# Nonparametric Directional Dependence Estimation and Its Application to Cryptocurrency

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## Abstract

**:**

## 1. Introduction

## 2. Copula and Directional Dependence

## 3. Nonparametric Directional Dependence and Numerical Study

- (1)
- Choose a measure-preserving function $f:[0,1]\to [0,1]$ that captures the desired dependence structure.
- (2)
- Choose a parametric family of copulas $\{{C}_{\alpha}:\phantom{\rule{3.33333pt}{0ex}}\alpha \phantom{\rule{3.33333pt}{0ex}}\mathrm{is}\phantom{\rule{3.33333pt}{0ex}}a\phantom{\rule{3.33333pt}{0ex}}\mathrm{copula}\phantom{\rule{3.33333pt}{0ex}}\mathrm{parameter}\}$ (the copulas ${C}_{\alpha}$ admit densities ${\varphi}_{\alpha}$).
- (3)
- The resulting copula density function is given as ${p}_{\alpha}({u}_{1},{u}_{2})={\varphi}_{\alpha}(f\left({x}_{1}\right),{x}_{2})$ and the corresponding distribution function is given as ${D}_{\alpha}({u}_{1},{u}_{2})={\int}_{0}^{{u}_{1}}{\int}_{0}^{{u}_{2}}{p}_{\alpha}({s}_{1},{s}_{2})d{s}_{1}d{s}_{2}$.

## 4. Real Data Illustration

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Regression functions between the transformed variables with their best approximations (dotted line) under the first scenario.

**Figure 2.**Regression functions between the transformed variables with their best approximations (dotted line) under the second scenario.

Sample Size | $\mathit{U}\to \mathit{V}$ | $\mathit{V}\to \mathit{U}$ | ||
---|---|---|---|---|

Nonparametric DD | Copula DD | Nonparametric DD | Copula DD | |

n = 100 | 0.077 | 0.083 | 0.076 | 0.056 |

n = 200 | 0.050 | 0.076 | 0.050 | 0.042 |

n = 400 | 0.035 | 0.072 | 0.033 | 0.031 |

n = 800 | 0.025 | 0.071 | 0.023 | 0.025 |

n = 1600 | 0.018 | 0.069 | 0.016 | 0.021 |

Sample Size | $\mathit{U}\to \mathit{V}$ | $\mathit{V}\to \mathit{U}$ | ||
---|---|---|---|---|

Nonparametric DD | Copula DD | Nonparametric DD | Copula DD | |

$n=100$ | 0.072 | 0.208 | 0.076 | 0.034 |

$n=200$ | 0.052 | 0.212 | 0.040 | 0.022 |

$n=400$ | 0.038 | 0.213 | 0.025 | 0.017 |

$n=800$ | 0.028 | 0.214 | 0.015 | 0.011 |

$n=1600$ | 0.021 | 0.214 | 0.010 | 0.008 |

LBTC | LETH | LXRP | LXLM | |
---|---|---|---|---|

Mean | 0.213 | 0.277 | 0.242 | 0.260 |

Standard Deviation | 4.133 | 5.661 | 7.896 | 9.118 |

Sample variance | 17.080 | 32.046 | 62.342 | 83.145 |

Kurtosis | 11.885 | 8.674 | 34.177 | 16.897 |

Skewness | −0.797 | −0.418 | 2.135 | 1.133 |

Range | 68.985 | 84.084 | 179.176 | 138.629 |

Minimum | −46.473 | −55.071 | −69.315 | −69.315 |

Maximum | 22.512 | 29.013 | 109.861 | 69.315 |

Sum | 387.253 | 503.490 | 439.491 | 472.758 |

Count | 1817 | 1817 | 1817 | 1817 |

**Table 4.**Directional dependence (DD) computed from the nonparametric and copula methods (the direction is from the row to the column).

Nonparametric DD Method | ||||

Coin | LBTC | LETH | LXRP | LXLM |

LBTC | 0.45 | 0.26 | 0.18 | |

LETH | 0.44 | 0.36 | 0.21 | |

LXRP | 0.26 | 0.35 | 0.26 | |

LXLM | 0.18 | 0.23 | 0.27 | |

Copula DD Method | ||||

Coin | LBTC | LETH | LXRP | LXLM |

LBTC | 0.39 | 0.21 | 0.14 | |

LETH | 0.4 | 0.29 | 0.18 | |

LXRP | 0.22 | 0.3 | 0.23 | |

LXLM | 0.14 | 0.17 | 0.21 |

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**MDPI and ACS Style**

Noh, H.; Jang, H.; Kim, K.H.; Kim, J.-M.
Nonparametric Directional Dependence Estimation and Its Application to Cryptocurrency. *Axioms* **2023**, *12*, 293.
https://doi.org/10.3390/axioms12030293

**AMA Style**

Noh H, Jang H, Kim KH, Kim J-M.
Nonparametric Directional Dependence Estimation and Its Application to Cryptocurrency. *Axioms*. 2023; 12(3):293.
https://doi.org/10.3390/axioms12030293

**Chicago/Turabian Style**

Noh, Hohsuk, Hyuna Jang, Kun Ho Kim, and Jong-Min Kim.
2023. "Nonparametric Directional Dependence Estimation and Its Application to Cryptocurrency" *Axioms* 12, no. 3: 293.
https://doi.org/10.3390/axioms12030293