  In many applications, parameters of interest are estimated by solving some
non-smooth estimating equations with $U$-statistic structure. Jackknife
empirical likelihood (JEL) approach can solve this problem efficiently by
reducing the computation complexity of the empirical likelihood (EL) method.
However, as EL, JEL suffers the sensitivity problem to outliers. In this paper,
we propose a weighted jackknife empirical likelihood (WJEL) to tackle the above
limitation of JEL. The proposed WJEL tilts the JEL function by assigning
smaller weights to outliers. The asymptotic of the WJEL ratio statistic is
derived. It converges in distribution to a multiple of a chi-square random
variable. The multiplying constant depends on the weighting scheme. The
self-normalized version of WJEL ratio does not require to know the constant and
hence yields the standard chi-square distribution in the limit. Robustness of
the proposed method is illustrated by simulation studies and one real data
application.
