  We explored here the near-infrared $H$-band atmospheric window aiming to
provide quantitative diagnostic tools for deriving stellar parameters, for
instance, effective temperature ($T_{eff}$) and metallicity ([$Fe/H$]), of cool
giants ($T_{eff}$ $<$ 5000 K) using low-resolution spectra. We obtained 177
cool giants from the X-shooter spectral library covering a wider metallicity
range ($-$2.35 dex $<$ [$Fe/H$] $<$ 0.5 dex) than in earlier works. Degrading
the spectral resolution to R$\sim$ 1200, we estimated equivalent widths of
several important spectral features, and the behavior of spectral features with
stellar parameters are studied. Also, the empirical relations for deriving
$T_{eff}$ and [$Fe/H$] are established in the Bayesian framework. We found that
$^{12}$CO at 1.56 $\mu$m and 1.62 $\mu$m, and $^{12}$CO+MgI at 1.71 $\mu$m are
the best three $T_{eff}$ indicators with a typical accuracy of 153 K, 123 K and
107 K, respectively. The cubic Bayesian model provides the best metallicity
estimator with a typical accuracy of 0.22 dex, 0.28 dex, and 0.24 dex for FeH
at 1.62 $\mu$m, $^{12}$CO at 1.64 $\mu$m, and Fe I at 1.66 $\mu$m,
respectively. We also showed a detailed quantitative metallicity dependence of
$T_{eff}-$EWs correlations defining three metallicity groups, supersolar
([$Fe/H$] $>$ 0.0 dex), solar ($-$0.3 dex $<$ [$Fe/H$] $<$ 0.3 dex), and
subsolar ([$Fe/H$] $<$ $-$0.3 dex), from Hierarchical Bayesian modelling. The
difference between the solar and subsolar relationship is statistically
significant, but such difference is not evident between the solar and
supersolar groups.
