# A New Kernel Estimator of Copulas Based on Beta Quantile Transformations

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Kernel Estimation of Copulas

#### Theoretical Results

**Theorem**

**1.**

**Proof**

**of Theorem 1.**

**Proposition**

**1.**

**Proof**

**of Proposition 1.**

- The statistics ${\widehat{S}}^{{r}_{l}}$ are approximated using a uniformly spaced grid $({u}_{j1},{u}_{j2})$, $j=1,\dots ,m$, of points on ${(0,1)}^{2}$, i.e., ${\widehat{S}}^{{r}_{l}}\approx \frac{1}{m}{\sum}_{j=1}^{m}{\left({\widehat{\mathbb{D}}}^{{r}_{l}}({u}_{j1},{u}_{j2})\right)}^{2}$.
- R independent copies of ${\widehat{\mathbb{D}}}^{{r}_{l}}$, ${\widehat{\mathbb{D}}}^{{r}_{l},\left(1\right)},\dots ,{\widehat{\mathbb{D}}}^{{r}_{l}\left(R\right)}$ are generated, such that$$\left({\widehat{\mathbb{D}}}^{{r}_{l}},{\widehat{\mathbb{D}}}^{{r}_{l},\left(1\right)},\dots {\widehat{\mathbb{D}}}^{{r}_{l},\left(R\right)}\right)\u27fc\left({\mathbb{D}}^{{r}_{l}},{\mathbb{D}}^{{r}_{l},\left(1\right)},\dots {\mathbb{D}}^{{r}_{l},\left(R\right)}\right),$$
- To calculate the copies of ${\widehat{S}}^{{r}_{l}}$ as ${\widehat{S}}^{{r}_{l},\left(k\right)}=\frac{1}{m}{\sum}_{j=1}^{m}{\widehat{\mathbb{D}}}^{{r}_{l},\left(k\right)}({u}_{j1},{u}_{j2})$ and to obtain the p-value of the statistics as:$$\frac{1}{R}\sum _{s=1}^{R}\mathbf{I}({\widehat{S}}^{{r}_{l},\left(s\right)}\ge {\widehat{S}}^{{r}_{l}}).$$

## 3. Simulation Study

#### 3.1. Analysing the Errors of Kernel Estimators

#### 3.2. Test for Extreme Value Copula

## 4. Data Analysis

- The fit of non parametric copulas to estimate the probability that the observed losses of two stock market indexes together exceed some percentiles, i.e., we estimate the value of $1-C(q,q)$, $q=0.9,0.95,0.99,0.995$, with the analysed kernels estimators.
- The test to analyse if the data is generated by an extreme value copula.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Procedure to Obtain Independent Copies

## Appendix B. Simulation Study

**Table A1.**Approximate MISE for the empirical copula $\times 1000$ (d.f. indicates degree of freedom).

Copula | Gaussian | Student’s t (d.f. = 1) | Student’s t (d.f. = 2) | Student’s t (d.f. = 3) | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Parameter | 0.9 | 0.5 | 0.3 | 0.9 | 0.5 | 0.3 | 0.9 | 0.5 | 0.3 | 0.9 | 0.5 | 0.3 |

n = 50 | 3.2473 | 3.0454 | 2.9277 | 3.3064 | 3.1751 | 2.9481 | 3.4585 | 3.1341 | 2.8430 | 3.3167 | 3.0923 | 2.9834 |

n = 500 | 0.3152 | 0.3104 | 0.2984 | 0.3356 | 0.3101 | 0.2967 | 0.3272 | 0.3104 | 0.2984 | 0.3276 | 0.3098 | 0.2971 |

Copula | Frank | Clayton | Gumbel | |||||||||

Parameter | 1 | 2 | 3 | 1 | 2 | 3 | 2 | 3 | 4 | |||

n = 50 | 3.1141 | 3.0819 | 3.3664 | 3.2126 | 3.3294 | 3.4487 | 7.8489 | 3.4600 | 3.4147 | |||

n = 500 | 0.2873 | 0.2966 | 0.3040 | 0.3089 | 0.3201 | 0.3238 | 0.3081 | 0.3058 | 0.3029 |

## Appendix C. Application

Model | |
---|---|

Spain | $ARMA(0,0)-GARCH(0,0)$ |

Germany | $ARMA(0,0)-GARCH(0,0)$ |

France | $ARMA(0,0)-GARCH(1,1)$ |

Italy | $ARMA(0,0)-GARCH(0,0)$ |

Portugal | $ARMA(1,0)-GARCH(0,0)$ |

UK | $ARMA(0,0)-GARCH(1,1)$ |

USA (DJ) | $ARMA(0,0)-GARCH(1,1)$ |

USA (SP) | $ARMA(0,0)-GARCH(1,1)$ |

Hong Kong | $ARMA(0,0)-GARCH(1,1)$ |

**Table A3.**Values of $1-C(q,q)$ for Spain and th countriese analysed, estimated with the empirical copula and Gaussian transformed kernel estimator.

Empirical Copula | ${\widehat{\mathit{C}}}^{\mathit{G}}$ | |||||||
---|---|---|---|---|---|---|---|---|

$\mathit{q}$ | 0.9 | 0.95 | 0.99 | 0.995 | 0.9 | 0.95 | 0.99 | 0.995 |

Germany | 0.1294 | 0.0706 | 0.0157 | 0.0078 | 0.1490 | 0.0794 | 0.0176 | 0.0081 |

France | 0.1339 | 0.0709 | 0.0118 | 0.0079 | 0.1469 | 0.0807 | 0.0171 | 0.0077 |

Italy | 0.1294 | 0.0667 | 0.0118 | 0.0067 | 0.1464 | 0.0774 | 0.0150 | 0.0039 |

Portugal | 0.1417 | 0.0669 | 0.0118 | 0.0079 | 0.1566 | 0.0797 | 0.0162 | 0.0076 |

UK | 0.1378 | 0.0709 | 0.0157 | 0.0079 | 0.1549 | 0.0820 | 0.0184 | 0.0082 |

USA (DJ) | 0.1417 | 0.0630 | 0.0118 | 0.0079 | 0.1546 | 0.0807 | 0.0169 | 0.0078 |

USA (SP) | 0.1378 | 0.0709 | 0.0118 | 0.0079 | 0.1538 | 0.0836 | 0.0165 | 0.0075 |

Hong Kong | 0.1378 | 0.0709 | 0.0118 | 0.0079 | 0.1538 | 0.0836 | 0.0179 | 0.0075 |

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**Figure 1.**Losses of Spanish stock market index (dashed line in blue) and stock market indexes of four countries in the European Union (solid line in black).

**Figure 2.**Losses of Spanish stock market index (dashed line in blue) and stock market indexes of four countries outside of the European Union (solid line in black).

**Figure 3.**Pseudo-data for each bivariate copula between Spain and Germany, Italy, France and Portugal.

**Figure 4.**Pseudo-data for each bivariate copula between Spain and UK, USA (DJ and SP) and Hong Kong.

**Table 1.**Quotient betweenthe approximate MISE of kernel estimators of the copula (numerator) and approximate MISE of the empirical copula (denominator) for elliptical copulas (* indicates optimal bandwidth).

$\mathit{n}=50$ | |||||||

Copula | Parameter | ${\widehat{\mathit{C}}}^{\mathit{B}}$ | ${\widehat{\mathit{C}}}^{\mathit{B}}$ * | ${\widehat{\mathit{C}}}^{\mathit{B}}$ | ${\widehat{\mathit{C}}}^{\mathit{G}}$ * | $\widehat{\mathit{C}}$ | $\widehat{\mathit{C}}$ * |

Gaussian | 0.9 | 0.7727 | 0.6910 | 0.7861 | 0.7423 | 3.8676 | 0.7079 |

0.5 | 0.6649 | 0.5879 | 0.6795 | 0.6993 | 2.7843 | 0.6108 | |

0.3 | 0.6394 | 0.5889 | 0.6553 | 0.6572 | 2.5860 | 0.6046 | |

Student’s t (d.f. = 1) | 0.9 | 0.8186 | 0.6516 | 0.8350 | 0.7120 | 3.8347 | 0.6626 |

0.5 | 0.7211 | 0.6294 | 0.7368 | 0.6734 | 2.9403 | 0.6679 | |

0.3 | 0.7005 | 0.6793 | 0.7176 | 0.7254 | 2.9272 | 0.6467 | |

Student’s t (d.f. = 2) | 0.9 | 0.7907 | 0.6405 | 0.8055 | 0.6769 | 3.6564 | 0.6650 |

0.5 | 0.7022 | 0.6338 | 0.7163 | 0.6569 | 2.8777 | 0.6432 | |

0.3 | 0.6592 | 0.6676 | 0.6746 | 0.7255 | 2.8678 | 0.6519 | |

Student’s t (d.f. = 3) | 0.9 | 0.7962 | 0.6671 | 0.8112 | 0.6782 | 3.8211 | 0.7376 |

0.5 | 0.6912 | 0.6354 | 0.7083 | 0.6506 | 2.8528 | 0.5949 | |

0.3 | 0.6626 | 0.6243 | 0.6790 | 0.6636 | 2.6940 | 0.6084 | |

n = 500 | |||||||

Copula | Parameter | ${\widehat{\mathit{C}}}^{\mathit{B}}$ | ${\widehat{\mathit{C}}}^{\mathit{B}}$ * | ${\widehat{\mathit{C}}}^{\mathit{B}}$ | ${\widehat{\mathit{C}}}^{\mathit{G}}$ * | $\widehat{\mathit{C}}$ | $\widehat{\mathit{C}}$ * |

Gaussian | 0.9 | 0.9099 | 0.8032 | 0.9199 | 0.8347 | 6.3718 | 0.8162 |

0.5 | 0.8445 | 0.7300 | 0.8532 | 0.7915 | 3.8651 | 0.7603 | |

0.3 | 0.8298 | 0.7316 | 0.8388 | 0.7905 | 3.4954 | 0.7757 | |

Student’s t (d.f. = 1) | 0.9 | 0.9232 | 0.8285 | 0.9337 | 0.8479 | 6.0640 | 0.8557 |

0.5 | 0.8513 | 0.7903 | 0.8602 | 0.8267 | 4.1395 | 0.8244 | |

0.3 | 0.8361 | 0.7852 | 0.8454 | 0.8201 | 3.8260 | 0.8224 | |

Student’s t (d.f. = 2) | 0.9 | 0.9129 | 0.8110 | 0.9231 | 0.8371 | 6.1449 | 0.8448 |

0.5 | 0.8445 | 0.7733 | 0.8532 | 0.8229 | 3.8651 | 0.8109 | |

0.3 | 0.8298 | 0.7710 | 0.8388 | 0.8172 | 3.4954 | 0.8123 | |

Student’s t (d.f. = 3) | 0.9 | 0.9289 | 0.8202 | 0.9389 | 0.8438 | 6.2509 | 0.8560 |

0.5 | 0.8529 | 0.7772 | 0.8619 | 0.8286 | 3.8559 | 0.8145 | |

0.3 | 0.8384 | 0.7748 | 0.8479 | 0.8243 | 3.4502 | 0.8147 |

**Table 2.**Quotient between the approximate MISE of kernel estimators of the copula (numerator) and approximate MISE of the empirical copula (denominator) for archimedean copulas (* indicates optimal bandwidth).

n = 50 | |||||||

Copula | Parameter | ${\widehat{\mathit{C}}}^{\mathit{B}}$ | ${\widehat{\mathit{C}}}^{\mathit{B}}$ * | ${\widehat{\mathit{C}}}^{\mathit{B}}$ | ${\widehat{\mathit{C}}}^{\mathit{G}}$ * | $\widehat{\mathit{C}}$ | $\widehat{\mathit{C}}$ * |

Frank | 1 | 0.6469 | 0.1641 | 0.6620 | 0.5961 | 2.3603 | 0.5979 |

2 | 0.6710 | 0.5689 | 0.6858 | 0.6412 | 2.5716 | 0.5896 | |

3 | 0.6784 | 0.5684 | 0.6925 | 0.6099 | 2.5097 | 0.5584 | |

Clayton | 1 | 0.6859 | 0.5703 | 0.7012 | 0.7032 | 2.4038 | 0.6370 |

2 | 0.7645 | 0.5924 | 0.7803 | 0.7263 | 2.8428 | 0.5983 | |

3 | 0.7839 | 0.6616 | 0.7968 | 0.7124 | 3.1101 | 0.6493 | |

Gumbel | 2 | 0.6531 | 0.2448 | 0.6645 | 0.2707 | 2.3911 | 0.2469 |

3 | 0.7739 | 0.5976 | 0.7862 | 0.6517 | 3.6354 | 0.5600 | |

4 | 0.7781 | 0.6332 | 0.7959 | 0.6510 | 3.8202 | 0.6145 | |

n = 500 | |||||||

Copula | Parameter | ${\widehat{\mathit{C}}}^{\mathit{B}}$ | ${\widehat{\mathit{C}}}^{\mathit{B}}$ * | ${\widehat{\mathit{C}}}^{\mathit{B}}$ | ${\widehat{\mathit{C}}}^{\mathit{G}}$ * | $\widehat{\mathit{C}}$ | $\widehat{\mathit{C}}$* |

Frank | 1 | 0.8117 | 0.7460 | 0.8216 | 0.8067 | 3.0013 | 0.7950 |

2 | 0.8204 | 0.7417 | 0.8304 | 0.8096 | 3.1943 | 0.7854 | |

3 | 0.8298 | 0.7596 | 0.8401 | 0.8117 | 3.5081 | 0.7807 | |

Clayton | 1 | 0.8393 | 0.7561 | 0.8484 | 0.8253 | 3.3645 | 0.8131 |

2 | 0.8640 | 0.7674 | 0.8724 | 0.8291 | 4.3028 | 0.8225 | |

3 | 0.8844 | 0.7811 | 0.8934 | 0.8303 | 5.0583 | 0.8290 | |

Gumbel | 2 | 0.8577 | 0.7698 | 0.8662 | 0.8180 | 4.8203 | 0.8009 |

3 | 0.8996 | 0.7982 | 0.9102 | 0.8286 | 6.1713 | 0.8298 | |

4 | 0.9294 | 0.8148 | 0.9421 | 0.8357 | 6.9994 | 0.8444 |

Empirical Copula | ||||||||

Gumbel | Student’s $\mathit{t}$ (d.f. = 1) | Student’s $\mathit{t}$ (d.f. = 2) | Student’s $\mathit{t}$ (d.f. = 3) | |||||

Parameter | 2 | 4 | 0.9 | 0.3 | 0.9 | 0.3 | 0.9 | 0.3 |

$\alpha =0.10$ | 0.08 | 0.02 | 0.25 | 0.38 | 0.59 | 0.86 | 0.70 | 0.56 |

$\alpha =0.05$ | 0.01 | 0.00 | 0.16 | 0.22 | 0.39 | 0.71 | 0.48 | 0.36 |

$\alpha =0.01$ | 0.00 | 0.00 | 0.03 | 0.07 | 0.14 | 0.39 | 0.23 | 0.13 |

${\widehat{\mathit{C}}}^{\mathit{B}}$ | ||||||||

Gumbel | Student’s $\mathit{t}$ (d.f. = 1) | Student’s $\mathit{t}$ (d.f. = 2) | Student’s $\mathit{t}$ (d.f. = 3) | |||||

Parameter | 2 | 4 | 0.9 | 0.3 | 0.9 | 0.3 | 0.9 | 0.3 |

$\alpha =0.10$ | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | |

$\alpha =0.05$ | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | |

$\alpha =0.01$ | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

Empirical Copula | ||||||

Gaussian | Frank | Clayton | ||||

Parameter | 0.9 | 0.3 | 1 | 3 | 1 | 3 |

$\alpha =0.10$ | 0.50 | 0.18 | 0.34 | 0.32 | 0.70 | 0.69 |

$\alpha =0.05$ | 0.61 | 0.37 | 0.54 | 0.53 | 0.80 | 0.87 |

$\alpha =0.01$ | 0.81 | 0.63 | 0.73 | 0.79 | 0.98 | 0.95 |

${\widehat{\mathit{C}}}^{\mathit{B}}$ | ||||||

Gaussian | Frank | Clayton | ||||

Parameter | 0.9 | 0.3 | 1 | 3 | 1 | 3 |

$\alpha =0.10$ | 0.67 | 0.50 | 0.6 | 0.62 | 0.80 | 0.81 |

$\alpha =0.05$ | 0.85 | 0.77 | 0.80 | 0.79 | 0.95 | 0.98 |

$\alpha =0.01$ | 0.92 | 0.90 | 0.82 | 0.78 | 1.00 | 1.00 |

**Table 5.**Descriptive statistics of monthly losses in percentage: Means, Standard Deviation (STD), Minimum (Min), First Quantile (Q25), Median, Third Quantile (Q75), Maximum (Mas), Skewness and Kurtosis.

Means | STD | Min | Q25 | Median | Q75 | Max | Skewness | Kurtosis | |
---|---|---|---|---|---|---|---|---|---|

Spain | 0.0565 | 2.5931 | −9.7537 | −1.3088 | −0.2354 | 1.3759 | 10.9100 | 0.3025 | 2.0327 |

Germany | −0.1196 | 2.6469 | −8.4139 | −1.7341 | −0.3275 | 1.1770 | 12.7390 | 0.8978 | 2.9100 |

France | 0.0043 | 2.2675 | −7.9611 | −1.3871 | −0.3142 | 1.2895 | 8.3501 | 0.5444 | 1.4149 |

Italy | 0.1044 | 2.6915 | −8.9727 | −1.5668 | −0.2481 | 1.5654 | 11.0363 | 0.4579 | 1.5547 |

Portugal | 0.1564 | 2.3768 | −7.2752 | −1.2973 | 0.0391 | 1.4856 | 10.1398 | 0.5301 | 1.6257 |

UK | 0.0080 | 1.7670 | −5.0582 | −1.1605 | −0.3326 | 0.9555 | 6.4533 | 0.6996 | 1.1175 |

USA (DJ) | −0.1711 | 1.8644 | −4.8586 | −1.2329 | −0.3472 | 0.7277 | 6.5807 | 0.6714 | 1.3456 |

USA (SP) | −0.1650 | 1.9120 | −5.1864 | −1.2883 | −0.4313 | 0.7868 | 8.0621 | 0.7009 | 1.4097 |

Hong Kong | −0.0919 | 2.6021 | −6.8459 | −1.6894 | −0.4514 | 1.2787 | 11.0508 | 0.5713 | 1.1150 |

q | 0.9 | 0.95 | 0.99 | 0.995 |
---|---|---|---|---|

Germany | 0.1533 | 0.0848 | 0.0246 | 0.0151 |

France | 0.1500 | 0.0853 | 0.0244 | 0.0150 |

Italy | 0.1497 | 0.0827 | 0.0234 | 0.0143 |

Portugal | 0.1583 | 0.0858 | 0.0234 | 0.0145 |

UK | 0.1585 | 0.0879 | 0.0253 | 0.0155 |

USA (DJ) | 0.1583 | 0.0862 | 0.0242 | 0.0149 |

USA (SP) | 0.1579 | 0.0880 | 0.0246 | 0.0149 |

Hong Kong | 0.1662 | 0.0928 | 0.0255 | 0.0155 |

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Bolancé, C.; Acuña, C.A.
A New Kernel Estimator of Copulas Based on Beta Quantile Transformations. *Mathematics* **2021**, *9*, 1078.
https://doi.org/10.3390/math9101078

**AMA Style**

Bolancé C, Acuña CA.
A New Kernel Estimator of Copulas Based on Beta Quantile Transformations. *Mathematics*. 2021; 9(10):1078.
https://doi.org/10.3390/math9101078

**Chicago/Turabian Style**

Bolancé, Catalina, and Carlos Alberto Acuña.
2021. "A New Kernel Estimator of Copulas Based on Beta Quantile Transformations" *Mathematics* 9, no. 10: 1078.
https://doi.org/10.3390/math9101078