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

Maximum likelihood estimation for bivariate dependent competing risks data under the SNBP distribution

Description

Maximum likelihood estimation for bivariate dependent competing risks data under the SNBP distribution (Sankaran and Nair, 1993).

Usage

```MLE.SN.Pareto(
t.event,
event1,
event2,
Alpha0,
Alpha1.0 = 1,
Alpha2.0 = 1,
Gamma.0 = 1,
epsilon = 1e-05,
d = exp(10),
r.1 = 6,
r.2 = 6,
r.3 = 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. `Alpha0` Copula parameter α_{0} with restricted range. `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. `Gamma.0` Initial guess for the common shape parameter γ 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.

Details

The admissible range of `Alpha0` (α_{0}) is 0 ≤q α_{0} ≤q (γ+1) α_{1} α_{2}.

To adapt our functions to dependent censoring models in Emura and Chen (2018), one can simply set `event2` = `1-event1`.

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). `Gamma` Common positive shape parameter for the Pareto margins. `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

Sankaran PG, Nair NU (1993), A bivariate Pareto model and its applications to reliability, Naval Research Logistics, 40(7): 1013-1020.

Emura T, Chen Y-H (2018) Analysis of Survival Data with Dependent Censoring, Copula-Based Approaches, JSS Research Series in Statistics, Springer, Singapore.

Shih J-H, Lee W, Sun L-H, Emura T (2019), Fitting competing risks data to bivariate Pareto models, Communications in Statistics - Theory and Methods, 48:1193-1220.

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.SN.Pareto(t.event,event1,event2,Alpha0 = 7e-5)
```

[Package Bivariate.Pareto version 1.0.3 Index]