• Bernoulli
  • Volume 24, Number 1 (2018), 449-464.

Jackknife empirical likelihood goodness-of-fit tests for U-statistics based general estimating equations

Hanxiang Peng and Fei Tan

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Motivated by applications to goodness of fit U-statistic testing, the jackknife empirical likelihood (JEL) for vector U-statistics is justified with two approaches and the Wilks theorems are proved. This generalizes empirical likelihood (EL) for general estimating equations (GEE’s) to U-statistics based GEE’s. The results are extended to allow for the use of estimated constraints and for the number of constraints to grow with the sample size. It is demonstrated that the JEL can be used to construct EL tests for moment based distribution characteristics (e.g., skewness, coefficient of variation) with less computational burden and more flexibility than the usual EL. This can be done in the U-statistic representation approach and the vector U-statistic approach which were illustrated with several examples including JEL tests for Pearson’s correlation, Goodman–Kruskal’s Gamma, overdisperson, U-quantiles, variance components, and the simplicial depth function. The JEL tests are asymptotically distribution free. Simulations were run to exhibit power improvement of the JEL tests with incorporation of side information.

Article information

Bernoulli, Volume 24, Number 1 (2018), 449-464.

Received: November 2015
Revised: June 2016
First available in Project Euclid: 27 July 2017

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Zentralblatt MATH identifier

empirical likelihood infinitely many constraints Kendall’s tau linear mixed effects model overdisperson side information simplicial depth U-statistics


Peng, Hanxiang; Tan, Fei. Jackknife empirical likelihood goodness-of-fit tests for U-statistics based general estimating equations. Bernoulli 24 (2018), no. 1, 449--464. doi:10.3150/16-BEJ884.

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Supplemental materials

  • Supplement to “Jackknife empirical likelihood goodness-of-fit tests for U-statistics based general estimating equations”. In this Supplement, we introduce the notation and prove the theorems and provide the details to the examples.