Semiparametric Inference for Copula Models with Dependent Truncation Data

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Description

Nonparametric maximum likelihood estimation for dependent truncation data under the Plackett copula models (Emura & Wang, 2012). The forms of the marginal distributions for X and Y are completely unspecified, but the parametric form of copula is specified as the Plackett copula.

Usage

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NPMLE.Plackett(x.trunc, y.trunc,
x.fix = median(x.trunc), y.fix = median(y.trunc), plotX = TRUE)

Arguments

x.trunc

vector of variables satisfying x.trunc<=y.trunc

y.trunc

vector of variables satisfying x.trunc<=y.trunc

x.fix

vector of fixed points at which marginal distribution function of X is calculated

y.fix

vector of fixed points at which marginal survival function of Y are calculated

plotX

if TRUE, the culumative distribution function for X is plotted

Details

The function produces the nonparametric maximum likelihood estimate (NPMLE) for the marginal distributions and the estimate of the association parameter under the Plackett copula model. The method follows Emura & Wang (2012). The maximization of the likelihood (L) is conducted by minimizing -logL by "nlm". Standard errors are calculated from the diagonal elements in the observed Fisher information matrix, which is obtained from the output of the "nlm". The method can handle ties but cannot handle right-censoring.

Value

alpha

estimate of association parameter

alpha_se

standard error of the estimate of association parameter

Hx

estimate of the marginal cumulative reverse-hazard function for X

Hx_se

standard error of the estimate of the marginal cumulative reverse-hazard function for X

Ay

estimate of the marginal cumulative hazard function for Y

Ay_se

standard error of the estimate of the marginal cumulative hazard function for Y

Fx

estimate of the marginal distribution function for X

Fx_se

standard error of the estimate of the marginal distribution function for X

Sy

estimate of the marginal survival function for Y

Sy_se

standard error of the estimate of the marginal survival function for Y

conv

an integer indicating why the optimization process terminated in "nlm": If conv=1, the maximization of the likelihood function is properly done. Please refer R function "nlm" for more details.

iteration

the number of iterations until convergence

Grad

L_2 norm for the gradient vector at the solution (close to zero if the solution is proper)

MinEigen

Minimum eigenvalue of the Hessian matrix at the solution (positive if the solution is proper)

Author(s)

Takeshi EMURA

References

Emura T, Wang W (2012) Nonparametric maximum likelihood estimation for dependent truncation data based on copulas, Journal of Multivariate Analysis 110, 171-88

Examples

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##### Generate truncated data from Plackett copula with unit exponential marginals #####
set.seed(1)
m=25 #### sample size ####
alpha_true=1/5.11 #### Kendall's tau on (X,Y) = 0.5 ####
x.trunc=y.trunc=numeric(m)
l=1
while(l<=m){
  u=runif(1,min=0,max=1)
  v=runif(1,min=0,max=1)
  K=function(w){
    A=sqrt( (1+(alpha_true-1)*(u+w))^2-4*alpha_true*(alpha_true-1)*u*w )  
    2*v+(  1+(alpha_true-1)*u-(alpha_true+1)*w  )/A-1
  }
  low=0.00001*u
  up=1-0.00001*v
  for(j in 1:100){ #### Invert K(.) by the bisection method to ####
    mid=(low+up)/2;M<-K(mid);U<-K(up)
    if(M*U>0){up<-mid}else{low<-mid}
  }
  w=mid
  x=-log(1-u);y=-log(w)
  if((x<y)){
  x.trunc[l]=x;y.trunc[l]=y
  l=l+1
  }
}

NPMLE.Plackett(x.trunc, y.trunc, x.fix =1, y.fix = 1, plotX = TRUE)

#### compare the above estimate with the true value #####
c(alpha_true=alpha_true,Fx_true=1-exp(-1),Sy_true=exp(-1))