  Ongoing and future wide-field galaxy surveys can be used to locate a number
of clusters of galaxies with cosmic shear measurement alone. We study
constraints on cosmological models using statistics of weak lensing selected
galaxy clusters. We extend our previous theoretical framework to model the
statistical properties of clusters in variants of cosmological models as well
as in the standard $\Lambda$CDM model. Weak lensing selection of clusters does
not rely on the conventional assumption such as the relation between luminosity
and mass and/or hydrostatic equilibrium, but a number of observational effects
compromise robust identification. We use a large set of realistic mock
weak-lensing catalogs as well as analytic models to perform a Fisher analysis
and make forecast for constraining two competing cosmological models, $w$CDM
model and $f(R)$ model proposed by Hu & Sawicki, with our lensing statistics.
We show that weak lensing selected clusters are excellent probe of cosmology
when combined with cosmic shear power spectrum even in presence of galaxy shape
noise and masked regions. With the information of weak lensing selected
clusters, the precision of cosmological parameter estimate can be improved by a
factor of $\sim1.6$ and $\sim8$ for $w$CDM model and $f(R)$ model,
respectively. Hyper Suprime-Cam survey with sky coverage of $1250$ squared
degrees can constrain the equation of state of dark energy $w_{0}$ with a level
of $\Delta w_0 \sim0.1$. It can also constrain the additional scalar degree of
freedom in $f(R)$ model with a level of $|f_{R0}| \sim5\times10^{-6}$, when
constraints from cosmic microwave background measurements are incorporated.
Future weak lensing surveys with sky coverage of $20,000$ squared degrees will
place tighter constraints on $w_{0}$ and $|f_{R0}|$ even without cosmic
microwave background measurements.
