# Likelihood Inference for Copula Models Based on Left-Truncated and Competing Risks Data from Field Studies

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## Abstract

**:**

## 1. Introduction

## 2. Left-Truncation and Competing Risks

- ${l}_{1}=\{i:{T}_{1i}<{T}_{2i},{T}_{1i}<{C}_{i}\}$,
- ${l}_{2}=\{i:{T}_{2i}<{T}_{1i},{T}_{2i}<{C}_{i}\}$,
- ${l}_{0}=\{i:{C}_{i}<{T}_{1i},{C}_{i}<{T}_{2i}\}$.

## 3. Proposed Methods

#### 3.1. Copula Model for Competing Risks

#### 3.2. Likelihood Function

- (i)
- ${L}_{i}=P({T}_{1i}={t}_{i},{t}_{i}<{T}_{2i},{t}_{i}<{C}_{i})$ if $i\in {l}_{1}=\{i:{T}_{1i}<{T}_{2i},{T}_{1i}<{C}_{i}\}$ and ${\nu}_{i}=1$,
- (ii)
- ${L}_{i}=P({T}_{2i}={t}_{i},{t}_{i}<{T}_{1i},{t}_{i}<{C}_{i})$ if $i\in {l}_{2}=\{i:{T}_{2i}<{T}_{1i},{T}_{2i}<{C}_{i}\}\mathrm{and}{\nu}_{i}=1$,
- (iii)
- ${L}_{i}=P({C}_{i}={t}_{i},{t}_{i}<{T}_{1i},{t}_{i}<{T}_{2i})$ if $i\in {l}_{0}=\{i:{C}_{i}<{T}_{1i},{C}_{i}<{T}_{2i}\}\mathrm{and}{\nu}_{i}=1$,

- (iv)
- ${L}_{i}=P({T}_{1i}={t}_{i},{t}_{i}<{T}_{2i},{t}_{i}<{C}_{i}|{T}_{1i}\ge {\tau}_{i},{T}_{2i}\ge {\tau}_{i},{C}_{i}\ge {\tau}_{i})$ if $i\in {l}_{1}$ and ${\nu}_{i}=0$,
- (v)
- ${L}_{i}=P({T}_{2i}={t}_{i},{t}_{i}<{T}_{1i},{t}_{i}<{C}_{i}|{T}_{1i}\ge {\tau}_{i},{T}_{2i}\ge {\tau}_{i},{C}_{i}\ge {\tau}_{i})$ if $i\in {l}_{2}\mathrm{and}{\nu}_{i}=0$,
- (vi)
- ${L}_{i}=P({C}_{i}={t}_{i},{t}_{i}<{T}_{1i},{t}_{i}<{T}_{2i}|{T}_{1i}\ge {\tau}_{i},{T}_{2i}\ge {\tau}_{i},{C}_{i}\ge {\tau}_{i})$ if $i\in {l}_{0}\mathrm{and}{\nu}_{i}=0$.

#### 3.3. Weibull Model

## 4. Simulation Studies

#### 4.1. Simulation Settings

- (a)
**Decreasing hazard**: $\left({\alpha}_{i},{\lambda}_{i}\right)=\left(0.4,0.1\right),i\in \left\{1,2\right\}$; $s=10$; $e=13$,- (b)
**Constant hazard**: $\left({\alpha}_{i},{\lambda}_{i}\right)=\left(1.0,0.4\right),i\in \left\{1,2\right\}$; $s=10$; $e=13$,- (c)
**Increasing hazard**: $\left({\alpha}_{i},{\lambda}_{i}\right)=\left(1.5,1.0\right),i\in \left\{1,2\right\}$; $s=3$; $e=4$.

#### 4.2. Simulation Results

## 5. Data Analysis

## 6. Conclusions and Future Work

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Likelihood for the Gamma Model

## Appendix B. Likelihood for the Lognormal Model

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**Figure 1.**Left-truncated and competing risks data from a field study. Event 1 and Event 2 are subject to competing risks. Solid curves show observable times, and dashed curves show unobservable times. One can observe Event 1, Event 2, or Censoring, whichever occurs first, between the starting time (s) and the ending time (e). One cannot observe anything if Event 1 or Event 2 occurs before time s.

**Figure 2.**The shape parameter ${\alpha}_{i}$ and the scale parameter ${\lambda}_{i}$ used in the simulation study for the Weibull model: ${T}_{1}~{S}_{1}\left({t}_{1}\right)=\mathrm{exp}\left(-{\lambda}_{1}{t}_{1}^{{\alpha}_{1}}\right)$ and ${T}_{2}~{S}_{2}\left({t}_{2}\right)=\mathrm{exp}\left(-{\lambda}_{2}{t}_{2}^{{\alpha}_{2}}\right)$.

True Model | Fitted Model | Event 1 | Event 2 | ||
---|---|---|---|---|---|

${\alpha}_{1}$ = 0.4 | ${\lambda}_{1}$ = 0.1 | ${\alpha}_{2}$ = 0.4 | ${\lambda}_{2}$ = 0.1 | ||

Indep. ($\theta =0$) | Indep. ($\theta =0$) | 0.406 (0.006) | 0.099 (−0.001) | 0.403 (0.003) | 0.100 (−0.000) |

Clayton ($\theta =2$) | 0.441 (0.041) | 0.113 (0.011) | 0.437 (0.037) | 0.112 (0.012) | |

Clayton ($\theta =$ MLE) | 0.457 (0.057) | 0.149 (0.049) | 0.455 (0.055) | 0.149 (0.049) | |

Clayton ($\theta =$ PMLE) | 0.412 (0.012) | 0.101 (0.001) | 0.408 (0.008) | 0.101 (0.001) | |

Clayton ($\theta =2$) | Indep. ($\theta =0$) | 0.375 (−0.025) | 0.090 (−0.010) | 0.379 (−0.021) | 0.089 (−0.011) |

Clayton ($\theta =2$) | 0.404 (0.004) | 0.100 (−0.000) | 0.408 (0.008) | 0.100 (−0.000) | |

Clayton ($\theta =$ MLE) | 0.426 (0.026) | 0.132 (0.032) | 0.428 (0.028) | 0.132 (0.032) | |

Clayton ($\theta =$ PMLE) | 0.382 (−0.018) | 0.092 (−0.008) | 0.385 (−0.015) | 0.091 (−0.009) | |

${\alpha}_{1}$ = 1.0 | ${\lambda}_{1}$ = 0.4 | ${\alpha}_{2}$ = 1.0 | ${\lambda}_{2}$ = 0.4 | ||

Indep. ($\theta =0$) | Indep. ($\theta =0$) | 1.003 (0.003) | 0.398 (−0.002) | 1.001 (0.001) | 0.401 (0.001) |

Clayton ($\theta =2$) | 1.133 (0.133) | 0.557 (0.157) | 1.131 (0.131) | 0.559 (0.159) | |

Clayton ($\theta =$ MLE) | 1.082 (0.082) | 0.630 (0.230) | 1.081 (0.081) | 0.631 (0.231) | |

Clayton ($\theta =$ PMLE) | 1.109 (0.109) | 0.425 (0.025) | 1.107 (0.107) | 0.428 (0.028) | |

Clayton ($\theta =2$) | Indep. ($\theta =0$) | 0.869 (−0.131) | 0.298 (−0.102) | 0.869 (−0.131) | 0.300 (−0.101) |

Clayton ($\theta =2$) | 1.002 (0.002) | 0.399 (−0.001) | 1.001 (0.001) | 0.400 (0.000) | |

Clayton ($\theta =$ MLE) | 0.952 (−0.048) | 0.470 (0.070) | 0.951 (−0.049) | 0.471 (0.071) | |

Clayton ($\theta =$ PMLE) | 0.934 (−0.066) | 0.313 (−0.087) | 0.934 (−0.066) | 0.314 (−0.086) | |

${\alpha}_{1}$ = 1.5 | ${\lambda}_{1}$ = 1.0 | ${\alpha}_{2}$ = 1.5 | ${\lambda}_{2}$ = 1.0 | ||

Indep. ($\theta =0$) | Indep. ($\theta =0$) | 1.500 (−0.000) | 1.002 (0.002) | 1.501 (0.001) | 1.001 (0.001) |

Clayton ($\theta =2$) | 1.684 (0.184) | 1.575 (0.575) | 1.685 (0.185) | 1.574 (0.574) | |

Clayton ($\theta =$ MLE) | 1.642 (0.142) | 1.225 (0.225) | 1.643 (0.143) | 1.223 (0.223) | |

Clayton ($\theta =$ PMLE) | 1.671 (0.171) | 1.193 (0.193) | 1.672 (0.172) | 1.191 (0.191) | |

Clayton ($\theta =2$) | Indep. ($\theta =0$) | 1.315 (−0.185) | 0.666 (−0.334) | 1.316 (−0.184) | 0.665 (−0.335) |

Clayton ($\theta =2$) | 1.502 (0.001) | 1.002 (0.002) | 1.503 (0.003) | 1.002 (0.002) | |

Clayton ($\theta =$ MLE) | 1.496 (−0.004) | 0.991 (−0.009) | 1.496 (−0.004) | 0.990 (−0.010) | |

Clayton ($\theta =$ PMLE) | 1.426 (−0.073) | 0.747 (−0.025) | 1.426 (−0.073) | 0.746 (−0.253) |

**NOTE**: The simulation is based on simulated samples of $\left\{\left({t}_{i},{\tau}_{i},{\nu}_{i}\right);i=1,2,\dots ,1000\right\}$, consisting of 500 truncated samples and another 500 untruncated samples. The Clayton copula with $\theta =2$ yields Kendall’s tau = 0.5 for two failure times.

True Model | Fitted Model | Event 1 | Event 2 | ||
---|---|---|---|---|---|

${\alpha}_{1}$ = 0.4 | ${\lambda}_{1}$ = 0.1 | ${\alpha}_{2}$ = 0.4 | ${\lambda}_{2}$ = 0.1 | ||

$\mathrm{Independent}(\theta =0)$ | $\mathrm{Indep}.(\theta =0)$ | 0.00186 | 0.00017 | 0.00170 | 0.00017 |

$\mathrm{Clayton}(\theta =2)$ | 0.00374 | 0.00035 | 0.00335 | 0.00036 | |

Clayton ($\theta =$ MLE) | 0.00517 | 0.00320 | 0.00498 | 0.00321 | |

Clayton ($\theta =$ PMLE) | 0.00204 | 0.00018 | 0.00184 | 0.00017 | |

$\mathrm{Clayton}(\theta =2)$ | $\mathrm{Indep}.(\theta =0)$ | 0.00229 | 0.00025 | 0.00221 | 0.00028 |

$\mathrm{Clayton}(\theta =2)$ | 0.00195 | 0.00018 | 0.00207 | 0.00020 | |

Clayton ($\theta =$ MLE) | 0.00261 | 0.00178 | 0.00274 | 0.00178 | |

Clayton ($\theta =$ PMLE) | 0.00210 | 0.00022 | 0.00204 | 0.00025 | |

${\alpha}_{1}$ = 1.0 | ${\lambda}_{1}$ = 0.4 | ${\alpha}_{2}$ = 1.0 | ${\lambda}_{2}$ = 0.4 | ||

$\mathrm{Independent}(\theta =0)$ | $\mathrm{Indep}.(\theta =0)$ | 0.00111 | 0.00093 | 0.00111 | 0.00087 |

$\mathrm{Clayton}(\theta =2)$ | 0.01847 | 0.02621 | 0.01799 | 0.02673 | |

Clayton ($\theta =$ MLE) | 0.00115 | 0.07285 | 0.01124 | 0.07301 | |

Clayton ($\theta =$ PMLE) | 0.01323 | 0.00175 | 0.01266 | 0.00184 | |

$\mathrm{Clayton}(\theta =2)$ | $\mathrm{Indep}.(\theta =0)$ | 0.01831 | 0.01095 | 0.01814 | 0.01066 |

$\mathrm{Clayton}(\theta =2)$ | 0.00102 | 0.00090 | 0.00088 | 0.00081 | |

Clayton ($\theta =$ MLE) | 0.00629 | 0.01394 | 0.00622 | 0.01385 | |

Clayton ($\theta =$ PMLE) | 0.00575 | 0.00828 | 0.00560 | 0.00803 | |

${\alpha}_{1}$ = 1.5 | ${\lambda}_{1}$ = 1.0 | ${\alpha}_{2}$ = 1.5 | ${\lambda}_{2}$ = 1.0 | ||

$\mathrm{Independent}(\theta =0)$ | $\mathrm{Indep}.(\theta =0)$ | 0.00229 | 0.00346 | 0.00244 | 0.00314 |

$\mathrm{Clayton}(\theta =2)$ | 0.03547 | 0.33575 | 0.03570 | 0.33428 | |

Clayton ($\theta =$ MLE) | 0.02966 | 0.08040 | 0.02991 | 0.08363 | |

Clayton ($\theta =$ PMLE) | 0.03181 | 0.04223 | 0.03254 | 0.04108 | |

$\mathrm{Clayton}(\theta =2)$ | $\mathrm{Indep}.(\theta =0)$ | 0.03659 | 0.11357 | 0.03642 | 0.11415 |

$\mathrm{Clayton}(\theta =2)$ | 0.00193 | 0.00300 | 0.00179 | 0.00297 | |

Clayton ($\theta =$ MLE) | 0.00294 | 0.00875 | 0.00282 | 0.00867 | |

Clayton ($\theta =$ PMLE) | 0.00830 | 0.00664 | 0.00808 | 0.06680 |

**NOTE**: The simulation is based on simulated samples of $\left\{\left({t}_{i},{\tau}_{i},{\nu}_{i}\right);i=1,2,\dots ,1000\right\}$, consisting of 500 truncated samples and another 500 untruncated samples. The Clayton copula with $\theta =2$ yields Kendall’s tau = 0.5 for two failure times.

**Table 3.**The MSEs of the estimates of ${\alpha}_{1}$ under various sample sizes $n$ based on 1000 runs.

True Par. | True Model | Fitted Model | $\mathit{n}\mathbf{=}\mathbf{500}$ | $\mathit{n}\mathbf{=}\mathbf{1000}$ | $\mathit{n}\mathbf{=}\mathbf{1500}$ | $\mathit{n}\mathbf{=}\mathbf{2000}$ |
---|---|---|---|---|---|---|

${\alpha}_{1}=0.40$ | Indep. ($\theta =0$) | Indep. ($\theta =0$) | 0.00369 | 0.00186 | 0.00114 | 0.00096 |

Clayton ($\theta =2$) | 0.00614 | 0.00374 | 0.00261 | 0.00240 | ||

Clayton ($\theta =$ MLE) | 0.00706 | 0.00517 | 0.00404 | 0.00417 | ||

Clayton ($\theta =$ PMLE) | 0.00412 | 0.00204 | 0.00122 | 0.00104 | ||

Clayton ($\theta =2$) | Indep. ($\theta =0$) | 0.00394 | 0.00229 | 0.00174 | 0.00145 | |

Clayton ($\theta =2$) | 0.00404 | 0.00195 | 0.00134 | 0.00094 | ||

Clayton ($\theta =$ MLE) | 0.00408 | 0.00261 | 0.00215 | 0.00180 | ||

Clayton ($\theta =$ PMLE) | 0.00382 | 0.00209 | 0.00153 | 0.00122 | ||

${\alpha}_{1}=0.10$ | Indep. ($\theta =0$) | Indep. ($\theta =0$) | 0.00239 | 0.00111 | 0.00073 | 0.00053 |

Clayton ($\theta =2$) | 0.01950 | 0.01847 | 0.01738 | 0.01743 | ||

Clayton ($\theta =$ MLE) | 0.01295 | 0.00115 | 0.00955 | 0.00927 | ||

Clayton ($\theta =$ PMLE) | 0.00147 | 0.01323 | 0.01204 | 0.01179 | ||

Clayton ($\theta =2$) | Indep. ($\theta =0$) | 0.01967 | 0.01831 | 0.01818 | 0.01800 | |

Clayton ($\theta =2$) | 0.00195 | 0.00102 | 0.00065 | 0.00047 | ||

Clayton ($\theta =$ MLE) | 0.00678 | 0.00629 | 0.00728 | 0.00720 | ||

Clayton ($\theta =$ PMLE) | 0.00717 | 0.00575 | 0.00543 | 0.00523 | ||

${\alpha}_{1}=0.15$ | Indep. ($\theta =0$) | Indep. ($\theta =0$) | 0.00457 | 0.00229 | 0.00161 | 0.00110 |

Clayton ($\theta =2$) | 0.03838 | 0.03547 | 0.03474 | 0.03482 | ||

Clayton ($\theta =$ MLE) | 0.03402 | 0.02966 | 0.02840 | 0.02907 | ||

Clayton ($\theta =$ PMLE) | 0.03612 | 0.03181 | 0.03141 | 0.03109 | ||

Clayton ($\theta =2$) | Indep. ($\theta =0$) | 0.03807 | 0.03659 | 0.03572 | 0.03527 | |

Clayton ($\theta =2$) | 0.00394 | 0.00193 | 0.00127 | 0.00091 | ||

Clayton ($\theta =$ MLE) | 0.00605 | 0.00294 | 0.00179 | 0.00129 | ||

Clayton ($\theta =$ PMLE) | 0.01083 | 0.008300. | 0.00729 | 0.00676 |

**NOTE**: The simulation is based on 1000 runs. The Clayton copula uses $\theta =2$ (Kendall’s tau = 0.5).

True Model | Fitted Model | ${\mathit{\alpha}}_{\mathbf{1}}$ = 0.4 | ${\mathit{\lambda}}_{\mathbf{1}}$ = 0.1 | ${\mathit{\alpha}}_{\mathbf{2}}$ = 0.4 | ${\mathit{\lambda}}_{\mathbf{2}}$ = 0.1 | ||||
---|---|---|---|---|---|---|---|---|---|

SD | SE | SD | SE | SD | SE | SD | SE | ||

$\mathrm{Independent}(\theta =0)$ | $\mathrm{Independent}(\theta =0)$ | 0.043 | 0.042 | 0.013 | 0.013 | 0.041 | 0.042 | 0.013 | 0.013 |

$\mathrm{Clayton}(\theta =2)$ | 0.046 | 0.044 | 0.015 | 0.014 | 0.044 | 0.044 | 0.014 | 0.014 | |

Clayton ($\theta =$ PMLE) | 0.044 | 0.043 | 0.013 | 0.013 | 0.042 | 0.042 | 0.013 | 0.013 | |

$\mathrm{Clayton}(\theta =2)$ | $\mathrm{Independent}(\theta =0)$ | 0.041 | 0.043 | 0.012 | 0.012 | 0.042 | 0.043 | 0.013 | 0.012 |

$\mathrm{Clayton}(\theta =2)$ | 0.044 | 0.044 | 0.013 | 0.014 | 0.045 | 0.044 | 0.014 | 0.014 | |

Clayton ($\theta =$ PMLE) | 0.042 | 0.043 | 0.012 | 0.013 | 0.043 | 0.043 | 0.013 | 0.013 | |

${\mathit{\alpha}}_{\mathbf{1}}$= 1.0 | ${\mathit{\lambda}}_{\mathbf{1}}$= 0.4 | ${\mathit{\alpha}}_{\mathbf{2}}$= 1.0 | ${\mathit{\lambda}}_{\mathbf{2}}$= 0.4 | ||||||

SD | SE | SD | SE | SD | SE | SD | SE | ||

$\mathrm{Independent}(\theta =0)$ | $\mathrm{Independent}(\theta =0)$ | 0.033 | 0.033 | 0.030 | 0.029 | 0.033 | 0.033 | 0.029 | 0.029 |

$\mathrm{Clayton}(\theta =2)$ | 0.029 | 0.028 | 0.038 | 0.035 | 0.028 | 0.028 | 0.036 | 0.035 | |

Clayton ($\theta =$ PMLE) | 0.036 | 0.034 | 0.034 | 0.031 | 0.036 | 0.034 | 0.033 | 0.031 | |

$\mathrm{Clayton}(\theta =2)$ | $\mathrm{Independent}(\theta =0)$ | 0.035 | 0.035 | 0.024 | 0.024 | 0.033 | 0.035 | 0.024 | 0.024 |

$\mathrm{Clayton}(\theta =2)$ | 0.032 | 0.031 | 0.030 | 0.029 | 0.030 | 0.031 | 0.029 | 0.029 | |

Clayton ($\theta =$ PMLE) | 0.037 | 0.036 | 0.026 | 0.025 | 0.035 | 0.036 | 0.025 | 0.025 | |

${\mathit{\alpha}}_{\mathbf{1}}$= 1.5 | ${\mathit{\lambda}}_{\mathbf{1}}$= 1.0 | ${\mathit{\alpha}}_{\mathbf{2}}$= 1.5 | ${\mathit{\lambda}}_{\mathbf{2}}$= 1.0 | ||||||

SD | SE | SD | SE | SD | SE | SD | SE | ||

$\mathrm{Independent}(\theta =0)$ | $\mathrm{Independent}(\theta =0)$ | 0.048 | 0.049 | 0.059 | 0.058 | 0.049 | 0.049 | 0.056 | 0.058 |

$\mathrm{Clayton}(\theta =2)$ | 0.040 | 0.040 | 0.075 | 0.076 | 0.040 | 0.040 | 0.073 | 0.076 | |

Clayton ($\theta =$ PMLE) | 0.051 | 0.050 | 0.071 | 0.067 | 0.053 | 0.050 | 0.068 | 0.067 | |

$\mathrm{Clayton}(\theta =2)$ | $\mathrm{Independent}(\theta =0)$ | 0.049 | 0.050 | 0.043 | 0.043 | 0.049 | 0.050 | 0.042 | 0.043 |

$\mathrm{Clayton}(\theta =2)$ | 0.044 | 0.043 | 0.055 | 0.055 | 0.042 | 0.043 | 0.055 | 0.055 | |

Clayton ($\theta =$ PMLE) | 0.054 | 0.051 | 0.050 | 0.048 | 0.052 | 0.051 | 0.049 | 0.048 |

**NOTE**: The simulation is based on simulated samples of $\left\{\left({t}_{i},{\tau}_{i},{\nu}_{i}\right);i=1,2,\dots ,1000\right\}$, consisting of 500 truncated samples and another 500 untruncated samples. The Clayton copula with $\theta =2$ yields Kendall’s tau = 0.5 for two failure times.

**Table 5.**Artificial data from Kundu et al. [26], which consist of $n=100$ power transformers observed from the starting year (s = 1980) to the ending year (e = 2008). ${B}_{i}$ is the installation year. ${E}_{i}$ is either failure year or censoring year (2008). The 30 transformers are truncated (${\nu}_{i}=0$) and other 70 transformers are untruncated (${\nu}_{i}=1$). Observed events are shown in Event (=0 for censoring; =1 for Event 1; =2 for Event 2).

$\mathit{i}$ (Index) | ${\mathit{B}}_{\mathit{i}}$ (Install) | ${\mathit{E}}_{\mathit{i}}$ (End) | ${\mathit{\nu}}_{\mathit{i}}$ (Truncation) | Event | ${\mathit{\tau}}_{\mathit{i}}\mathbf{=}\mathbf{1980}\mathbf{-}{\mathit{B}}_{\mathit{i}}$ (Truncation Time) | ${\mathit{t}}_{\mathit{i}}\mathbf{=}{\mathit{E}}_{\mathit{i}}\mathbf{-}{\mathit{B}}_{\mathit{i}}$ (Failure Time) |
---|---|---|---|---|---|---|

1 | 1961 | 1996 | 0 | 2 | 19 | 35 |

2 | 1964 | 1985 | 0 | 1 | 16 | 21 |

3 | 1962 | 2007 | 0 | 2 | 18 | 45 |

4 | 1962 | 1986 | 0 | 2 | 18 | 24 |

5 | 1961 | 1992 | 0 | 2 | 19 | 31 |

: | : | : | : | : | : | : |

30 | 1963 | 1994 | 0 | 1 | 17 | 31 |

31 | 1987 | 2008 | 1 | 0 | Undefined | 21 |

32 | 1980 | 2008 | 1 | 0 | Undefined | 28 |

33 | 1988 | 2008 | 1 | 0 | Undefined | 20 |

34 | 1985 | 2008 | 1 | 0 | Undefined | 23 |

: | : | : | : | : | : | : |

100 | 1989 | 2008 | 1 | 0 | Undefined | 19 |

**Table 6.**The estimates (SEs in parenthesis) for the Clayton copula model with the Weibull failure times based on the dataset from Kundu et al. [26]. The Clayton copula with fixed $\theta $ or estimated $\theta $ are fitted.

$\mathit{\theta}$ | $\widehat{\mathit{\theta}}$ | $\widehat{{\mathit{\alpha}}_{\mathbf{1}}}$; Shape | $\widehat{{\mathit{\lambda}}_{\mathbf{1}}}$; Scale | $\widehat{{\mathit{\alpha}}_{\mathbf{2}}}$; Shape | $\widehat{{\mathit{\lambda}}_{\mathbf{2}}}$; Scale | logL |
---|---|---|---|---|---|---|

$\theta =0$ * | - | 2.82 (0.61) | 6.93 (5.24) | 2.79 (0.39) | 15.77 (7.73) | −8.984 |

$\theta =0.5$ | - | 3.18 (0.65) | 12.86 (10.56) | 2.88 (0.40) | 18.74 (9.29) | −9.024 |

$\theta =2$ | - | 3.68 (0.57) | 33.12 (22.90) | 2.93 (0.38) | 21.93 (10.19) | −9.389 |

$\theta =8$ | - | 3.39 (0.40) | 35.97 (16.21) | 2.92 (0.35) | 24.65 (10.64) | −9.239 |

MLE | 0.004 (0.11) | 2.82 (0.60) | 6.96 (5.16) | 2.79 (0.39) | 15.79 (7.70) | −8.983 |

PMLE (${\theta}^{*}=0.5$) | 0.402 | 3.12 (0.65) | 11.57 (9.42) | 2.87 (0.40) | 18.31 (9.10) | −9.008 |

PMLE (${\theta}^{*}=2$) | 1.853 | 3.66 (0.59) | 31.57 (22.44) | 2.93 (0.38) | 21.73 (10.15) | −9.384 |

PMLE (${\theta}^{*}=8$) | 8.179 | 3.38 (0.40) | 35.77 (16.07) | 2.92 (0.35) | 24.67 (10.63) | −9.238 |

Results of [26] | - | 2.80 (-) | 6.76 (-) | 2.80 (-) | 15.93 (-) | - |

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**MDPI and ACS Style**

Michimae, H.; Emura, T.
Likelihood Inference for Copula Models Based on Left-Truncated and Competing Risks Data from Field Studies. *Mathematics* **2022**, *10*, 2163.
https://doi.org/10.3390/math10132163

**AMA Style**

Michimae H, Emura T.
Likelihood Inference for Copula Models Based on Left-Truncated and Competing Risks Data from Field Studies. *Mathematics*. 2022; 10(13):2163.
https://doi.org/10.3390/math10132163

**Chicago/Turabian Style**

Michimae, Hirofumi, and Takeshi Emura.
2022. "Likelihood Inference for Copula Models Based on Left-Truncated and Competing Risks Data from Field Studies" *Mathematics* 10, no. 13: 2163.
https://doi.org/10.3390/math10132163