Empirical Likelihood - Southern Methodist University

Empirical Likelihood Dario Nappa Jon Sanders What we are going to talk about Parametric Likelihood Empirical likelihood Empirical Likelihood Statistical properties Pseudo Empirical Likelihood

Use of auxiliary information Comparison with other statistical confidence intervals Application 1: Wu and Rao 2004 Application 2: Chen and Quin 1993 Available Packages Bibliography Parametric Likelihood Parametric likelihood for a set of n independent

observations X1 , X 2 ,....X n ~ f(x) n l(X) f(X i ) i 1 density

Empirical Likelihood Statistical Properties We can define a likelihood ratio MLE Under some reasonable conditions sets of the form

Can be used as confidence region for T(F0) Owen 1988 Likelihood Ratio Pseudo Empirical 1 Use Horvitz-Thompson estimator for the empirical

log-likelihood di 1 i 2 Defined as pseudo empirical likelihood

3. Once again the likelihood ratio follows a Chi-square distribution Auxiliary information 1. We have to maximize the EL 2. Using the auxiliary information

3. Subject to the constrains 4. Solution ^ Y iS pi yi Note In this slide X is the auxiliary information, Y is the entity of interest

Comparison with other statistical confidence intervals Parametric Method Bootstrap Applications Application 1: Wu and Rao (2004)

Create confidence intervals for complex sampling designs Use Pseudo Empirical Likelihood: Authors show asymptotic results similar to empirical likelihood Coverage probability

Tails Error Rates Average Length NT: Normality Test EL1: Empirical Likelihood EL2: Empirical Likelihood with mean constrain

EL has coverage probability closer to the nominal value The error rates are more symmetrical for the EL EL2 has shorter length, worse coverage probability than L1 Application 2: Chen and Qin (1993) Interested in efficient use of auxiliary information Derive results involving auxiliary data seen earlier

Simulation comparing MELE to ratio estimators Models Used Results Available Packages el.S

S-Plus function for calculating ELR for mean http://www-stat.stanford.edu/~owen/empirical/el.S emplik

R package for EL CRAN Mirror or http://www.ms.uky.edu/~mai/splus/library/emplik/ elm.m/plog.m

ELR and PELR functions in Matlab http://www-stat.stanford.edu/~owen/empirical/elm.m http://www-stat.stanford.edu/~owen/empirical/plog.m GAUSS

http://www.ssc.wisc.edu/~bhansen/progs/progs_gmm.html Bibliography

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