# Empirical likelihood ratio for mean with left truncated and right censored data, by EM algorithm

### Description

This program uses EM algorithm to compute the maximized (wrt p_i) empirical log likelihood function for left truncated and right censored data with the MEAN constraint:

∑_{d_i=1} p_i f(x_i) = \int f(t) dF(t) = μ ~.

Where p_i = Δ F(x_i) is a probability, d_i is the censoring indicator, 1(uncensored), 0(right censored). The d for the largest observation x, is always (automatically) changed to 1. μ is a given constant. This function also returns those p_i.

The log empirical likelihood function been maximized is

∑_{d_i=1} \log \frac{ Δ F(x_i)}{1-F(y_i)} + ∑_{d_i=0} \log \frac{1-F(x_i)}{1-F(y_i)}.

### Usage

 1 el.ltrc.EM(y,x,d,fun=function(t){t},mu,maxit=30,error=1e-9) 

### Arguments

 y an optional vector containing the observed left truncation times. x a vector containing the censored survival times. d a vector containing the censoring indicators, 1-uncensored; 0-right censored. 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

We return the -2 log likelihood ratio, and the constrained NPMLE of CDF. The un-constrained NPMLE should be WJT or Lynden-Bell estimator.

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.

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 this version is easier to port to Splus, and seems faster enough).

### Value

A list with the following components:

 times locations of CDF that have positive mass. prob the probability of the constrained NPMLE of CDF at those locations. "-2LLR" It is Minus two times the Empirical Log Likelihood Ratio. Should be approximate chi-square distributed under Ho.

Mai Zhou

### References

Zhou, M. (2002). Computing censored and truncated empirical likelihood ratio by EM algorithm. Tech Report, Univ. of Kentucky, Dept of Statistics

Tsai, W. Y., Jewell, N. P., and Wang, M. C. (1987). A note on product-limit estimator under right censoring and left truncation. Biometrika, 74, 883-886.

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

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

### Examples

 1 2 3 4 5 6 7 8 9 ## example with tied observations y <- c(0, 0, 0.5, 0, 1, 2, 2, 0, 0, 0, 0, 0 ) x <- c(1, 1.5, 2, 3, 4, 5, 6, 5, 4, 1, 2, 4.5) d <- c(1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1) el.ltrc.EM(y,x,d,mu=3.5) ypsy <- c(51, 58, 55, 28, 25, 48, 47, 25, 31, 30, 33, 43, 45, 35, 36) xpsy <- c(52, 59, 57, 50, 57, 59, 61, 61, 62, 67, 68, 69, 69, 65, 76) dpsy <- c(1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 0, 1 ) el.ltrc.EM(ypsy,xpsy,dpsy,mu=64) 

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