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Testing symmetry based on empirical likelihood. (English) Zbl 1516.62053

Summary: In this paper, we propose a general \(k\)th 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 \(k\)th correlation coefficient and present its asymptotic results. Next, we consider statistical inference of the \(k\)th 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 \(k\)th 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 \(k\)th 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.

MSC:

62G10 Nonparametric hypothesis testing
62G20 Asymptotic properties of nonparametric inference
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