  We present the statistical analysis of the properties of gamma-ray bursts
with measured host galaxy redshifts and peaked optical light curves in proper
frames of reference. The optical transients are classified by comparing the
time lag of the optical peak relative to the GRB trigger with the duration of
the gamma-ray emission itself. The results of the correlation analysis of all
possible pairs of energy, spectral, and temporal characteristics of both
gamma-ray and optical emissions are given. We specify the pairs of the
parameters with correlation coefficients greater than 50 % at significance
levels better than 1 %. The following empirical relations, obtained for the
first time, are specifically discussed: a correlation between the peak optical
afterglow $R$ band luminosity and redshift $L_{R} \propto (z+1)^{5.39 \pm
0.74}$ and a correlation between the peak luminosity of the prompt optical
emissions and the time of the peak $L_{R} \propto T_{\rm peak}^{-3.85 \pm
1.22}$. We also analyze the similarity of the relationships between the peak
optical luminosity and the isotropic equivalent of the total energy of
gamma-ray bursts for afterglows ($L_{R} \propto E_{\rm iso}^{1.06 \pm 0.22}$)
and for prompt optical emissions ($L_{R} \propto E_{\rm iso}^{1.59 \pm 0.21}$).
