# Copula Model Selection for Vehicle Component Failures Based on Warranty Claims

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## Abstract

**:**

## 1. Introduction

- simultaneous failures of two or more components caused by a common event;
- long-term maintenance and exploitation conditions shared by the entire system causing excessive wear and tear of all its components;
- failure of one component putting other components under additional pressure and causing their excessive wear and tear.

## 2. Data Description

#### 2.1. Hyundai Warranty Claims

#### 2.2. Engine Assembly Components

## 3. Copula Models of Dependence

#### 3.1. Types of Copulas

**Hypothesis 1**

**(H1).**

**Hypothesis 2**

**(H2).**

**Hypothesis 3**

**(H3).**

**Hypothesis 4**

**(H4).**

#### 3.2. Tail Dependence

## 4. Estimation of Copula Parameters

#### 4.1. Right Censoring (I)

#### 4.2. Conditioning to Failures in the Warranty Period (II)

## 5. Comparison of Copula Classes

#### 5.1. Tail Dependence

#### 5.2. Information Criteria

#### 5.3. Kolmogorov–Smirnov Statistic

## 6. Bayesian Model Selection

## 7. Higher Dimensions

## 8. Conclusions

- simultaneous failure as a result of one critical event;
- similar exploitation and maintenance patterns;
- wear and tear of engine components due to other components’ malfunction;

**R**environment (Brechmann and Schepsmeier 2013; Kojadinovic and Yan 2010) allow for a straightforward implementation of parametric and semi-parametric estimation for different classes of copulas including those discussed in the paper: direct and dual Archimedean copulas (Clayton and Gumbel–Hougaard classes).

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

TTF | Time-to-failure |

MPLE | Maximum pseudolikelihood estimate |

AIC | Akaike information criterion |

BIC | Bayes information criterion |

c.d.f. | Cumulative distribution function |

e.c.d.f. | Empirical cumulative distribution function |

KS | Kolmogorov–Smirnov |

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Component | All | A | B | C | D | E |
---|---|---|---|---|---|---|

A | 1883 | 467 | 441 | 152 | 307 | |

B | 1745 | 260 | 160 | 258 | ||

C | 1646 | 271 | 686 | |||

D | 1860 | 267 | ||||

E | 10,178 |

H1 $\widehat{\mathit{\alpha}}$ | H1 $\widehat{\mathit{\tau}}$ | |
---|---|---|

AB | 2.64 | 0.57 |

AC | −0.05 | −0.03 |

AD | −0.20 | −0.33 |

AE | −0.57 | −0.40 |

BC | −0.27 | −0.16 |

BD | 0.16 | 0.08 |

BE | −0.51 | −0.34 |

CD | −0.41 | −0.25 |

CE | −0.59 | −0.42 |

DE | −0.61 | −044 |

Shape | Scale | |
---|---|---|

A | 1.25 | 393 |

(0.02) | (6.4) | |

B | 1.52 | 521 |

(0.03) | (8.2) | |

C | 1.09 | 351 |

(0.01) | (4.4) | |

D | 1.92 | 643 |

(0.03) | (7.1) | |

E | 1.35 | 480 |

(0.01) | (4.1) |

**Table 4.**Estimates of association conditioned to two failures (good matches indicated by asterisks).

H1 $\widehat{\mathit{\tau}}$ | H2 $\widehat{\mathit{\tau}}$ | H3 $\widehat{\mathit{\tau}}$ | H4 $\widehat{\mathit{\tau}}$ | Sample $\mathit{\tau}$ | Sample Size | |
---|---|---|---|---|---|---|

AB | 0.43 | 0.63 | 0.58 | 0.56 | 0.707 | 467 |

(0.02) | (0.01) | (0.02) | (0.02) | |||

AC | 0.12 | 0.28* | 0.28 * | 0.14 | 0.269 | 441 |

(0.02) | (0.01) | (0.03) | (0.02) | |||

AD | 0.20 | 0.36 * | 0.32 * | 0.26 | 0.375 | 152 |

(0.03) | (0.04) | (0.04) | (0.04) | |||

AE | 0.12 | 0.24 | 0.22 | 0.12 | 0.314 | 307 |

(0.01) | (0.03) | (0.03) | (0.02) | |||

BC | 0.16 | 0.32 * | 0.28 * | 0.20 | 0.307 | 260 |

(0.03) | (0.03) | (0.03) | (0.04) | |||

BD | 0.24 | 0.40 * | 0.36 * | 0.33 * | 0.400 | 160 |

(0.04) | (0.04) | (0.04) | (0.04) | |||

BE | 0.12 | 0.33 * | 0.32 | 0.15 | 0.386 | 258 |

(0.02) | (0.03) | (0.03) | (0.03) | |||

CD | 0.12 | 0.31 | 0.32 | 0.15 | 0.452 | 271 |

(0.02) | (0.03) | (0.03) | (0.03) | |||

CE | 0.17 | 0.30 * | 0.28 | 0.23 | 0.318 | 686 |

(0.02) | (0.02) | (0.02) | (0.02) | |||

DE | 0.18 * | 0.27 | 0.25 | 0.23 * | 0.188 | 267 |

(0.03) | (0.04) | (0.03) | (0.04) |

H1 (Lower) | H2 (Upper) | H3 (Upper) | H4 (Lower) | |
---|---|---|---|---|

AB | 0.63 | 0.71 | 0.77 | 0.65 |

(0.03) | (0.02) | (0.01) | (0.03) | |

AC | 0.07 | 0.35 | 0.40 | 0.19 |

(0.03) | (0.09) | (0.05) | (0.15) | |

AD | 0.26 | 0.44 | 0.47 | 0.33 |

(0.07) | (0.09) | (0.07) | (0.14) | |

AE | 0.07 | 0.30 | 0.29 | 0.16 |

(0.02) | (0.11) | (0.06) | (0.14) | |

BC | 0.16 | 0.39 | 0.42 | 0.26 |

(0.06) | (0.09) | (0.06) | (0.17) | |

BD | 0.33 | 0.48 | 0.54 | 0.41 |

(0.08) | (0.09) | (0.05) | (0.13) | |

BE | 0.08 | 0.41 | 0.47 | 0.19 |

(0.04) | (0.07) | (0.05) | (0.16) | |

CD | 0.07 | 0.38 | 0.47 | 0.20 |

(0.04) | (0.09) | (0.04) | (0.20) | |

CE | 0.19 | 0.38 | 0.41 | 0.29 |

(0.04) | (0.06) | (0.04) | (0.08) | |

DE | 0.21 | 0.14 | 0.35 | 0.28 |

(0.07) | (0.04) | (0.06) | (0.14) |

H1 | H2 | H3 | H4 | |
---|---|---|---|---|

AB | 0 | 1 | 0 | 0 |

AC | 0 | 0.996 | 0.004 | 0 |

AD | 0 | 0.995 | 0.001 | 0.004 |

AE | 0 | 0.997 | 0 | 0.003 |

BC | 0 | 0.999 | 0.0005 | 0.0005 |

BD | 0 | 0.989 | 0.010 | 0.001 |

BE | 0 | 1 | 0 | 0 |

CD | 0 | 0.136 | 0.864 | 0 |

CE | 0 | 1 | 0 | 0 |

DE | 0.001 | 0.983 | 0.009 | 0.007 |

Parameter | Estimate | St.error | z-value | Pr(>|z|) |
---|---|---|---|---|

correlation (AB) | 0.8015 | 0.0019 | 430.2 | $<2\times {10}^{-16}$ |

correlation (AC) | 0.8073 | 0.0016 | 506.5 | $<2\times {10}^{-16}$ |

correlation (AD) | 0.7609 | 0.0028 | 276.2 | $<2\times {10}^{-16}$ |

correlation (AE) | 0.7241 | 0.0034 | 212.8 | $<2\times {10}^{-16}$ |

degrees of freedom | 1.9329 | NA | NA | NA |

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**MDPI and ACS Style**

Wifvat, K.; Kumerow, J.; Shemyakin, A.
Copula Model Selection for Vehicle Component Failures Based on Warranty Claims. *Risks* **2020**, *8*, 56.
https://doi.org/10.3390/risks8020056

**AMA Style**

Wifvat K, Kumerow J, Shemyakin A.
Copula Model Selection for Vehicle Component Failures Based on Warranty Claims. *Risks*. 2020; 8(2):56.
https://doi.org/10.3390/risks8020056

**Chicago/Turabian Style**

Wifvat, Kathryn, John Kumerow, and Arkady Shemyakin.
2020. "Copula Model Selection for Vehicle Component Failures Based on Warranty Claims" *Risks* 8, no. 2: 56.
https://doi.org/10.3390/risks8020056