  The nuclear matrix elements (NMEs) of the neutrinoless and two-neutrino
double-$\beta$ decays of $^{48}$Ca are calculated by the quasiparticle
random-phase approximation (QRPA) with emphasis on the consistency examinations
of this calculation method. The main new examination points are the consistency
of two ways to treat the intermediate-state energies in the two-neutrino
double-$\beta$ NME and comparison with the experimental charge-exchange
strength functions obtained from $^{48}$Ca$(p,n)$ and $^{48}$Ti$(n,p)$
reactions. No decisive problem preventing the QRPA approach is found. The
obtained neutrinoless double-$\beta$ NME adjusted by the ratio of the effective
and bare axial-vector current couplings is lowest in those calculated by
several groups and close to one of the QRPA values obtained by another group.
