MLE.Frank.Pareto {Bivariate.Pareto} R Documentation

## Maximum likelihood estimation for bivariate dependent competing risks data under the Frank copula with the Pareto margins and fixed θ

### Description

Maximum likelihood estimation for bivariate dependent competing risks data under the Frank copula with the Pareto margins and fixed θ.

### Usage

```MLE.Frank.Pareto(
t.event,
event1,
event2,
Theta,
Alpha1.0 = 1,
Alpha2.0 = 1,
Gamma1.0 = 1,
Gamma2.0 = 1,
epsilon = 1e-05,
d = exp(10),
r.1 = 6,
r.2 = 6,
r.3 = 6,
r.4 = 6
)
```

### Arguments

 `t.event` Vector of the observed failure times. `event1` Vector of the indicators for the failure cause 1. `event2` Vector of the indicators for the failure cause 2. `Theta` Copula parameter θ. `Alpha1.0` Initial guess for the scale parameter α_{1} with default value 1. `Alpha2.0` Initial guess for the scale parameter α_{2} with default value 1. `Gamma1.0` Initial guess for the shape parameter γ_{1} with default value 1. `Gamma2.0` Initial guess for the shape parameter γ_{2} with default value 1. `epsilon` Positive tunning parameter in the NR algorithm with default value 10^{-5}. `d` Positive tunning parameter in the NR algorithm with default value e^{10}. `r.1` Positive tunning parameter in the NR algorithm with default value 1. `r.2` Positive tunning parameter in the NR algorithm with default value 1. `r.3` Positive tunning parameter in the NR algorithm with default value 1. `r.4` Positive tunning parameter in the NR algorithm with default value 1.

### Value

 `n` Sample size. `count` Iteration number. `random` Randomization number. `Alpha1` Positive scale parameter for the Pareto margin (failure cause 1). `Alpha2` Positive scale parameter for the Pareto margin (failure cause 2). `Gamma1` Positive shape parameter for the Pareto margin (failure cause 1). `Gamma2` Positive shape parameter for the Pareto margin (failure cause 2). `MedX` Median lifetime due to failure cause 1. `MedY` Median lifetime due to failure cause 2. `MeanX` Mean lifetime due to failure cause 1. `MeanY` Mean lifetime due to failure cause 2. `logL` Log-likelihood value under the fitted model. `AIC` AIC value under the fitted model. `BIC` BIC value under the fitted model.

### References

Shih J-H, Lee W, Sun L-H, Emura T (2018), Fitting competing risks data to bivariate Pareto models, Communications in Statistics - Theory and Methods, doi: 10.1080/03610926.2018.1425450.

### Examples

```t.event = c(72,40,20,65,24,46,62,61,60,60,59,59,49,20, 3,58,29,26,52,20,
51,51,31,42,38,69,39,33, 8,13,33, 9,21,66, 5,27, 2,20,19,60,
32,53,53,43,21,74,72,14,33, 8,10,51, 7,33, 3,43,37, 5, 6, 2,
5,64, 1,21,16,21,12,75,74,54,73,36,59, 6,58,16,19,39,26,60,
43, 7, 9,67,62,17,25, 0, 5,34,59,31,58,30,57, 5,55,55,52, 0,
51,17,70,74,74,20, 2, 8,27,23, 1,52,51, 6, 0,26,65,26, 6, 6,
68,33,67,23, 6,11, 6,57,57,29, 9,53,51, 8, 0,21,27,22,12,68,
21,68, 0, 2,14,18, 5,60,40,51,50,46,65, 9,21,27,54,52,75,30,
70,14, 0,42,12,40, 2,12,53,11,18,13,45, 8,28,67,67,24,64,26,
57,32,42,20,71,54,64,51, 1, 2, 0,54,69,68,67,66,64,63,35,62,
7,35,24,57, 1, 4,74, 0,51,36,16,32,68,17,66,65,19,41,28, 0,
46,63,60,59,46,63, 8,74,18,33,12, 1,66,28,30,57,50,39,40,24,
6,30,58,68,24,33,65, 2,64,19,15,10,12,53,51, 1,40,40,66, 2,
21,35,29,54,37,10,29,71,12,13,27,66,28,31,12, 9,21,19,51,71,
76,46,47,75,75,49,75,75,31,69,74,25,72,28,36, 8,71,60,14,22,
67,62,68,68,27,68,68,67,67, 3,49,12,30,67, 5,65,24,66,36,66,
40,13,40, 0,14,45,64,13,24,15,26, 5,63,35,61,61,50,57,21,26,
11,59,42,27,50,57,57, 0, 1,54,53,23, 8,51,27,52,52,52,45,48,
18, 2, 2,35,75,75, 9,39, 0,26,17,43,53,47,11,65,16,21,64, 7,
38,55, 5,28,38,20,24,27,31, 9, 9,11,56,36,56,15,51,33,70,32,
5,23,63,30,53,12,58,54,36,20,74,34,70,25,65, 4,10,58,37,56,
6, 0,70,70,28,40,67,36,23,23,62,62,62, 2,34, 4,12,56, 1, 7,
4,70,65, 7,30,40,13,22, 0,18,64,13,26, 1,16,33,22,30,53,53,
7,61,40, 9,59, 7,12,46,50, 0,52,19,52,51,51,14,27,51, 5, 0,
41,53,19)

event1 = c(0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,
0,0,1,0,0,0,1,0,1,1,0,1,1,1,1,0,0,1,1,0,
1,0,0,1,1,0,0,1,0,0,0,1,0,1,0,0,1,0,1,1,
1,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,1,1,0,0,0,0,0,1,1,0,0,1,0,0,
0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,1,0,1,0,
0,0,0,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,1,1,0,1,0,0,0,0,1,0,0,0,0,0,
1,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,1,0,0,1,1,0,1,0,0,1,1,0,0,
1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,
0,0,1,0,1,0,0,0,0,1,1,1,1,0,0,0,1,1,0,0,
1,1,1,1,0,0,1,0,1,1,1,1,1,1,1,0,1,1,0,1,
0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,
0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,
1,0,0,0,0,0,0,1,0,0,0,0,1,0,1,0,1,0,0,1,
1,1,0,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,
0,0,0,1,0,0,0,0,1,0,0,1,0,1,0,1,1,0,1,0,
1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,
1,0,0,1,0,0,0,1,0,1,0,0,1,0,0,0,1,1,0,1,
1,1,1,0,0,0,1,0,0,0,0,0,0,0,0,1,1,0,0,0,
0,0,1)

event2 = c(0,1,1,0,0,1,0,0,0,0,0,0,0,1,1,0,1,1,0,1,
0,0,0,1,1,0,0,1,0,0,1,0,0,0,0,1,1,0,0,0,
0,0,0,0,0,0,0,0,1,1,1,0,1,0,1,1,0,1,0,0,
0,0,1,0,1,1,1,0,0,0,0,1,1,1,1,1,1,1,1,1,
1,1,1,0,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,1,
0,1,1,0,0,1,0,0,1,1,1,0,0,0,0,1,1,0,1,1,
0,1,0,0,1,1,0,0,0,1,1,0,0,1,1,1,0,1,0,0,
1,0,1,0,0,1,0,0,1,0,1,1,0,1,1,1,0,0,0,1,
0,1,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,1,0,1,
0,0,1,1,0,1,0,1,1,1,0,1,0,0,0,0,0,0,1,0,
1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0,0,0,1,1,
0,0,0,0,1,0,1,0,1,1,1,1,0,1,1,1,0,1,1,1,
1,1,0,0,0,1,0,1,0,0,0,0,0,0,0,1,0,0,0,1,
0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,
0,0,1,0,0,1,0,0,1,0,0,1,0,1,1,0,0,1,1,1,
1,1,0,0,1,0,0,0,0,1,1,1,1,0,1,1,1,0,1,0,
1,1,1,1,1,1,0,1,1,1,1,0,0,1,0,0,1,1,1,0,
1,0,0,1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,1,1,
0,1,1,1,0,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0,
0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,1,0,1,0,1,
1,1,0,0,1,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,
0,1,0,0,1,1,0,1,1,1,0,0,0,1,0,1,0,0,1,1,
0,0,0,0,1,1,1,0,1,0,1,1,0,1,1,1,0,0,1,0,
0,0,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0,1,1,
1,0,0)

library(Bivariate.Pareto)
set.seed(10)
MLE.Frank.Pareto(t.event,event1,event2,Theta = -5)
```

[Package Bivariate.Pareto version 1.0.3 Index]