Under the existence of nuisance parameters,
we consider a class of tests $\mathscr{S}$
which contains the likelihood ratio,
Wald and Rao's score tests as special cases.
To investigate the influence of nuisance parameters,
we derive the second order asymptotic expansion
of the distribution of $T \in \mathscr{S}$
under a sequence of local alternatives.
This result and concrete examples illuminate
some interesting features of effects due
to nuisance parameters. Optimum properties
for a modified likelihood ratio test proposed
in Mukerjee [\ref{M-1994}] are shown under the criteria
of second order local maximinity.