# Copula Approach for Dependent Competing Risks of Generalized Half-Logistic Distributions under Accelerated Censoring Data

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

**Model considerations**

- 1.
- The number of I/Us which is failed with respect to stress level ${\mathbf{S}}_{k}$ and cause j is denoted by$${n}_{kj}=\sum _{i=1}^{{m}_{k}}{I}_{j}\left({\delta}_{ki}\right),k,j=1,2.$$
- 2.
- The number of I/Us which is failed with respect cause j is denoted by$${J}_{j}=\sum _{k=1}^{2}\sum _{i=1}^{{m}_{k}}{I}_{j}\left({\delta}_{ki}\right),k,j=1,2.$$
- 3.
- For any stress levels ${\mathbf{S}}_{k},$$k=1,2$, only two dependence causes of failure are exist.
- 4.
- The time-to-failure ${T}_{kj}\phantom{\rule{4pt}{0ex}}$ respected to stress level ${\mathbf{S}}_{k}$ and the cause of failure j distribute with GHL distribution with scale parameters ${\theta}_{kj}$ and shape parameter ${\beta}_{j}\phantom{\rule{4pt}{0ex}}$ with CDFs given by$${F}_{kj}\left(t\right|{\beta}_{j},{\theta}_{kj})=1-{\left(\right)}^{\frac{2{e}^{-\frac{t}{{\theta}_{kj}}}}{1+{e}^{-\frac{t}{{\theta}_{kj}}}}}{\beta}_{j}$$The corresponding PDFs$${f}_{kj}\left(t\right|{\beta}_{j},{\theta}_{kj})=\frac{{\beta}_{j}}{{\theta}_{kj}(1+{e}^{-\frac{t}{{\theta}_{kj}}})}{\left(\right)}^{\frac{2{e}^{-\frac{t}{{\theta}_{kj}}}}{1+{e}^{-\frac{t}{{\theta}_{kj}}}}}{\beta}_{j}$$$${S}_{kj}\left(t\right|{\beta}_{j},{\theta}_{kj})={\left(\right)}^{\frac{2{e}^{-\frac{t}{{\theta}_{kj}}}}{1+{e}^{-\frac{t}{{\theta}_{kj}}}}}{\beta}_{j}$$$${H}_{kj}\left(t\right|{\beta}_{j},{\theta}_{kj})=\frac{{\beta}_{j}}{{\theta}_{kj}(1+{e}^{-\frac{t}{{\theta}_{kj}}})}.$$
- 5.
- The shape parameters is common for stress levels ${\mathbf{S}}_{k},$$k=1,2$ and different for causes of failure.
- 6.
- The joint survival function under BPC is given by$${S}_{k}\left(t\right)={\left(\right)}^{{\left(\right)}^{\frac{2{e}^{-\frac{t}{{\theta}_{k1}}}}{1+{e}^{-\frac{t}{{\theta}_{k1}}}}}}+{\left(\right)}^{\frac{2{e}^{-\frac{t}{{\theta}_{k2}}}}{1+{e}^{-\frac{t}{{\theta}_{k2}}}}}\frac{-{\beta}_{2}}{\gamma}-\gamma $$
- 7.
- The scale parameters ${\theta}_{k1}$ is log-linear function of the stress function $\varphi \left({\mathbf{S}}_{k}\right)$ of the j-th competing failure mode$$\mathrm{log}{\theta}_{kj}={a}_{j}+{b}_{j}\varphi \left({\mathbf{S}}_{k}\right),\phantom{\rule{4.pt}{0ex}}k;\phantom{\rule{4.pt}{0ex}}j=1,2,$$

## 3. Result and Discussion

#### 3.1. Copula Function

**Archimedean copula**

**Measure of association**

#### 3.2. The Point ML Estimate

#### 3.3. Approximate Confidence Intervals (ACIs)

#### 3.4. Bootstrap Confidence Intervals (BCIs)

**Algorithms 1 (Generate bootstrap sample of estimates)**

**Step 1:**- For given ${n}_{1}$, ${n}_{2},$${m}_{1},$${m}_{2}$, stress levels ${\mathbf{S}}_{1}$ and ${\mathbf{S}}_{2}$ and two censoring schemes ${\mathbf{R}}_{1}=\{{R}_{11},$${R}_{12},$$\dots ,$${R}_{1{m}_{1}}\}$ and ${\mathbf{R}}_{2}=\{{R}_{21},$${R}_{22},$$\dots ,$${R}_{2{m}_{2}}\}$ with the original competing risks type-II PCS ${\mathbf{t}}_{k}{|}_{k=1,2}=\left\{\right({T}_{k1;{m}_{k},{n}_{k}},$${\delta}_{k1}),$$({T}_{k2;{m}_{k},{n}_{k}},$${\delta}_{k2}),$$\dots ,$$({T}_{k{m}_{k};{m}_{k},{n}_{k}},$${\delta}_{k{m}_{k}}\left)\right\}$ compute ${J}_{1},$${J}_{2},$${n}_{11},$${n}_{12},$${n}_{11}\phantom{\rule{4pt}{0ex}}$ and ${n}_{12}.$ Then, the estimate values of the model parameters $\widehat{\Omega}=\{{\beta}_{1},$${\widehat{\beta}}_{2},$${\widehat{\theta}}_{11},$${\widehat{\theta}}_{12},$${\widehat{\theta}}_{21},$${\widehat{\theta}}_{22}\}$ are computed.
**Step 2:**- Based on ${n}_{1}$, ${m}_{1}\phantom{\rule{4pt}{0ex}}$ and ${\mathbf{R}}_{1}$ using the algorithms given by Balakrishnan and Sandhu [42], we generate two type-II PC samples of size ${m}_{1}$ from GHL distributions with parameters (${\beta}_{1}$, ${\widehat{\theta}}_{11})$ and (${\beta}_{2}$, ${\widehat{\theta}}_{12}),$ respectively. The competing risks type-II PC sample is difened by (${T}_{1i},$${\delta}_{1i})=min({T}_{11i},$${T}_{12i}),$ $i=$1, 2, …, ${m}_{1}.$
**Step 3:**- Based on ${n}_{2}$, ${m}_{2}\phantom{\rule{4pt}{0ex}}$ and ${\mathbf{R}}_{2}$ generate two type-II PC samples of size ${m}_{2}$ from GHL distributions with parameters (${\beta}_{1}$, ${\widehat{\theta}}_{21})$ and (${\beta}_{2}$, ${\widehat{\theta}}_{22}),$ respectively. The competing risks type-II PC sample is difened by (${T}_{2i},$ ${\delta}_{2i})=min({T}_{21i},$${T}_{22i}),$$i=$1, 2, …, ${m}_{2}.$
**Step 4:**- From two Step 2 and 3 the joint sample ${\mathbf{t}}_{k}^{\ast}{|}_{k=1,2}=\left\{\right({T}_{k1;{m}_{k},{n}_{k}}^{\ast},$${\delta}_{k1}^{\ast}),$$({T}_{k2;{m}_{k},{n}_{k}}^{\ast},$${\delta}_{k2}^{\ast}),$$\dots ,$$({T}_{k{m}_{k};{m}_{k},{n}_{k}}^{\ast},$${\delta}_{k{m}_{k}}^{\ast}\left)\right\}$ is formulated.
**Step 5:**- Based on ${\mathbf{t}}_{k}^{\ast}{|}_{k=1,2}=\left\{\right({T}_{k1;{m}_{k},{n}_{k}}^{\ast},$${\delta}_{k1}^{\ast}),$$({T}_{k2;{m}_{k},{n}_{k}}^{\ast},$${\delta}_{k2}^{\ast}),$$\dots ,$$({T}_{k{m}_{k};{m}_{k},{n}_{k}}^{\ast},$${\delta}_{k{m}_{k}}^{\ast}\left)\right\}$ compute the MLE estimate ${\widehat{\Omega}}^{\ast}=\{{\widehat{\beta}}_{1}^{\ast},$${\widehat{\beta}}_{2}^{\ast},$${\widehat{\theta}}_{11}^{\ast},$${\widehat{\theta}}_{12},$${\widehat{\theta}}_{21}^{\ast},$${\widehat{\theta}}_{22}^{\ast}\}.$

**Bootstrap-p confidence interval (Boot-P CIs)**

**Bootstrap-t confidence intervals (Boot-t CIs)**

#### 3.5. Reliability Estimation

#### 3.6. Simulation Study

**Discussion:**The results of the numerical computation for the Monte Carlo simulation study have revealed the following points:

- 1.
- The proposed model and the proposed methods of estimation serve well for all of the parameter values and censoring schemes.
- 2.
- The values of MSEs decrease when the sample size and affected sample size increase.
- 3.
- The results show that the value of the copula parameter $\gamma =2$ has a small MSEs than value $\gamma =1$. Hence, a stronger dependent serves better than a weaker dependent.
- 4.
- Finally, the coverage percentages of ACIs are always less than the nominal level when the sample size is less or equivalent to 60. For a sample size as large as 70, the coverage percentages of ACIs improve, which can maintain the pre-fixed nominal level.
- 5.
- Bootstrap-t serve well than Bootstarp-p and MLE.

**Table 1.**Estimated MESs when $\gamma =1$ and $\Omega $ = (0.2, 0.5, 1.3238, 2.4865, 0.8227, 1.1142).

(${\mathit{n}}_{1},{\mathit{m}}_{1}$) | (${\mathit{n}}_{2},{\mathit{m}}_{2}$) | Scheme | ${\mathit{\beta}}_{1}$ | ${\mathit{\beta}}_{2}$ | ${\mathit{\theta}}_{11}$ | ${\mathit{\theta}}_{12}$ | ${\mathit{\theta}}_{21}$ | ${\mathit{\theta}}_{22}$ |
---|---|---|---|---|---|---|---|---|

(25,10) | (25,10) | ${\mathbf{R}}_{1}=(6,1,\dots ,1)$ | 0.0873 | 0.1242 | 0.3214 | 0.5621 | 0.2741 | 0.2987 |

${\mathbf{R}}_{2}=(6,1,\dots ,1)$ | ||||||||

(25,20) | (25,20) | ${\mathbf{R}}_{1}=(2,2,1,0,\dots ,0)$ | 0.0745 | 0.1115 | 0.3098 | 0.5428 | 0.2622 | 0.2777 |

${\mathbf{R}}_{2}=(2,2,1,0,\dots ,0)$ | ||||||||

(50,20) | (50,20) | ${\mathbf{R}}_{1}=(2,1,2,1,\dots ,2,1)$ | 0.0768 | 0.1103 | 0.3111 | 0.5414 | 0.2611 | 0.2792 |

${\mathbf{R}}_{2}=(2,1,2,1,\dots ,2,1)$ | ||||||||

(50,20) | (50,20) | ${\mathbf{R}}_{1}=(30,0,0,\dots ,0)$ | 0.0722 | 0.1089 | 0.3102 | 0.5399 | 0.2601 | 0.2774 |

${\mathbf{R}}_{2}=(30,0,0,\dots ,0)$ | ||||||||

(50,20) | (50,35) | ${\mathbf{R}}_{1}=(0,0,\dots ,0,30)$ | 0.0715 | 0.1045 | 0.3111 | 0.5389 | 0.2541 | 0.2730 |

${\mathbf{R}}_{2}=(0,0,\dots ,0,15)$ | ||||||||

(50,35) | (50,20) | ${\mathbf{R}}_{1}=(0,0,\dots ,0,15)$ | 0.0692 | 0.1093 | 0.3045 | 0.5352 | 0.2613 | 0.2771 |

${\mathbf{R}}_{2}=(0,0,\dots ,0,30)$ | ||||||||

(80,40) | (80,40) | ${\mathbf{R}}_{1}=\left({1}^{40}\right)$ | 0.0601 | 0.0875 | 0.3003 | 0.5211 | 0.2492 | 0.2665 |

${\mathbf{R}}_{2}=\left({1}^{40}\right)$ | ||||||||

(80,40) | (80,40) | ${\mathbf{R}}_{1}=({0}^{20},{2}^{20})$ | 0.0614 | 0.0879 | 0.3012 | 0.5209 | 0.2489 | 0.2671 |

${\mathbf{R}}_{2}=({0}^{20},{2}^{20})$ | ||||||||

(80,60) | (80,40) | ${\mathbf{R}}_{1}=({1}^{20},{0}^{40})$ | 0.0541 | 0.0869 | 0.2985 | 0.5154 | 0.2494 | 0.2653 |

${\mathbf{R}}_{2}=\left({1}^{40}\right)$ | ||||||||

(80,40) | (80,60) | ${\mathbf{R}}_{1}=\left({1}^{40}\right)$ | 0.0608 | 0.0833 | 0.3007 | 0.5207 | 0.2448 | 0.2618 |

${\mathbf{R}}_{2}=({1}^{20},{0}^{40})$ | ||||||||

(80,60) | (80,60) | ${\mathbf{R}}_{1}=({0}^{40},{1}^{20})$ | 0.0518 | 0.0782 | 0.2945 | 0.5105 | 0.2399 | 0.2559 |

${\mathbf{R}}_{2}=({0}^{40},{1}^{20})$ |

(${\mathit{n}}_{1},{\mathit{m}}_{1}$) | (${\mathit{n}}_{2},{\mathit{m}}_{2}$) | Scheme | Method | ${\mathit{\beta}}_{1}$ | ${\mathit{\beta}}_{2}$ | ${\mathit{\theta}}_{11}$ | ${\mathit{\theta}}_{12}$ | ${\mathit{\theta}}_{21}$ | ${\mathit{\theta}}_{22}$ |
---|---|---|---|---|---|---|---|---|---|

(25,10) | (25,10) | ${\mathbf{R}}_{1}=(6,1,\dots ,1)$ | MLE | 0.87 | 0.89 | 0.86 | 0.88 | 0.88 | 0.89 |

${\mathbf{R}}_{2}=(6,1,\dots ,1)$ | Boot-p | 0.87 | 0.88 | 0.89 | 0.89 | 0.86 | 0.89 | ||

Boot-t | 0.89 | 0.89 | 0.90 | 0.90 | 0.89 | 0.90 | |||

(25,20) | (25,20) | ${\mathbf{R}}_{1}=(2,2,1,0,\dots ,0)$ | MLE | 0.89 | 0.91 | 0.88 | 0.89 | 0.91 | 0.91 |

${\mathbf{R}}_{2}=(2,2,1,0,\dots ,0)$ | Boot-p | 0.89 | 0.88 | 0.91 | 0.89 | 0.79 | 0.94 | ||

Boot-t | 0.93 | 0.92 | 0.92 | 0.92 | 0.93 | 0.94 | |||

(50,20) | (50,20) | ${\mathbf{R}}_{1}=(2,1,2,1,\dots ,2,1)$ | MLE | 0.91 | 0.90 | 0.89 | 0.88 | 0.92 | 0.90 |

${\mathbf{R}}_{2}=(2,1,2,1,\dots ,2,1)$ | Boot-p | 0.91 | 0.88 | 0.89 | 0.90 | 0.91 | 0.92 | ||

Boot-t | 0.94 | 0.93 | 0.92 | 0.92 | 0.91 | 0.93 | |||

(50,20) | (50,20) | ${\mathbf{R}}_{1}=(30,0,0,\dots ,0)$ | MLE | 0.91 | 0.91 | 0.89 | 0.91 | 0.91 | 0.91 |

${\mathbf{R}}_{2}=(30,0,0,\dots ,0)$ | Boot-p | 0.89 | 0.89 | 0.90 | 0.92 | 0.92 | 0.92 | ||

Boot-t | 0.93 | 0.93 | 0.95 | 0.92 | 0.92 | 0.92 | |||

(50,20) | (50,35) | ${\mathbf{R}}_{1}=(0,0,\dots ,0,30)$ | MLE | 0.91 | 0.90 | 0.92 | 0.90 | 0.93 | 0.92 |

${\mathbf{R}}_{2}=(0,0,\dots ,0,15)$ | Boot-p | 0.91 | 0.91 | 0.89 | 0.92 | 0.90 | 0.91 | ||

Boot-t | 0.93 | 0.93 | 0.94 | 0.94 | 0.91 | 0.92 | |||

(50,35) | (50,20) | ${\mathbf{R}}_{1}=(0,0,\dots ,0,15)$ | MLE | 0.90 | 0.91 | 0.89 | 0.90 | 0.91 | 0.91 |

${\mathbf{R}}_{2}=(0,0,\dots ,0,30)$ | Boot-p | 0.90 | 0.91 | 0.88 | 0.91 | 0.92 | 0.91 | ||

Boot-t | 0.94 | 0.93 | 0.93 | 0.92 | 0.92 | 0.93 | |||

(80,40) | (80,40) | ${\mathbf{R}}_{1}=\left({1}^{40}\right)$ | MLE | 0.91 | 0.92 | 0.90 | 0.92 | 0.91 | 0.92 |

${\mathbf{R}}_{2}=\left({1}^{40}\right)$ | Boot-p | 0.91 | 0.90 | 0.92 | 0.96 | 0.90 | 0.92 | ||

Boot-t | 0.94 | 0.93 | 0.92 | 0.94 | 0.92 | 0.93 | |||

(80,40) | (80,40) | ${\mathbf{R}}_{1}=({0}^{20},{2}^{20})$ | MLE | 0.92 | 0.92 | 0.91 | 0.91 | 0.94 | 0.92 |

${\mathbf{R}}_{2}=({0}^{20},{2}^{20})$ | Boot-p | 0.91 | 0.92 | 0.93 | 0.91 | 0.92 | 0.91 | ||

Boot-t | 0.91 | 0.92 | 0.92 | 0.92 | 0.94 | 0.90 | |||

(80,60) | (80,40) | ${\mathbf{R}}_{1}=({1}^{20},{0}^{40})$ | MLE | 0.92 | 0.92 | 0.92 | 0.94 | 0.91 | 0.94 |

${\mathbf{R}}_{2}=\left({1}^{40}\right)$ | Boot-p | 0.92 | 0.92 | 0.92 | 0.92 | 0.92 | 0.93 | ||

Boot-t | 0.92 | 0.92 | 0.95 | 0.91 | 0.92 | 0.94 | |||

(80,40) | (80,60) | ${\mathbf{R}}_{1}=\left({1}^{40}\right)$ | MLE | 0.90 | 0.90 | 0.92 | 0.91 | 0.95 | 0.92 |

${\mathbf{R}}_{2}=({1}^{20},{0}^{40})$ | Boot-p | 0.91 | 0.93 | 0.91 | 0.90 | 0.94 | 0.90 | ||

Boot-t | 0.94 | 0.95 | 0.92 | 0.92 | 0.92 | 0.93 | |||

(80,60) | (80,60) | ${\mathbf{R}}_{1}=({0}^{40},{1}^{20})$ | MLE | 0.91 | 0.97 | 0.91 | 0.93 | 0.91 | 0.92 |

${\mathbf{R}}_{2}=({0}^{40},{1}^{20})$ | Boot-p | 0.92 | 0.90 | 0.92 | 0.94 | 0.92 | 0.91 | ||

Boot-t | 0.94 | 0.92 | 0.95 | 0.93 | 0.95 | 0.94 |

**Table 3.**Estimated MESs when $\gamma =2$ and $\Omega $ = (0.2, 0.5, 1.3238, 2.4865, 0.8227, 1.1142).

(${\mathit{n}}_{1},{\mathit{m}}_{1}$) | (${\mathit{n}}_{2},{\mathit{m}}_{2}$) | Scheme | ${\mathit{\beta}}_{1}$ | ${\mathit{\beta}}_{2}$ | ${\mathit{\theta}}_{11}$ | ${\mathit{\theta}}_{12}$ | ${\mathit{\theta}}_{21}$ | ${\mathit{\theta}}_{22}$ |
---|---|---|---|---|---|---|---|---|

(25,10) | (25,10) | ${\mathbf{R}}_{1}=(6,1,\dots ,1)$ | 0.0825 | 0.1200 | 0.3162 | 0.5572 | 0.2741 | 0.2987 |

${\mathbf{R}}_{2}=(6,1,\dots ,1)$ | ||||||||

(25,20) | (25,20) | ${\mathbf{R}}_{1}=(2,2,1,0,\dots ,0)$ | 0.0701 | 0.1072 | 0.3045 | 0.5401 | 0.2584 | 0.2719 |

${\mathbf{R}}_{2}=(2,2,1,0,\dots ,0)$ | ||||||||

(50,20) | (50,20) | ${\mathbf{R}}_{1}=(2,1,2,1,\dots ,2,1)$ | 0.0725 | 0.1055 | 0.3061 | 0.5382 | 0.2562 | 0.2701 |

${\mathbf{R}}_{2}=(2,1,2,1,\dots ,2,1)$ | ||||||||

(50,20) | (50,20) | ${\mathbf{R}}_{1}=(30,0,0,\dots ,0)$ | 0.0682 | 0.1051 | 0.349 | 0.5354 | 0.2571 | 0.2748 |

${\mathbf{R}}_{2}=(30,0,0,\dots ,0)$ | ||||||||

(50,20) | (50,35) | ${\mathbf{R}}_{1}=(0,0,\dots ,0,30)$ | 0.0677 | 0.1002 | 0.349 | 0.5341 | 0.2500 | 0.2701 |

${\mathbf{R}}_{2}=(0,0,\dots ,0,15)$ | ||||||||

(50,35) | (50,20) | ${\mathbf{R}}_{1}=(0,0,\dots ,0,15)$ | 0.0651 | 0.1048 | 0.3007 | 0.5313 | 0.2582 | 0.2729 |

${\mathbf{R}}_{2}=(0,0,\dots ,0,30)$ | ||||||||

(80,40) | (80,40) | ${\mathbf{R}}_{1}=\left({1}^{40}\right)$ | 0.0571 | 0.0824 | 0.2890 | 0.5142 | 0.2433 | 0.2619 |

${\mathbf{R}}_{2}=\left({1}^{40}\right)$ | ||||||||

(80,40) | (80,40) | ${\mathbf{R}}_{1}=({0}^{20},{2}^{20})$ | 0.0572 | 0.0841 | 0.2975 | 0.5162 | 0.2417 | 0.2614 |

${\mathbf{R}}_{2}=({0}^{20},{2}^{20})$ | ||||||||

(80,60) | (80,40) | ${\mathbf{R}}_{1}=({1}^{20},{0}^{40})$ | 0.0508 | 0.0821 | 0.2929 | 0.5118 | 0.2451 | 0.2614 |

${\mathbf{R}}_{2}=\left({1}^{40}\right)$ | ||||||||

(80,40) | (80,60) | ${\mathbf{R}}_{1}=\left({1}^{40}\right)$ | 0.0555 | 0.0800 | 0.2952 | 0.5144 | 0.2403 | 0.2581 |

${\mathbf{R}}_{2}=({1}^{20},{0}^{40})$ | ||||||||

(80,60) | (80,60) | ${\mathbf{R}}_{1}=({0}^{40},{1}^{20})$ | 0.0488 | 0.0728 | 0.2901 | 0.5044 | 0.2362 | 0.2511 |

${\mathbf{R}}_{2}=({0}^{40},{1}^{20})$ |

(${\mathit{n}}_{1},{\mathit{m}}_{1}$) | (${\mathit{n}}_{2},{\mathit{m}}_{2}$) | Scheme | Method | ${\mathit{\beta}}_{1}$ | ${\mathit{\beta}}_{2}$ | ${\mathit{\theta}}_{11}$ | ${\mathit{\theta}}_{12}$ | ${\mathit{\theta}}_{21}$ | ${\mathit{\theta}}_{22}$ |
---|---|---|---|---|---|---|---|---|---|

(25,10) | (25,10) | ${\mathbf{R}}_{1}=(6,1,\dots ,1)$ | MLE | 0.88 | 0.89 | 0.87 | 0.88 | 0.86 | 0.90 |

${\mathbf{R}}_{2}=(6,1,\dots ,1)$ | Boot-p | 0.87 | 0.89 | 0.88 | 0.88 | 0.86 | 0.89 | ||

Boot-t | 0.90 | 0.89 | 0.90 | 0.91 | 0.89 | 0.90 | |||

(25,20) | (25,20) | ${\mathbf{R}}_{1}=(2,2,1,0,\dots ,0)$ | MLE | 0.89 | 0.90 | 0.90 | 0.89 | 0.91 | 0.90 |

${\mathbf{R}}_{2}=(2,2,1,0,\dots ,0)$ | Boot-p | 0.89 | 0.88 | 0.90 | 0.89 | 0.90 | 0.90 | ||

Boot-t | 0.91 | 0.92 | 0.90 | 0.91 | 0.91 | 0.93 | |||

(50,20) | (50,20) | ${\mathbf{R}}_{1}=(2,1,2,1,\dots ,2,1)$ | MLE | 0.90 | 0.90 | 0.89 | 0.90 | 0.91 | 0.91 |

${\mathbf{R}}_{2}=(2,1,2,1,\dots ,2,1)$ | Boot-p | 0.91 | 0.90 | 0.89 | 0.90 | 0.90 | 0.92 | ||

Boot-t | 0.92 | 0.93 | 0.96 | 0.92 | 0.91 | 0.94 | |||

(50,20) | (50,20) | ${\mathbf{R}}_{1}=(30,0,0,\dots ,0)$ | MLE | 0.90 | 0.90 | 0.89 | 0.90 | 0.91 | 0.91 |

${\mathbf{R}}_{2}=(30,0,0,\dots ,0)$ | Boot-p | 0.89 | 0.90 | 0.91 | 0.92 | 0.91 | 0.90 | ||

Boot-t | 0.92 | 0.91 | 0.89 | 0.96 | 0.93 | 0.92 | |||

(50,20) | (50,35) | ${\mathbf{R}}_{1}=(0,0,\dots ,0,30)$ | MLE | 0.90 | 0.90 | 0.89 | 0.90 | 0.91 | 0.91 |

${\mathbf{R}}_{2}=(0,0,\dots ,0,15)$ | Boot-p | 0.91 | 0.90 | 0.89 | 0.92 | 0.91 | 0.90 | ||

Boot-t | 0.93 | 0.92 | 0.94 | 0.92 | 0.91 | 0.95 | |||

(50,35) | (50,20) | ${\mathbf{R}}_{1}=(0,0,\dots ,0,15)$ | MLE | 0.88 | 0.90 | 0.89 | 0.89 | 0.91 | 0.90 |

${\mathbf{R}}_{2}=(0,0,\dots ,0,30)$ | Boot-p | 0.90 | 0.91 | 0.90 | 0.91 | 0.91 | 0.90 | ||

Boot-t | 0.94 | 0.92 | 0.93 | 0.94 | 0.92 | 0.91 | |||

(80,40) | (80,40) | ${\mathbf{R}}_{1}=\left({1}^{40}\right)$ | MLE | 0.92 | 0.93 | 0.90 | 0.96 | 0.92 | 0.92 |

${\mathbf{R}}_{2}=\left({1}^{40}\right)$ | Boot-p | 0.91 | 0.90 | 0.90 | 0.96 | 0.92 | 0.93 | ||

Boot-t | 0.96 | 0.93 | 0.92 | 0.96 | 0.92 | 0.93 | |||

(80,40) | (80,40) | ${\mathbf{R}}_{1}=({0}^{20},{2}^{20})$ | MLE | 0.93 | 0.92 | 0.92 | 0.90 | 0.94 | 0.93 |

${\mathbf{R}}_{2}=({0}^{20},{2}^{20})$ | Boot-p | 0.90 | 0.92 | 0.91 | 0.90 | 0.92 | 0.91 | ||

Boot-t | 0.92 | 0.92 | 0.93 | 0.92 | 0.94 | 0.92 | |||

(80,60) | (80,40) | ${\mathbf{R}}_{1}=({1}^{20},{0}^{40})$ | MLE | 0.95 | 0.90 | 0.92 | 0.94 | 0.91 | 0.90 |

${\mathbf{R}}_{2}=\left({1}^{40}\right)$ | Boot-p | 0.91 | 0.92 | 0.92 | 0.91 | 0.91 | 0.93 | ||

Boot-t | 0.93 | 0.92 | 0.94 | 0.91 | 0.92 | 0.94 | |||

(80,40) | (80,60) | ${\mathbf{R}}_{1}=\left({1}^{40}\right)$ | MLE | 0.93 | 0.92 | 0.92 | 0.94 | 0.95 | 0.93 |

${\mathbf{R}}_{2}=({1}^{20},{0}^{40})$ | Boot-p | 0.91 | 0.92 | 0.91 | 0.90 | 0.92 | 0.90 | ||

Boot-t | 0.94 | 0.92 | 0.93 | 0.92 | 0.92 | 0.91 | |||

(80,60) | (80,60) | ${\mathbf{R}}_{1}=({0}^{40},{1}^{20})$ | MLE | 0.94 | 0.97 | 0.92 | 0.93 | 0.92 | 0.94 |

${\mathbf{R}}_{2}=({0}^{40},{1}^{20})$ | Boot-p | 0.91 | 0.90 | 0.92 | 0.92 | 0.92 | 0.92 | ||

Boot-t | 0.93 | 0.92 | 0.92 | 0.93 | 0.95 | 0.94 |

#### 3.7. Data Analysis

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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0.0512 | 0.3533 | 0.5253 | 0.5297 | 0.7615 | 0.7737 | 0.7941 | 1.2752 | 1.3143 | 1.8658 | |

1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | |

${\mathbf{S}}_{1}$ | 2.0817 | 2.6984 | 2.7525 | 2.8935 | 2.9355 | 3.0838 | 3.9839 | 4.3776 | 4.4603 | 4.9003 |

1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | |

5.0907 | 5.1854 | 5.2346 | 5.2654 | 5.4079 | 5.8034 | 5.9266 | 5.9658 | 6.7407 | 7.1575 | |

2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | |

0.0793 | 0.1826 | 0.2967 | 0.3620 | 0.5833 | 0.7460 | 0.8963 | 1.0052 | 1.0378 | 1.1951 | |

2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | |

${\mathbf{S}}_{2}$ | 1.1955 | 1.3332 | 1.4629 | 1.5213 | 1.5498 | 1.7281 | 1.8321 | 1.9088 | 2.0058 | 2.2701 |

1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | |

2.3247 | 2.3734 | 2.4229 | 2.8080 | 3.5315 | 3.7945 | 3.9892 | 4.1753 | 4.1884 | 7.7951 | |

1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 |

Exact | MLE | 95% ACI | 95% Boot-p | 95%Boot-t | ||||
---|---|---|---|---|---|---|---|---|

${\beta}_{1}$ | 0.2000 | 0.0792 | (0.0094, 0.6644) | (0.0478, 1.4254) | (0.0113, 0.5478) | |||

${\beta}_{2}$ | 0.3000 | 0.4475 | (0.1034, 1.9365) | (0.1220, 2.8412) | (0.0047, 0.8745) | |||

${\theta}_{11}$ | 0.9517 | 0.6328 | (0.0878, 4.5614) | (0.2345, 4.9994) | (0.2473, 2.9982) | |||

${\theta}_{12}$ | 1.3503 | 2.8562 | (0.8957, 9.1080) | (0.7845, 13.1457) | (0.7412, 5.6547) | |||

${\theta}_{21}$ | 0.6093 | 0.2959 | (0.0421, 2.0789) | (0.1240, 4.2145) | (0.2314, 1.9879) | |||

${\theta}_{22}$ | 1.0030 | 4.8317 | (0.9872, 20.3486) | (0.4521, 22.3874) | (0.5462, 10.8754) |

t | $\mathit{S}\left(\mathit{t}\right)$ | t | $\mathit{S}\left(\mathit{t}\right)$ | ||
---|---|---|---|---|---|

0.5 | 0.916947 | 3.0 | 0.515836 | ||

1.0 | 0.831423 | 3.5 | 0.451519 | ||

1.5 | 0.746312 | 4.0 | 0.394082 | ||

2.0 | 0.664164 | 4.5 | 0.343289 | ||

2.5 | 0.586922 | 5.0 | 0.298688 |

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**MDPI and ACS Style**

Al-Essa, L.A.; Soliman, A.A.; Abd-Elmougod, G.A.; Alshanbari, H.M.
Copula Approach for Dependent Competing Risks of Generalized Half-Logistic Distributions under Accelerated Censoring Data. *Symmetry* **2023**, *15*, 564.
https://doi.org/10.3390/sym15020564

**AMA Style**

Al-Essa LA, Soliman AA, Abd-Elmougod GA, Alshanbari HM.
Copula Approach for Dependent Competing Risks of Generalized Half-Logistic Distributions under Accelerated Censoring Data. *Symmetry*. 2023; 15(2):564.
https://doi.org/10.3390/sym15020564

**Chicago/Turabian Style**

Al-Essa, Laila A., Ahmed A. Soliman, Gamal A. Abd-Elmougod, and Huda M. Alshanbari.
2023. "Copula Approach for Dependent Competing Risks of Generalized Half-Logistic Distributions under Accelerated Censoring Data" *Symmetry* 15, no. 2: 564.
https://doi.org/10.3390/sym15020564