# 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:

∑ p_i f(x_i) = \int f(t) dF(t) = μ ~.

Where p_i = Δ F(x_i) is a probability. μ 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

∑_{i=1}^n \log \frac{Δ F(x_i)}{1-F(y_i)} .

### Usage

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

### 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.

### 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 μ 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 —

∑_{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.

### 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.

Mai Zhou

### 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

 1 2 3 4 5 ## 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) 

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