  Asymptotic expansion of a variation with anticipative weights is derived by
the theory of asymptotic expansion for Skorohod integrals having a mixed normal
limit. The expansion formula is expressed with the quasi-torsion, quasi-tangent
and other random symbols. To specify these random symbols, it is necessary to
classify the level of the effect of each term appearing in the stochastic
expansion of the variable in question. To solve this problem, we consider a
class ${\cal L}$ of certain sequences $({\cal I}_n)_{n\in{\mathbb N}}$ of
Wiener functionals and we give a systematic way of estimation of the order of
$({\cal I}_n)_{n\in{\mathbb N}}$. Based on this method, we introduce a notion
of exponent of the sequence $({\cal I}_n)_{n\in{\mathbb N}}$, and investigate
the stability and contraction effect of the operators $D_{u_n}$ and $D$ on
${\cal L}$, where $u_n$ is the integrand of a Skorohod integral. After
constructed these machineries, we derive asymptotic expansion of the variation
having anticipative weights. An application to robust volatility estimation is
mentioned.
