1. Two Types of Empirical Likelihood Zheng, Yan Department of Biostatistics University of California, Los Angeles

2. Introduction For the statistics of mean, we can define two types of EL for right censored data  Expression by CDF  Expression by hazard rate

3. Introduction In the context of tests, the two types have corresponding null hypothesis  Where is a constant and p i ’s are positive and sum to 1.  Where is a constant.

4. Differences between two types The expression for S(x).  The variation in constraints When h(t) is NAE, two types’ constraints are same. for continuous distributions for discrete distributions

5. Similarities between two types Asymptotically, they both converge to a 1-degree-of-freedom chi-square distribution under the null hypothesis. Point estimation of theta is same when the hazard function is the NAE.

6. Questions Which type outperforms in the confidence interval coverage and chisquare approximation? Which type has a narrower confidence interval? What’s the impact of continuous and discrete distribution on their performance?

7. Empirical Likelihood Ratio Theorem 1 For the right censored data with a F, suppose the constraint equation is, where is the true value. When n goes to infinity, where the constant

8. Empirical Likelihood Ratio Under the maximization of is more complicated than straight use of Lagrange multiplier. The application usually doesn’t give the simple solution for p i.. A modified EM/self-consistent algorithm is proposed.

9. EM Algorithm in ELR Theorem 2 The maximization of w.r.t. p i s.t. the two constraints: and is given by where satisfies

10. EM Algorithm in ELR E-Step: Given F, the weight w j at location t j can be computed as where t j is either a jump point for the given distribution F or an uncensored observation. The w j is zero at other locations. E F =

11. EM Algorithm in ELR M-Step: With the uncensored pseudo observations X=t j and weights w j from E- step, we then find the probability p j by using Theorem 2. Those probabilities give rise to a new distribution F. A good initial F to start the EM is the NPMLE without the constraint. For right censored data, KME will be the choice.

12. ELR Computation Suppose is the NPMLE from EM algorithm under H 0 and is the NPMLE without any constraint, Find the p-value by Chi square distribution. Thus we can test the hypothesis and construct the confidence intervals.

13. Poisson Extension of the L AL is a function of hazard function Linear Constraint Notice we have used a formula for S(t)=exp(- H(t)) that is only valid for continuous distribution in the case of a discrete distribution. The difference is small for large n.

14. MLE for AL Apply Lagrange multiplier, we get Where the is the solution to

15. ALR Properties Notice the summation are only over the uncensored locations. The last jump of a discrete cumulative hazard function must be one if survival function decreased to zero at last point. ALR has an asymptotic 1-df Chi-square distribution when the constraint is We can construct confidence interval for by chi-square distribution of -2LLR

16. Simulation for Continuous Cases Suppose our F=1- e -t, G=1- e -0.035t. X i =Min{T i, C i }, d i =1 if X i <=C i and d i =0 else. S i =KME= and S 0 =1 where D i =sum(d i ) at same x i. Suppose g(x)=e -x, Sample size=50

17. Simulation for Continuous Cases QQ-plot for ELR QQ-plot for ALR

18. Simulation for Continuous Cases  QQ-plot for ALR with constraint of

19. Simulation for Continuous Cases Confidence Interval Coverage For 1000 runs, ELR gives 949 confidence intervals covering 0.5 and ALR gives 1000 confidence intervals covering 0.5. The above observation indicated that ALR has wider confidence interval than ELR

20. Simulation for Continuous Cases Plots of -2LLR vs. Mu

21. Simulation for Continuous Cases Confidence Intervals from ALR

22. Simulation for Discrete Cases Suppose F=poisson(5), G=poisson(7) Xi=Min{Ti,Ci}, d i =1 if X i <=C i and d i =0 else. S i =KME= and S 0 =1 where D i =sum(d i ) at same x i. Suppose g(x)=x, Sample size=50, 1000 runs

23. Simulation for Discrete Cases QQ-plot for ALR

24. Simulation for Discrete Cases Confidence Intervals: In 1000 runs, 985 confidence intervals covered Mu0=5.0067.

25. Conclusions ELR has narrower confidence interval than ALR which indicates that LRT for ELR should be more powerful than LRT for ALR at same alpha level. On the other hand, the ALR has more accurate coverage on true Mu than ELR. They had similar performance on approximation to Chi Square distribution when n is large.

26. Conclusions Discrete cases have comparable performance on chi-square approximation and confidence interval coverage as continuous cases when sample size is rather large. However, theoretical insight explores that ALR’s perform will be inferior to the ELR in the discrete cases.