  This paper analyzes several different biases that emerge from the (possibly)
low-precision nonparametric ingredient in a semiparametric model. We show that
both the variance part and the bias part of the nonparametric ingredient can
lead to some biases in the semiparametric estimator, under conditions weaker
than typically required in the literature. We then propose two bias-robust
inference procedures, based on multi-scale jackknife and analytical bias
correction, respectively. We also extend our framework to the case where the
semiparametric estimator is constructed by some discontinuous functionals of
the nonparametric ingredient. Simulation study shows that both bias-correction
methods have good finite-sample performance.
