Package 'Bivariate.Pareto'

Title: Bivariate Pareto Models
Description: Perform competing risks analysis under bivariate Pareto models. See Shih et al. (2019) <doi:10.1080/03610926.2018.1425450> for details.
Authors: Jia-Han Shih, Wei Lee
Maintainer: Jia-Han Shih <[email protected]>
License: GPL-2
Version: 1.0.3
Built: 2026-05-18 07:47:11 UTC
Source: https://github.com/cran/Bivariate.Pareto

Help Index

Bivariate Pareto Models

Description

Perform competing risks analysis under bivariate Pareto models. See Shih et al. (2018) for details.

Details

The functions in this package are based on latent failure time models with competing risks in Shih et al. (2018). However, they can be adapted to dependent censoring models in Emura and Chen (2018). See MLE.SN.Pareto for example.

Author(s)

Jia-Han Shih, Wei Lee

Maintainer: Jia-Han Shih <[email protected]>

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.

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

Generate samples from the Frank copula with the Pareto margins

Description

Generate samples from the Frank copula with the Pareto margins.

Usage

Frank.Pareto(n, Theta, Alpha1, Alpha2, Gamma1, Gamma2)

Arguments

n

Sample size.
Theta

Copula parameter θ\theta.
Alpha1

Positive scale parameter α1\alpha_{1} for the Pareto margin.
Alpha2

Positive scale parameter α2\alpha_{2} for the Pareto margin.
Gamma1

Positive shape parameter γ1\gamma_{1} for the Pareto margin.
Gamma2

Positive shape parameter γ2\gamma_{2} for the Pareto margin.

Value

X

X is asscoiated with the parameters Alpha1 and Gamma1.
Y

Y is asscoiated with the parameters Alpha2 and Gamma2.

References

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

library(Bivariate.Pareto)
Frank.Pareto(5,5,1,1,1,1)

Kendall's tau under the SNBP distribution

Description

Compute Kendall's tau under the Sankaran and Nair bivairate Pareto (SNBP) distribution (Sankaran and Nair, 1993) by numerical integration.

Usage

Kendall.SNBP(Alpha0, Alpha1, Alpha2, Gamma)

Arguments

Alpha0

Copula parameter α0\alpha_{0} with restricted range.
Alpha1

Positive scale parameter α1\alpha_{1} for the Pareto margin.
Alpha2

Positive scale parameter α2\alpha_{2} for the Pareto margin.
Gamma

Common positive shape parameter γ\gamma for the Pareto margins.

Details

The admissible range of Alpha0 (α0\alpha_{0}) is 0≤α0≤(γ+1)α1α2.0 \leq \alpha_{0} \leq (\gamma+1) \alpha_{1} \alpha_{2}.

Value

tau

Kendall's tau.

References

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

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

library(Bivariate.Pareto)
Kendall.SNBP(7e-5,0.0036,0.0075,1.8277)

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

Description

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

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\theta.
Alpha1.0

Initial guess for the scale parameter α1\alpha_{1} with default value 1.
Alpha2.0

Initial guess for the scale parameter α2\alpha_{2} with default value 1.
Gamma1.0

Initial guess for the shape parameter γ1\gamma_{1} with default value 1.
Gamma2.0

Initial guess for the shape parameter γ2\gamma_{2} with default value 1.
epsilon

Positive tunning parameter in the NR algorithm with default value 10−510^{-5}.
d

Positive tunning parameter in the NR algorithm with default value e10e^{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)

Maximum likelihood estimation for bivariate dependent competing risks data under the Frank copula with the common Pareto margins

Description

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

Usage

MLE.Frank.Pareto.com(
  t.event,
  event1,
  event2,
  Theta.0 = 1,
  Alpha.0 = 1,
  Gamma.0 = 1,
  epsilon = 1e-05,
  r.1 = 13,
  r.2 = 3,
  r.3 = 3,
  bootstrap = FALSE,
  B = 200
)

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.0

Initial guess for the copula parameter θ\theta.
Alpha.0

Initial guess for the common scale parameter α\alpha with default value 1.
Gamma.0

Initial guess for the common shape parameter γ\gamma with default value 1.
epsilon

Positive tunning parameter in the NR algorithm with default value 10−510^{-5}.
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.
bootstrap

Perform parametric bootstrap if TRUE.
B

Number of bootstrap replications.

Details

The parametric bootstrap method requires the assumption of the uniform censoring distribution. One must notice that such assumption is not always true in real data analysis.

Value

n

Sample size.
count

Iteration number.
random

Randomization number.
Theta

Copula parameter.
Theta.B

Copula parameter (SE and CI are calculated by parametric bootstrap method).
Alpha

Common positive scale parameter for the Pareto margin.
Alpha.B

Common positive scale parameter for the Pareto margin (SE and CI are calculated by parametric bootstrap method).
Gamma

Common positive shape parameter for the Pareto margin.
Gamma.B

Common positive shape parameter for the Pareto margin (SE and CI are calculated by parametric bootstrap method).
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 (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.Frank.Pareto.com(t.event,event1,event2,bootstrap = FALSE)

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 α0\alpha_{0} with restricted range.
Alpha1.0

Initial guess for the scale parameter α1\alpha_{1} with default value 1.
Alpha2.0

Initial guess for the scale parameter α2\alpha_{2} with default value 1.
Gamma.0

Initial guess for the common shape parameter γ\gamma with default value 1.
epsilon

Positive tunning parameter in the NR algorithm with default value 10−510^{-5}.
d

Positive tunning parameter in the NR algorithm with default value e10e^{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 (α0\alpha_{0}) is 0≤α0≤(γ+1)α1α2.0 \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.

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)

Generate samples from the SNBP distribution

Description

Generate samples from the Sankaran and Nair bivairate Pareto (SNBP) distribution (Sankaran and Nair, 1993).

Usage

SN.Pareto(n, Alpha0, Alpha1, Alpha2, Gamma)

Arguments

n

Sample size.
Alpha0

Copula parameter α0\alpha_{0} with restricted range.
Alpha1

Positive scale parameter α1\alpha_{1} for the Pareto margin.
Alpha2

Positive scale parameter α2\alpha_{2} for the Pareto margin.
Gamma

Common positive shape parameter γ\gamma for the Pareto margins.

Details

The admissible range of Alpha0 (α0\alpha_{0}) is 0≤α0≤(γ+1)α1α2.0 \leq \alpha_{0} \leq (\gamma+1) \alpha_{1} \alpha_{2}.

Value

X

X is asscoiated with the parameters Alpha1 and Gamma.
Y

Y is asscoiated with the parameters Alpha2 and Gamma.

References

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

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

library(Bivariate.Pareto)
SN.Pareto(5,2,1,1,1)