emplik (version 1.3-2)

emplikH1B: Return binomial empirical likelihood ratio for the given lambda, with right censored data

Description

Compute the binomial empirical likelihood ratio for the given tilt parameter lambda. Most useful for construct Wilks confidence intervals. The null hypothesis or constraint is defined by the parameter \(\theta\), where $$\int fung(t) d \log (1-H(t)) = \theta $$

Where \(H(t)\) is the unknown cumulative hazard function; \(fung(t)\) can be any given function. In the future, the function \(fung\) may replaced by the vector of \(fung(x)\), since this is more flexible.

Input data can be right censored. If no censoring, set d=rep(1, length(x)).

Usage

emplikH1B(lambda, x, d, fung, CIforTheta=FALSE)

Value

A list with the following components:

times

the location of the hazard jumps.

jumps

the jump size of hazard function at those locations.

lambda

the input lambda.

"-2LLR"

the -2Log Likelihood ratio.

IntHaz

The theta defined above, for the given lambda.

Arguments

lambda

a scalar. Can be positive or negative. The amount of tiling.

x

a vector of the censored survival times.

d

a vector of the censoring indicators, 1-uncensor; 0-right censor.

fung

a left continuous (weight) function used to calculate the weighted hazard in the parameter \(\theta\). fung must be able to take a vector input. See example below.

CIforTheta

an optional logical value. Default to FALSE. If set to TRUE, will return the integrated hazard value for the given lambda.

Author

Mai Zhou

Details

This function is used to calculate lambda confidence interval (Wilks type) for \(\theta\).

This function is designed for the case where the true distribution should be discrete. Ties in the data are OK.

The log empirical likelihood used here is the `binomial' version empirical likelihood: $$ \sum_{i=1}^n \delta_i \log (dH(x_i)) + (R_i - \delta_i)\log [1- dH(x_i) ] . $$

References

Pan, X. and Zhou, M. (2002), ``Empirical likelihood in terms of hazard for censored data''. Journal of Multivariate Analysis 80, 166-188.

Examples

## fun <- function(x) { as.numeric(x <= 6.5) }
## emplikH1.test( x=c(1,2,3,4,5), d=c(1,1,0,1,1), theta=2, fun=fun) 
## fun2 <- function(x) {exp(-x)}  
## emplikH1.test( x=c(1,2,3,4,5), d=c(1,1,0,1,1), theta=0.2, fun=fun2)