MLE.SN.Pareto: 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 \(\alpha_{0}\) with restricted range.

Alpha1.0

Initial guess for the scale parameter \(\alpha_{1}\) with default value 1.

Alpha2.0

Initial guess for the scale parameter \(\alpha_{2}\) with default value 1.

Gamma.0

Initial guess for the common shape parameter \(\gamma\) with default value 1.

epsilon

Positive tunning parameter in the NR algorithm with default value \(10^{-5}\).

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.

Value

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.

Details

The admissible range of Alpha0 (\(\alpha_{0}\)) is \(0 \leq \alpha_{0} \leq (\gamma+1) \alpha_{1} \alpha_{2}.\)

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

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

Run this code

# NOT RUN {
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)
# }

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