NPMLE.Normal: Semiparametric Inference under the Normal Copula with...

In depend.truncation: Statistical Methods for the Analysis of Dependently Truncated Data

Description

Nonparametric maximum likelihood estimation for dependent truncation data under the normal(Gaussian) 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 normal(Gaussian) copula.

Usage

NPMLE.Normal(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 normal(Gaussian) 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

x.trunc=c(0.53, 0.43, 1.63, 0.98, 0.62)
y.trunc=c(1.20, 4.93, 2.09, 2.57, 1.52)
# NPMLE.Normal(x.trunc, y.trunc, x.fix =1, y.fix = 2, plotX = TRUE)

depend.truncation documentation built on May 2, 2019, 3:04 a.m.