FrankFrank.Weibull.MLE {Copula.Markov.survival}

Parameter estimation based on the Frank copula for serial dependence and the Frank copula for dependent censoring with the Weibull distributions

Description

Perform two-stage estimation based on the Frank copula C_theta for serial dependence and the Frank copula tilde(C)_alpha for dependent censoring with the marginal distributions Weib(scale1, shape1) and Weib(scale2, shape2). The jackknife method estimates the asymptotic covariance matrix. Parametric bootstrap is applied while doing Kolmogorov-Smirnov tests and Cramer-von Mises test. The guide for using this function shall be explained by Huang (2019), and Huang, Wang and Emura (2020).

Usage

FrankFrank.Weibull.MLE(subject, t.event, event, t.death, death, stageI, Weibull.plot,
                              jackknife, plot, GOF, GOF.plot, rep.GOF, digit)

Arguments

subject

a vector for numbers of subject

t.event

a vector for event times

event

a vector for event indicator (=1 if recurrent; =0 if censoring)

t.death

a vector for death times

death

a vector for death vindicator (=1 if death; =0 if censoring)

stageI

an option to select MLE or LSE method for the 1st-stage optimization

Weibull.plot

if TRUE, show the Weibull probability plot

jackknife

if TRUE, the jackknife method is used for estimate covariance matrix (default = TRUE)

plot

if TRUE, the plots for marginal distributions are shown (default = FALSE)

GOF

if TRUE, show the p-values for KS-test and CvM-test

GOF.plot

if TRUE, show the model diagnostic plot

rep.GOF

repetition number of parametric bootstrap

digit

accurate to some decimal places

Details

When jackknife=FALSE, the corresponding standard error and confidence interval values are shown as NA.

Value

A list with the following elements:

Sample_size

Sample size N

Case

Count for event occurences

scale1

Scale parameter for Weib(scale1, shape1)

shape1

Shape parameter for Weib(scale1, shape1)

scale2

Scale parameter for Weib(scale2, shape2)

shape2

Shape parameter for Weib(scale2, shape2)

theta

Copula parameter for the Frank copula C_theta

alpha

Copula parameter for the Frank copula tilde(C)_alpha

COV

Asymptotic covariance estimated by the jackknife method

KS

Kolmogorov-Smirnov test statistics

p.KS

P-values for Kolmogorov-Smirnov tests

CM

Cramer-von Mises test statistics

p.CM

P-values for Cramer-von Mises tests

Convergence

Convergence results for each stage

Jackknife_error

Count for error in jackknife repititions

Log_likelihood

Log-likelihood values

Author(s)

Xinwei Huang

Examples

data = FrankFrank.Weibull.data(N = 300, scale1 = 1, shape1 =0.5, theta = 2,
                               scale2 = 0.45, shape2 = 0.5, alpha = 2, b = 10, l = 300)


FrankFrank.Weibull.MLE(subject = data$Subject,
                           t.event = data$T_ij, event = data$delta_ij,
                           t.death = data$T_i_star, death = data$delta_i_star,
                           jackknife= TRUE, plot = TRUE)