  Empirical likelihood is a well-known nonparametric method in statistics and
has been widely applied in statistical inference. The method has been employed
by Lu and Peng (2002) to constructing confidence intervals for the tail index
of a heavy-tailed distribution. It is demonstrated in Lu and Peng (2002) that
the empirical likelihood-based confidence intervals performs better than
confidence intervals based on normal approximation in terms of the coverage
probability. In general, the empirical likelihood method can be hindered by its
imprecision in the coverage probability when the sample size is small. This may
cause a serious undercoverage issue when we apply the empirical likelihood to
the tail index as only a very small portion of observations can be used in the
estimation of the tail index. In this paper, we employ an adjusted empirical
likelihood method, developed by Chen et al. (2008) and Liu and Chen (2010), to
constructing confidence intervals of the tail index so as to achieve a better
accuracy. We conduct a simulation study to compare the performance of the
adjusted empirical likelihood method and the normal approximation method. Our
simulation results indicate that the adjusted empirical likelihood method
outperforms other methods in terms of the coverage probability and length of
confidence intervals. We also apply the adjusted empirical likelihood method to
a real data set.
