  We present an analytic model for the splashback mass function of dark matter
halos, which is parameterized by a single coefficient and constructed in the
framework of the generalized excursion set theory and the self-similar
spherical infall model. The value of the single coefficient that quantifies the
diffusive nature of the splashback boundary is determined at various redshifts
by comparing the model with the numerical results from the Erebos N-body
simulations for the Planck and the WMAP7 cosmologies. Showing that the analytic
model with the best-fit coefficient provides excellent matches to the numerical
results in the mass range of $5\le M/(10^{12}h^{-1}\,M_{\odot})< 10^{3}$, we
employ the Bayesian and Akaike Information Criterion tests to confirm that our
model is most preferred by the numerical results to the previous models at
redshifts of $0.3\le z\le 3$ for both of the cosmologies. It is also found that
the diffusion coefficient decreases almost linearly with redshifts, converging
to zero at a certain threshold redshift, $z_{c}$, whose value significantly
differs between the Planck and WMAP7 cosmologies. Our result implies that the
splashback mass function of dark matter halos at $z\ge z_{c}$ is well described
by a parameter-free analytic formula and that $z_{c}$ may have a potential to
independently constrain the initial conditions of the universe.
