emplik (version 1.3-1)

el.trun.test: Empirical likelihood ratio for mean with left truncated data

Description

This program uses EM algorithm to compute the maximized (wrt \(p_i\)) empirical log likelihood function for left truncated data with the MEAN constraint: $$ \sum p_i f(x_i) = \int f(t) dF(t) = \mu ~. $$ Where \(p_i = \Delta F(x_i)\) is a probability. \(\mu\) is a given constant. It also returns those \(p_i\) and the \(p_i\) without constraint, the Lynden-Bell estimator.

The log likelihood been maximized is $$ \sum_{i=1}^n \log \frac{\Delta F(x_i)}{1-F(y_i)} .$$

Usage

el.trun.test(y,x,fun=function(t){t},mu,maxit=20,error=1e-9)

Value

A list with the following components:

"-2LLR"

the maximized empirical log likelihood ratio under the constraint.

NPMLE

jumps of NPMLE of CDF at ordered x.

NPMLEmu

same jumps but for constrained NPMLE.

Arguments

y

a vector containing the left truncation times.

x

a vector containing the survival times. truncation means x>y.

fun

a continuous (weight) function used to calculate the mean as in \(H_0\). fun(t) must be able to take a vector input t. Default to the identity function \(f(t)=t\).

mu

a real number used in the constraint, mean value of \(f(X)\).

error

an optional positive real number specifying the tolerance of iteration error. This is the bound of the \(L_1\) norm of the difference of two successive weights.

maxit

an optional integer, used to control maximum number of iterations.

Author

Mai Zhou

Details

This implementation is all in R and have several for-loops in it. A faster version would use C to do the for-loop part. But it seems faster enough and is easier to port to Splus.

When the given constants \(\mu\) is too far away from the NPMLE, there will be no distribution satisfy the constraint. In this case the computation will stop. The -2 Log empirical likelihood ratio should be infinite.

The constant mu must be inside \(( \min f(x_i) , \max f(x_i) ) \) for the computation to continue. It is always true that the NPMLE values are feasible. So when the computation stops, try move the mu closer to the NPMLE --- $$ \sum_{d_i=1} p_i^0 f(x_i) $$ \(p_i^0\) taken to be the jumps of the NPMLE of CDF. Or use a different fun.

References

Zhou, M. (2005). Empirical likelihood ratio with arbitrary censored/truncated data by EM algorithm. Journal of Computational and Graphical Statistics, 14, 643-656.

Li, G. (1995). Nonparametric likelihood ratio estimation of probabilities for truncated data. JASA 90, 997-1003.

Turnbull (1976). The empirical distribution function with arbitrarily grouped, censored and truncated data. JRSS B 38, 290-295.

Examples

## example with tied observations
vet <- c(30, 384, 4, 54, 13, 123, 97, 153, 59, 117, 16, 151, 22, 56, 21, 18,
         139, 20, 31, 52, 287, 18, 51, 122, 27, 54, 7, 63, 392, 10)
vetstart <- c(0,60,0,0,0,33,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)
el.trun.test(vetstart, vet, mu=80, maxit=15)