Testing symmetry based on empirical likelihood

Jun Zhang, Jing Zhang, Xuehu Zhu, Tao Lu

Research output: Contribution to journalArticlepeer-review

10 Scopus citations


In this paper, we propose a general kth correlation coefficient between the density function and distribution function of a continuous variable as a measure of symmetry and asymmetry. We first propose a root-n moment-based estimator of the kth correlation coefficient and present its asymptotic results. Next, we consider statistical inference of the kth correlation coefficient by using the empirical likelihood (EL) method. The EL statistic is shown to be asymptotically a standard chi-squared distribution. Last, we propose a residual-based estimator of the kth correlation coefficient for a parametric regression model to test whether the density function of the true model error is symmetric or not. We present the asymptotic results of the residual-based kth correlation coefficient estimator and also construct its EL-based confidence intervals. Simulation studies are conducted to examine the performance of the proposed estimators, and we also use our proposed estimators to analyze the air quality dataset.

Original languageEnglish
Pages (from-to)2429-2454
Number of pages26
JournalJournal of Applied Statistics
Issue number13
StatePublished - Oct 3 2018


  • Correlation coefficient
  • empirical likelihood
  • kernel smoothing
  • residuals
  • symmetry


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