Two Types of Empirical Likelihood

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 pis 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 discrete distributions
for continuous 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?
• Whats 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 doesnt give
  the simple solution for pi..
• A modified EM/self-consistent algorithm is
  proposed.

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EM Algorithm in ELR

• Theorem 2
• The maximization of w.r.t. pi
  s.t. the two constraints and
  is given by
• where satisfies

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EM Algorithm in ELR

• E-Step Given F, the weight wj at location tj can
  be computed as
• where tj is either a jump point for the
  given distribution F or an uncensored
  observation.
• The wj is zero at other locations.
• EF

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EM Algorithm in ELR

• M-Step With the uncensored pseudo observations
  Xtj and weights wj from E-step, we then find the
  probability pj 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
  H0 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

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Simulation for Continuous Cases

• Suppose our F1- e-t, G1- e-0.035t.
• XiMinTi,Ci, di1 if XiltCi and di0 else.
• SiKME and S01 where
  Disum(di) at same xi.
• Suppose g(x)e-x,
• Sample size50

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Simulation for Continuous Cases

• QQ-plot for ELR

• QQ-plot for ALR

18
Simulation for Continuous Cases

• QQ-plot for ALR
• with constraint of

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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

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Simulation for Continuous Cases

• Confidence Intervals from ALR

22
Simulation for Discrete Cases

• Suppose Fpoisson(5), Gpoisson(7)
• XiMinTi,Ci, di1 if XiltCi and di0 else.
• SiKME and S01 where
  Disum(di) at same xi.
• Suppose g(x)x,
• Sample size50, 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 Mu05.0067.

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 ALRs perform will be inferior to
  the ELR in the discrete cases.