar] [CrossRef] - Segers, J.; Sibuya, M.; Tsukahara, H. The empirical beta copula. J. Multivar. Anal.
**2017**, 155, 35–51. [Google Scholar] [CrossRef] [Green Version] - Vil’chevskii, N.O.; Shevlyakov, G.L. On the Bernstein polynomial estimators of distribution and quantile functions. J. Math. Sci.
**2001**, 105, 2626–2629. [Google Scholar] [CrossRef] - Wang, T.; Guan, Z. Bernstein polynomial model for nonparametric multivariate density. Statistics
**2018**, 53, 321–338. [Google Scholar] [CrossRef] [Green Version] - Durante, F.; Sempi, C. Principles of Copula Theory; CRC Press: Boca Raton, FL, USA, 2018. [Google Scholar]
- Ibragimov, R.; Prokhorov, A. Heavy Tails and Copulas. Topics in Dependence Modelling in Economics and Finance; World Scientific: Singapore, 2017. [Google Scholar]
- Rose, D. Modeling and Estimating Multivariate Dependence Structures with the Bernstein Copula. Ph.D. Thesis, Ludwig-Maximilians-Universität, München, Germany, 8 October 2015. Available online: https://edoc.ub.uni-muenchen.de/18757/ (accessed on 8 December 2020).
- Avomo Ngomo, J.S.-L. Entwicklung und Implementierung eines Verfahrens zur Optimierung des Speicheraufwands bei Bernstein-und Verwandten Copulas. Ph.D. Thesis, Carl von Ossietzky Universität, Oldenburg, Germany, 18 December 2017. Available online: http://oops.uni-oldenburg.de/3457/ (accessed on 8 December 2020).
- Masuhr, A.; Trede, M. Bayesian estimation of generalized partition of unity copulas. Depend. Model.
**2020**, 8, 119–131. [Google Scholar] [CrossRef] - Deheuvels, P. La fonction de dépendance empirique et ses propriétés. Un test non paramétrique d’indépendance. Acad. R. Belg. Bull. Sci.
**1979**, 65, 274–292. [Google Scholar] [CrossRef] - Butzer, P.L. On two-dimensional Bernstein polynomials. Canad. J. Math.
**1953**, 5, 107–113. [Google Scholar] [CrossRef] - Pfeifer, D.; Ragulina, O. Generating VaR scenarios under Solvency II with product beta distributions. Risks
**2018**, 6, 122. [Google Scholar] [CrossRef] [Green Version] - Junker, R.R.; Griessenberger, F.; Trutschnig, W. Estimating scale-invariant directed dependence of bivariate distributions. Comput. Statist. Data Anal.
**2021**, 153, 107058. [Google Scholar] [CrossRef] - Pfeifer, D.; Mändle, A.; Ragulina, O.; Girschig, C. New copulas based on general partitions-of-unity (part III)—the continuous case. Depend. Model.
**2019**, 7, 181–201. [Google Scholar] [CrossRef] - Pfeifer, D.; Tsatedem, H.A.; Mändle, A.; Girschig, C. New copulas based on general partitions-of-unity and their applications to risk management. Depend. Model.
**2016**, 4, 123–140. [Google Scholar] [CrossRef] [Green Version] - Pfeifer, D.; Mändle, A.; Ragulina, O. New copulas based on general partitions-of-unity and their applications to risk management (part II). Depend. Model.
**2017**, 5, 246–255. [Google Scholar] [CrossRef]

**Figure 1.**Plots of the functions in Example 1. (

**a**) Plot of f(x,y). (

**b**) Plot of B

**f(x,y). (**

_{n}**c**) Plot of f(x,y) − B

**f(x,y).**

_{n}**Figure 2.**Plots of the functions in Example 2, n

_{1}= 3 and n

_{2}= 5. (

**a**) Plot of F(x,y). (

**b**) Plot of B

**F(x,y). (**

_{n}**c**) Plot of b

**F(x,y).**

_{n}**Figure 3.**Plots of the functions in Example 2, n

_{1}= 11 and n

_{2}= 7. (

**a**) Plot of F(x,y). (

**b**) Plot of B

**F(x,y). (**

_{n}**c**) Plot of b

**F(x,y).**

_{n}**Figure 4.**Plots of the functions in Example 2, n

_{1}= 50 and n

_{2}= 50. (

**a**) Plot of F(x,y). (

**b**) Plot of B

**F(x,y). (**

_{n}**c**) Plot of b

**F(x,y).**

_{n}**Figure 5.**Plots for the empirical Bernstein copula in Example 3. (

**a**) Support points of scaled skeleton. (

**b**) Empirical Bernstein copula density contour plot. (

**c**) Simulation of 5000 empirical Bernstein copula pairs.

**Figure 6.**Plots for the Gaussian copula in Example 3. (

**a**) Support points of scaled skeleton. (

**b**) Gaussian copula density contour plot. (

**c**) Simulation of 5000 Gaussian copula pairs.

**Figure 7.**Plots of the copula densities in Example 4. (

**a**) Copula density ${c}_{\mathbf{U}}({x}_{1},{x}_{2})$. (

**b**) Copula density ${c}_{{\mathbf{U}}^{\ast}}({x}_{1},{x}_{2})$

**Figure 8.**Contour plots for the Bernstein copula densities in Example 3. (

**a**) Bernstein copula density ${c}_{1}({x}_{1},{x}_{2})$. (

**b**) Bernstein copula density ${c}_{2}({x}_{1},{x}_{2})$

**Figure 9.**Contour plots for the adaptive Bernstein copula densities in Example 3. (

**a**) Adaptive Bernstein copula density ${c}_{3}({x}_{1},{x}_{2})$. (

**b**) Adaptive Bernstein copula density ${c}_{4}({x}_{1},{x}_{2})$

**Figure 10.**Plots related to the adaptive Bernstein copulas in Example 3. (

**a**) Support points of the adaptive scaled discrete skeleton, n

_{1}= n

_{2}= 10. (

**b**) Simulation of 5000 adapted Bernstein copula pairs, n

_{1}= n

_{2}= 10. (

**c**) Support points of the adaptive scaled discrete skeleton, n

_{1}= n

_{2}= 5. (

**d**) Simulation of 5000 adapted Bernstein copula pairs, n

_{1}= n

_{2}= 5. (

**e**) Support points of the adaptive scaled discrete skeleton, n

_{1}= n

_{2}= 5. (

**f**) Simulation of 5000 adapted Bernstein copula pairs, n

_{1}= n

_{2}= 5.

**Figure 11.**Empirical histograms for the densities of the aggregated risk in Example 3. (

**a**) Bernstein copula, grid type 34 × 34. (

**b**) Bernstein copula, grid type 10 × 10. (

**c**) Bernstein copula, grid type 5 × 5. (

**d**) Bernstein copula, grid type 4 × 4. (

**e**) Gaussian copula. (

**f**) Independence copula. (

**g**) Comonotonicity copula. (

**h**) Countermonotonicity copula.

**Figure 12.**Simulation of 5000 adaptive Bernstein copula points in Example 5, m = 100. (

**a**) Area 1 vs. Area 13. (

**b**) Area 3 vs. Area 4. (

**c**) Area 7 vs. Area 16. (

**d**) Area 3 vs. Area 18.

**Figure 13.**Simulation of 5000 adaptive Bernstein copula points in Example 5, m = 20. (

**a**) Area 1 vs. Area 13. (

**b**) Area 3 vs. Area 4. (

**c**) Area 7 vs. Area 16. (

**d**) Area 3 vs. Area 18.

**Figure 14.**Simulation of 5000 adaptive Bernstein copula points in Example 5, m = 17. (

**a**) Area 1 vs. Area 13. (

**b**) Area 3 vs. Area 4. (

**c**) Area 7 vs. Area 16. (

**d**) Area 3 vs. Area 18.

**Figure 15.**Simulation of 5000 adaptive Bernstein copula points in Example 5, m = 13. (

**a**) Area 1 vs. Area 13. (

**b**) Area 3 vs. Area 4. (

**c**) Area 7 vs. Area 16. (

**d**) Area 3 vs. Area 18.

**Figure 16.**Simulation of 5000 adaptive Bernstein copula points in Example 5, m = 7. (

**a**) Area 1 vs. Area 13. (

**b**) Area 3 vs. Area 4. (

**c**) Area 7 vs. Area 16. (

**d**) Area 3 vs. Area 18.

**Table 1.**Values of $\Delta {f}_{{\mathbf{a}}_{i}}^{{\mathbf{b}}_{i}}$ for the polynomial $f(x,y)$ in Example 1.

${i}_{1}$ | 0 | 1 | 0 | 1 | 0 | 1 |

${i}_{2}$ | 0 | 0 | 1 | 1 | 2 | 2 |

$\Delta {f}_{{\mathbf{a}}_{i}}^{{\mathbf{b}}_{i}}$ | $-\frac{145}{216}$ | $-\frac{151}{216}$ | $\frac{49}{216}$ | $-\frac{41}{216}$ | $\frac{149}{72}$ | $\frac{19}{72}$ |

**Table 2.**Values of $\Delta {g}_{{\mathbf{a}}_{i}}^{{\mathbf{b}}_{i}}$ for the modified polynomial $g(x,y)$.

${i}_{1}$ | 0 | 1 | 0 | 1 | 0 | 1 |

${i}_{2}$ | 0 | 0 | 1 | 1 | 2 | 2 |

$\Delta {g}_{{\mathbf{a}}_{i}}^{{\mathbf{b}}_{i}}$ | $\frac{71}{216}$ | $\frac{65}{216}$ | $\frac{265}{216}$ | $\frac{175}{216}$ | $\frac{221}{72}$ | $\frac{91}{72}$ |

$\mathbb{P}(\mathit{X}={\mathit{x}}_{\mathit{i}},\mathit{Y}={\mathit{y}}_{\mathit{j}})$ | ${\mathit{x}}_{1}=0.2$ | ${\mathit{x}}_{2}=0.7$ |
---|---|---|

${y}_{1}=0.3$ | 0.3 | 0.2 |

${y}_{2}=0.5$ | 0.2 | 0.3 |

**Table 4.**Ranks ${r}_{ij}$ for observed insurance data from windstorm ($i=1$) and flooding ($i=2$) losses in Example 3.

$\mathit{i}\setminus \mathit{j}$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 |

1 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 |

2 | 12 | 5 | 31 | 7 | 24 | 18 | 17 | 3 | 2 | 19 | 10 | 9 | 21 | 15 | 14 | 4 | 6 |

$\mathit{i}\setminus \mathit{j}$ | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 | 31 | 32 | 33 | 34 |

1 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 | 31 | 32 | 33 | 34 |

2 | 34 | 1 | 23 | 11 | 29 | 33 | 13 | 8 | 20 | 32 | 28 | 22 | 16 | 26 | 25 | 30 | 27 |

$\mathit{j}\setminus \mathit{i}$ | 1 | 2 | $\mathit{p}({\mathit{r}}_{1},{\mathit{r}}_{2})$ |
---|---|---|---|

1 | 1 | 3 | 0.2 |

2 | 2 | 4 | 0.2 |

3 | 3 | 1 | 0.2 |

4 | 4 | 2 | 0.2 |

5 | 5 | 5 | 0.2 |

**Table 6.**Resulting pseudo-ranks ${r}_{ij}^{+}$ and probabilities $p({r}_{1}^{+},{r}_{2}^{+})$ in Example 4.

$\mathit{j}\setminus \mathit{i}$ | 1 | 2 | $\mathit{p}({\mathit{r}}_{1}^{+},{\mathit{r}}_{2}^{+})$ |
---|---|---|---|

1 | 12 | 36 | $0.01\overline{6}$ |

2 | 11 | 35 | $0.01\overline{6}$ |

3 | 10 | 34 | $0.01\overline{6}$ |

⋮ | ⋮ | ⋮ | ⋮ |

13 | 24 | 48 | $0.01\overline{6}$ |

14 | 23 | 47 | $0.01\overline{6}$ |

15 | 22 | 46 | $0.01\overline{6}$ |

⋮ | ⋮ | ⋮ | ⋮ |

25 | 36 | 12 | $0.01\overline{6}$ |

26 | 35 | 11 | $0.01\overline{6}$ |

27 | 34 | 10 | $0.01\overline{6}$ |

⋮ | ⋮ | ⋮ | ⋮ |

58 | 51 | 51 | $0.01\overline{6}$ |

59 | 50 | 50 | $0.01\overline{6}$ |

60 | 49 | 49 | $0.01\overline{6}$ |

**Table 7.**Final ranks ${r}_{ij}^{\ast}$ and probabilities $p({r}_{1}^{\ast},{r}_{2}^{\ast})$ in Example 4.

$\mathit{j}\phantom{\rule{0.277778em}{0ex}}\setminus \phantom{\rule{0.277778em}{0ex}}\mathit{i}$ | 1 | 2 | $\mathit{p}({\mathit{r}}_{1}^{\ast},{\mathit{r}}_{2}^{\ast})$ |
---|---|---|---|

1 | 1 | 2 | $0.1$ |

2 | 1 | 3 | $0.2\overline{3}$ |

3 | 2 | 1 | $0.25$ |

4 | 2 | 2 | $0.01\overline{6}$ |

5 | 2 | 3 | $0.01\overline{6}$ |

6 | 2 | 4 | $0.05$ |

7 | 3 | 2 | $0.1\overline{3}$ |

8 | 3 | 4 | $0.2$ |

$\mathit{j}\phantom{\rule{0.277778em}{0ex}}\setminus \phantom{\rule{0.277778em}{0ex}}\mathit{i}$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
---|---|---|---|---|---|---|---|---|---|---|

9 | 0.0032 | 0.0000 | 0.0022 | 0.0000 | 0.0032 | 0.0266 | 0.0320 | 0.0274 | 0.0028 | 0.0028 |

8 | 0.0318 | 0.0000 | 0.0014 | 0.0000 | 0.0024 | 0.0000 | 0.0312 | 0.0000 | 0.0020 | 0.0314 |

7 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0204 | 0.0251 | 0.0545 |

6 | 0.0032 | 0.0275 | 0.0022 | 0.0000 | 0.0032 | 0.0265 | 0.0026 | 0.0000 | 0.0322 | 0.0028 |

5 | 0.0003 | 0.0246 | 0.0287 | 0.0215 | 0.0003 | 0.0000 | 0.0000 | 0.0246 | 0.0000 | 0.0000 |

4 | 0.0034 | 0.0278 | 0.0024 | 0.0246 | 0.0034 | 0.0000 | 0.0029 | 0.0000 | 0.0324 | 0.0030 |

3 | 0.0266 | 0.0000 | 0.0000 | 0.0000 | 0.0266 | 0.0206 | 0.0261 | 0.0000 | 0.0000 | 0.0000 |

2 | 0.0034 | 0.0000 | 0.0025 | 0.0540 | 0.0034 | 0.0000 | 0.0029 | 0.0277 | 0.0031 | 0.0031 |

1 | 0.0252 | 0.0201 | 0.0000 | 0.0000 | 0.0546 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |

0 | 0.0029 | 0.0000 | 0.0607 | 0.0000 | 0.0029 | 0.0263 | 0.0023 | 0.0000 | 0.0025 | 0.0025 |

$\mathit{j}\phantom{\rule{0.277778em}{0ex}}\setminus \phantom{\rule{0.277778em}{0ex}}\mathit{i}$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
---|---|---|---|---|---|---|---|---|---|---|

9 | 0.0118 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0294 | 0.0294 | 0.0294 | 0.0000 | 0.0000 |

8 | 0.0176 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0294 | 0.0000 | 0.0235 | 0.0294 |

7 | 0.0000 | 0.0059 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0059 | 0.0176 | 0.0706 |

6 | 0.0000 | 0.0235 | 0.0000 | 0.0176 | 0.0000 | 0.0294 | 0.0000 | 0.0000 | 0.0294 | 0.0000 |

5 | 0.0000 | 0.0294 | 0.0294 | 0.0118 | 0.0000 | 0.0000 | 0.0000 | 0.0294 | 0.0000 | 0.0000 |

4 | 0.0000 | 0.0235 | 0.0059 | 0.0176 | 0.0235 | 0.0000 | 0.0000 | 0.0000 | 0.0294 | 0.0000 |

3 | 0.0294 | 0.0000 | 0.0000 | 0.0000 | 0.0176 | 0.0059 | 0.0412 | 0.0059 | 0.0000 | 0.0000 |

2 | 0.0000 | 0.0059 | 0.0059 | 0.0529 | 0.0000 | 0.0059 | 0.0000 | 0.0294 | 0.0000 | 0.0000 |

1 | 0.0412 | 0.0118 | 0.0000 | 0.0000 | 0.0471 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |

0 | 0.0000 | 0.0000 | 0.0588 | 0.0000 | 0.0118 | 0.0294 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |

Grid Type | $34\times 34$ | $10\times 10$ | $5\times 5$ | $4\times 4$ | Gaussian | Independence | Comonotonic | Countermonotonic |
---|---|---|---|---|---|---|---|---|

${\mathrm{VaR}}_{0.005}$ | 1348 | 1334 | 1356 | 1369 | 1386 | 1349 | 1500 | 1327 |

Year | Area 1 | Area 2 | Area 3 | Area 4 | Area 5 | Area 6 | Area 7 | Area 8 | Area 9 | Area 10 |
---|---|---|---|---|---|---|---|---|---|---|

1 | 23.664 | 154.664 | 40.569 | 14.504 | 10.468 | 7.464 | 22.202 | 17.682 | 12.395 | 18.551 |

2 | 1.080 | 59.545 | 3.297 | 1.344 | 1.859 | 0.477 | 6.107 | 7.196 | 1.436 | 3.720 |

3 | 21.731 | 31.049 | 55.973 | 5.816 | 14.869 | 20.771 | 3.580 | 14.509 | 17.175 | 87.307 |

4 | 28.990 | 31.052 | 30.328 | 4.709 | 0.717 | 3.530 | 6.032 | 6.512 | 0.682 | 3.115 |

5 | 53.616 | 62.027 | 57.639 | 1.804 | 2.073 | 4.361 | 46.018 | 22.612 | 1.581 | 11.179 |

6 | 29.950 | 41.722 | 12.964 | 1.127 | 1.063 | 4.873 | 6.571 | 11.966 | 15.676 | 24.263 |

7 | 3.474 | 14.429 | 10.869 | 0.945 | 2.198 | 1.484 | 4.547 | 2.556 | 0.456 | 1.137 |

8 | 10.020 | 31.283 | 21.116 | 1.663 | 2.153 | 0.932 | 25.163 | 3.222 | 1.581 | 5.477 |

9 | 5.816 | 14.804 | 128.072 | 0.523 | 0.324 | 0.477 | 3.049 | 7.791 | 4.079 | 7.002 |

10 | 170.725 | 576.767 | 108.361 | 41.599 | 20.253 | 35.412 | 126.698 | 71.079 | 21.762 | 64.582 |

11 | 21.423 | 50.595 | 4.360 | 0.327 | 1.566 | 64.621 | 5.650 | 1.258 | 0.626 | 3.556 |

12 | 6.380 | 28.316 | 3.740 | 0.442 | 0.736 | 0.470 | 3.406 | 7.859 | 0.894 | 3.591 |

13 | 124.665 | 33.359 | 14.712 | 0.321 | 0.975 | 2.005 | 3.981 | 4.769 | 2.006 | 1.973 |

14 | 20.165 | 49.948 | 17.658 | 0.595 | 0.548 | 29.350 | 6.782 | 4.873 | 2.921 | 6.394 |

15 | 78.106 | 41.681 | 13.753 | 0.585 | 0.259 | 0.765 | 7.013 | 9.426 | 2.180 | 3.769 |

16 | 11.067 | 444.712 | 365.351 | 99.366 | 8.856 | 28.654 | 10.589 | 13.621 | 9.589 | 19.485 |

17 | 6.704 | 81.895 | 14.266 | 0.972 | 0.519 | 0.644 | 8.057 | 18.071 | 5.515 | 13.163 |

18 | 15.550 | 277.643 | 26.564 | 0.788 | 0.225 | 1.230 | 26.800 | 64.538 | 2.637 | 80.711 |

19 | 10.099 | 18.815 | 9.352 | 2.051 | 1.089 | 6.102 | 2.678 | 4.064 | 2.373 | 2.057 |

20 | 8.492 | 138.708 | 46.708 | 3.680 | 1.132 | 1.698 | 165.600 | 7.926 | 2.972 | 5.237 |

Year | Area 11 | Area 12 | Area 13 | Area 14 | Area 15 | Area 16 | Area 17 | Area 18 | Area 19 |
---|---|---|---|---|---|---|---|---|---|

1 | 1.842 | 4.100 | 46.135 | 14.698 | 44.441 | 7.981 | 35.833 | 10.689 | 7.299 |

2 | 0.429 | 1.026 | 7.469 | 7.058 | 4.512 | 0.762 | 14.474 | 9.337 | 0.740 |

3 | 0.209 | 2.344 | 22.651 | 4.117 | 26.586 | 3.920 | 13.804 | 2.683 | 3.026 |

4 | 0.521 | 0.696 | 31.126 | 1.878 | 29.423 | 6.394 | 18.064 | 1.201 | 0.894 |

5 | 2.715 | 1.327 | 40.156 | 4.655 | 104.691 | 28.579 | 17.832 | 1.618 | 3.402 |

6 | 4.832 | 0.701 | 16.712 | 11.852 | 29.234 | 7.098 | 17.866 | 5.206 | 5.664 |

7 | 0.268 | 0.580 | 11.851 | 2.057 | 11.605 | 0.282 | 16.925 | 2.082 | 1.008 |

8 | 0.741 | 0.369 | 3.814 | 1.869 | 8.126 | 1.032 | 14.985 | 1.390 | 1.703 |

9 | 0.524 | 6.554 | 5.459 | 3.007 | 8.528 | 1.920 | 5.638 | 2.149 | 2.908 |

10 | 9.882 | 6.401 | 106.197 | 44.912 | 191.809 | 90.559 | 154.492 | 36.626 | 36.276 |

11 | 1.052 | 8.277 | 22.564 | 8.961 | 19.817 | 16.437 | 25.990 | 2.364 | 6.434 |

12 | 0.136 | 0.364 | 28.000 | 7.574 | 3.213 | 1.749 | 12.735 | 1.744 | 0.558 |

13 | 1.990 | 15.176 | 57.235 | 23.686 | 110.035 | 17.373 | 7.276 | 2.494 | 0.525 |

14 | 0.630 | 0.762 | 25.897 | 3.439 | 8.161 | 3.327 | 24.733 | 2.807 | 1.618 |

15 | 0.770 | 15.024 | 36.068 | 1.613 | 6.127 | 8.103 | 12.596 | 4.894 | 0.822 |

16 | 0.287 | 0.464 | 24.211 | 38.616 | 51.889 | 1.316 | 173.080 | 3.557 | 11.627 |

17 | 0.590 | 2.745 | 16.124 | 2.398 | 20.997 | 2.515 | 5.161 | 2.840 | 3.002 |

18 | 0.245 | 0.217 | 12.416 | 4.972 | 59.417 | 3.762 | 24.603 | 7.404 | 19.107 |

19 | 0.415 | 0.351 | 10.707 | 2.468 | 10.673 | 1.743 | 27.266 | 1.368 | 0.644 |

20 | 0.566 | 0.708 | 22.646 | 6.652 | 14.437 | 63.692 | 113.231 | 7.218 | 2.548 |

**Table 13.**Values of the parameters ${\mu}_{k}$ and ${\sigma}_{k}$ estimated from the log data in Example 5, part I.

Parameter | Area 1 | Area 2 | Area 3 | Area 4 | Area 5 | Area 6 | Area 7 | Area 8 | Area 9 | Area 10 |
---|---|---|---|---|---|---|---|---|---|---|

${\mu}_{k}$ | 2.8063 | 4.0717 | 3.1407 | 0.6375 | 0.3984 | 1.2227 | 2.3210 | 2.2123 | 1.0783 | 2.1055 |

${\sigma}_{k}$ | 1.2161 | 1.0521 | 1.2110 | 1.5685 | 1.2998 | 1.5987 | 1.1980 | 0.9882 | 1.1445 | 1.2531 |

**Table 14.**Values of the parameters ${\mu}_{k}$ and ${\sigma}_{k}$ estimated from the log data in Example 5, part II.

Parameter | Area 11 | Area 12 | Area 13 | Area 14 | Area 15 | Area 16 | Area 17 | Area 18 | Area 19 |
---|---|---|---|---|---|---|---|---|---|

${\mu}_{k}$ | −0.3231 | 0.3815 | 3.0198 | 1.7488 | 3.0409 | 1.5501 | 3.0700 | 1.2444 | 0.9378 |

${\sigma}_{k}$ | 1.0881 | 1.3353 | 0.8027 | 1.0033 | 1.1221 | 1.4765 | 0.9622 | 0.8577 | 1.2141 |

Area | A1 | A2 | A3 | A4 | A5 | A6 | A7 | A8 | A9 | A10 | A11 | A12 | A13 | A14 | A15 | A16 | A17 | A18 | A19 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

A1 | 1 | 0.46 | 0.03 | 0.16 | 0.47 | 0.20 | 0.35 | 0.49 | 0.41 | 0.24 | 0.78 | 0.64 | 0.91 | 0.63 | 0.85 | 0.66 | 0.30 | 0.67 | 0.56 |

A2 | 0.46 | 1 | 0.64 | 0.78 | 0.67 | 0.36 | 0.51 | 0.76 | 0.57 | 0.51 | 0.58 | −0.04 | 0.59 | 0.84 | 0.68 | 0.58 | 0.87 | 0.77 | 0.90 |

A3 | 0.03 | 0.64 | 1 | 0.93 | 0.41 | 0.26 | 0.11 | 0.16 | 0.33 | 0.16 | 0.08 | −0.09 | 0.13 | 0.64 | 0.25 | 0.10 | 0.74 | 0.14 | 0.35 |

A4 | 0.16 | 0.78 | 0.93 | 1 | 0.54 | 0.36 | 0.16 | 0.25 | 0.43 | 0.19 | 0.22 | −0.10 | 0.30 | 0.79 | 0.36 | 0.19 | 0.84 | 0.32 | 0.49 |

A5 | 0.47 | 0.67 | 0.41 | 0.54 | 1 | 0.41 | 0.35 | 0.51 | 0.84 | 0.63 | 0.59 | 0.02 | 0.64 | 0.67 | 0.59 | 0.50 | 0.58 | 0.71 | 0.67 |

A6 | 0.20 | 0.36 | 0.26 | 0.36 | 0.41 | 1 | 0.07 | 0.11 | 0.28 | 0.19 | 0.28 | 0.14 | 0.31 | 0.42 | 0.24 | 0.27 | 0.39 | 0.27 | 0.40 |

A7 | 0.35 | 0.51 | 0.11 | 0.16 | 0.35 | 0.07 | 1 | 0.44 | 0.27 | 0.19 | 0.48 | −0.07 | 0.46 | 0.35 | 0.45 | 0.91 | 0.64 | 0.61 | 0.49 |

A8 | 0.49 | 0.76 | 0.16 | 0.25 | 0.51 | 0.11 | 0.44 | 1 | 0.50 | 0.75 | 0.61 | −0.03 | 0.54 | 0.47 | 0.71 | 0.53 | 0.40 | 0.75 | 0.90 |

A9 | 0.41 | 0.57 | 0.33 | 0.43 | 0.84 | 0.28 | 0.27 | 0.50 | 1 | 0.66 | 0.68 | −0.01 | 0.52 | 0.60 | 0.50 | 0.41 | 0.46 | 0.65 | 0.63 |

A10 | 0.24 | 0.51 | 0.16 | 0.19 | 0.63 | 0.19 | 0.19 | 0.75 | 0.66 | 1 | 0.33 | −0.12 | 0.27 | 0.28 | 0.43 | 0.24 | 0.23 | 0.45 | 0.65 |

A11 | 0.78 | 0.58 | 0.08 | 0.22 | 0.59 | 0.28 | 0.48 | 0.61 | 0.68 | 0.33 | 1 | 0.19 | 0.79 | 0.65 | 0.80 | 0.73 | 0.43 | 0.84 | 0.74 |

A12 | 0.64 | −0.04 | −0.09 | −0.10 | 0.02 | 0.14 | −0.07 | −0.03 | −0.01 | −0.12 | 0.19 | 1 | 0.44 | 0.21 | 0.28 | 0.17 | −0.12 | 0.13 | 0.03 |

A13 | 0.91 | 0.59 | 0.13 | 0.30 | 0.64 | 0.31 | 0.46 | 0.54 | 0.52 | 0.27 | 0.79 | 0.44 | 1 | 0.71 | 0.86 | 0.74 | 0.47 | 0.76 | 0.65 |

A14 | 0.63 | 0.84 | 0.64 | 0.79 | 0.67 | 0.42 | 0.35 | 0.47 | 0.60 | 0.28 | 0.65 | 0.21 | 0.71 | 1 | 0.74 | 0.54 | 0.79 | 0.68 | 0.72 |

A15 | 0.85 | 0.68 | 0.25 | 0.36 | 0.59 | 0.24 | 0.45 | 0.71 | 0.50 | 0.43 | 0.80 | 0.28 | 0.86 | 0.74 | 1 | 0.69 | 0.47 | 0.71 | 0.75 |

A16 | 0.66 | 0.58 | 0.10 | 0.19 | 0.50 | 0.27 | 0.91 | 0.53 | 0.41 | 0.24 | 0.73 | 0.17 | 0.74 | 0.54 | 0.69 | 1 | 0.63 | 0.77 | 0.64 |

A17 | 0.30 | 0.87 | 0.74 | 0.84 | 0.58 | 0.39 | 0.64 | 0.40 | 0.46 | 0.23 | 0.43 | −0.12 | 0.47 | 0.79 | 0.47 | 0.63 | 1 | 0.59 | 0.64 |

A18 | 0.67 | 0.77 | 0.14 | 0.32 | 0.71 | 0.27 | 0.61 | 0.75 | 0.65 | 0.45 | 0.84 | 0.13 | 0.76 | 0.68 | 0.71 | 0.77 | 0.59 | 1 | 0.86 |

A19 | 0.56 | 0.90 | 0.35 | 0.49 | 0.67 | 0.40 | 0.49 | 0.90 | 0.63 | 0.65 | 0.74 | 0.03 | 0.65 | 0.72 | 0.75 | 0.64 | 0.64 | 0.86 | 1 |

Area | A1 | A2 | A3 | A4 | A5 | A6 | A7 | A8 | A9 | A10 | A11 | A12 | A13 | A14 | A15 | A16 | A17 | A18 | A19 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

A1 | 1 | 0.27 | 0.30 | 0.16 | 0.17 | 0.45 | 0.28 | 0.32 | 0.32 | 0.29 | 0.67 | 0.51 | 0.76 | 0.34 | 0.67 | 0.74 | 0.18 | 0.21 | 0.29 |

A2 | 0.27 | 1 | 0.48 | 0.66 | 0.39 | 0.37 | 0.71 | 0.69 | 0.52 | 0.64 | 0.30 | −0.02 | 0.45 | 0.66 | 0.58 | 0.45 | 0.73 | 0.74 | 0.78 |

A3 | 0.30 | 0.48 | 1 | 0.70 | 0.40 | 0.31 | 0.42 | 0.51 | 0.58 | 0.53 | 0.18 | 0.07 | 0.21 | 0.32 | 0.54 | 0.26 | 0.47 | 0.21 | 0.57 |

A4 | 0.16 | 0.66 | 0.70 | 1 | 0.77 | 0.47 | 0.46 | 0.47 | 0.59 | 0.49 | 0.18 | −0.13 | 0.33 | 0.50 | 0.47 | 0.18 | 0.76 | 0.43 | 0.54 |

A5 | 0.17 | 0.39 | 0.40 | 0.77 | 1 | 0.59 | 0.30 | 0.20 | 0.49 | 0.39 | 0.28 | 0.08 | 0.35 | 0.56 | 0.44 | 0.16 | 0.55 | 0.36 | 0.41 |

A6 | 0.45 | 0.37 | 0.31 | 0.47 | 0.59 | 1 | 0.14 | 0.01 | 0.36 | 0.34 | 0.33 | 0.12 | 0.48 | 0.46 | 0.48 | 0.37 | 0.59 | 0.17 | 0.50 |

A7 | 0.28 | 0.71 | 0.42 | 0.46 | 0.30 | 0.14 | 1 | 0.52 | 0.27 | 0.40 | 0.45 | −0.07 | 0.31 | 0.31 | 0.46 | 0.62 | 0.63 | 0.58 | 0.57 |

A8 | 0.32 | 0.69 | 0.51 | 0.47 | 0.20 | 0.01 | 0.52 | 1 | 0.64 | 0.81 | 0.27 | −0.02 | 0.38 | 0.35 | 0.56 | 0.35 | 0.28 | 0.62 | 0.63 |

A9 | 0.32 | 0.52 | 0.58 | 0.59 | 0.49 | 0.36 | 0.27 | 0.64 | 1 | 0.78 | 0.40 | 0.19 | 0.27 | 0.50 | 0.44 | 0.30 | 0.33 | 0.57 | 0.61 |

A10 | 0.29 | 0.64 | 0.53 | 0.49 | 0.39 | 0.34 | 0.40 | 0.81 | 0.78 | 1 | 0.21 | −0.02 | 0.21 | 0.37 | 0.52 | 0.30 | 0.31 | 0.53 | 0.81 |

A11 | 0.67 | 0.30 | 0.18 | 0.18 | 0.28 | 0.33 | 0.45 | 0.27 | 0.40 | 0.21 | 1 | 0.47 | 0.49 | 0.45 | 0.60 | 0.67 | 0.20 | 0.45 | 0.39 |

A12 | 0.51 | −0.02 | 0.07 | −0.13 | 0.08 | 0.12 | −0.07 | −0.02 | 0.19 | −0.02 | 0.47 | 1 | 0.44 | 0.21 | 0.24 | 0.46 | −0.23 | 0.25 | 0.05 |

A13 | 0.76 | 0.45 | 0.21 | 0.33 | 0.35 | 0.48 | 0.31 | 0.38 | 0.27 | 0.21 | 0.49 | 0.44 | 1 | 0.55 | 0.60 | 0.71 | 0.37 | 0.39 | 0.24 |

A14 | 0.34 | 0.66 | 0.32 | 0.50 | 0.56 | 0.46 | 0.31 | 0.35 | 0.50 | 0.37 | 0.45 | 0.21 | 0.55 | 1 | 0.59 | 0.43 | 0.57 | 0.58 | 0.53 |

A15 | 0.67 | 0.58 | 0.54 | 0.47 | 0.44 | 0.48 | 0.46 | 0.56 | 0.44 | 0.52 | 0.60 | 0.24 | 0.60 | 0.59 | 1 | 0.59 | 0.36 | 0.35 | 0.63 |

A16 | 0.74 | 0.45 | 0.26 | 0.18 | 0.16 | 0.37 | 0.62 | 0.35 | 0.30 | 0.30 | 0.67 | 0.46 | 0.71 | 0.43 | 0.59 | 1 | 0.38 | 0.43 | 0.39 |

A17 | 0.18 | 0.73 | 0.47 | 0.76 | 0.55 | 0.59 | 0.63 | 0.28 | 0.33 | 0.31 | 0.20 | −0.23 | 0.37 | 0.57 | 0.36 | 0.38 | 1 | 0.52 | 0.56 |

A18 | 0.21 | 0.74 | 0.21 | 0.43 | 0.36 | 0.17 | 0.58 | 0.62 | 0.57 | 0.53 | 0.45 | 0.25 | 0.39 | 0.58 | 0.35 | 0.43 | 0.52 | 1 | 0.60 |

A19 | 0.29 | 0.78 | 0.57 | 0.54 | 0.41 | 0.50 | 0.57 | 0.63 | 0.61 | 0.81 | 0.39 | 0.05 | 0.24 | 0.53 | 0.63 | 0.39 | 0.56 | 0.60 | 1 |

m | 100 | 20 | 17 | 13 | 7 |
---|---|---|---|---|---|

${\mathrm{VaR}}_{0.005}$ | 2842 | 2247 | 2204 | 2105 | 1878 |

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**MDPI and ACS Style**

Pfeifer, D.; Ragulina, O.
Adaptive Bernstein Copulas and Risk Management. *Mathematics* **2020**, *8*, 2221.
https://doi.org/10.3390/math8122221

**AMA Style**

Pfeifer D, Ragulina O.
Adaptive Bernstein Copulas and Risk Management. *Mathematics*. 2020; 8(12):2221.
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**Chicago/Turabian Style**

Pfeifer, Dietmar, and Olena Ragulina.
2020. "Adaptive Bernstein Copulas and Risk Management" *Mathematics* 8, no. 12: 2221.
https://doi.org/10.3390/math8122221