  We consider the problem of the estimation of the mean function of an
inhomogeneous Poisson process when its intensity function is periodic. For the
mean integrated squared error (MISE) there is a classical lower bound for all
estimators and the empirical mean function attains that lower bound, thus it is
asymptotically efficient. Following the ideas of the work by Golubev and Levit,
we compare asymptotically efficient estimators and propose an estimator which
is second order asymptotically efficient. Second order efficiency is done over
Sobolev ellipsoids, following the idea of Pinsker.
