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In this paper, the problem of whether the left tail and the right tail of a distribution share the same extreme value index (EVI) is addressed and we propose two different test statistics. The first one is based on the result of the joint asymptotic normality of the two Hill estimators for the EVIs of both tails. And therefore, we can construct a quotient-type test statistic, which is asymptotic χ2(1) distributed after some standardization. The second test statistic proposed in this paper is inspired by the two-sample empirical likelihood methodology, and we prove its non parametric version of Wilk's theorem. At last, we compare the efficiencies of our two test statistics and the maximum likelihood (ML) ratio test statistic proposed by Jondeau and Rockinger (2003) in terms of empirical first type error and power through a number of simulation studies, which indicate that the performance of the ML ratio test statistic is worse than our two test statistics in most cases. [ABSTRACT FROM AUTHOR]