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Empirical Likelihood for Non‐Smooth Criterion Functions
Author(s) -
MOLANES LOPEZ ELISA M.,
KEILEGOM INGRID VAN,
VERAVERBEKE NOËL
Publication year - 2009
Publication title -
scandinavian journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.359
H-Index - 65
eISSN - 1467-9469
pISSN - 0303-6898
DOI - 10.1111/j.1467-9469.2009.00640.x
Subject(s) - mathematics , nuisance parameter , quantile , independent and identically distributed random variables , likelihood ratio test , multivariate random variable , empirical likelihood , limit (mathematics) , score test , function (biology) , limit of a function , sequence (biology) , statistics , likelihood function , inference , combinatorics , random variable , maximum likelihood , mathematical analysis , estimator , artificial intelligence , genetics , evolutionary biology , biology , computer science
. Suppose that X 1 ,…, X n is a sequence of independent random vectors, identically distributed as a d ‐dimensional random vector X . Let be a parameter of interest and be some nuisance parameter. The unknown, true parameters ( μ 0 , ν 0 ) are uniquely determined by the system of equations E { g ( X , μ 0 , ν 0 )} = 0 , where g = ( g 1 ,…, g p + q ) is a vector of p + q functions. In this paper we develop an empirical likelihood (EL) method to do inference for the parameter μ 0 . The results in this paper are valid under very mild conditions on the vector of criterion functions g . In particular, we do not require that g 1 ,…, g p + q are smooth in μ or ν . This offers the advantage that the criterion function may involve indicators, which are encountered when considering, e.g. differences of quantiles, copulas, ROC curves, to mention just a few examples. We prove the asymptotic limit of the empirical log‐likelihood ratio, and carry out a small simulation study to test the performance of the proposed EL method for small samples.