  Cosmological applications of HII galaxies (HIIGx) and giant extragalactic HII
regions (GEHR) to construct the Hubble diagram at higher redshifts require
knowledge of the "$L$--$\sigma$" relation of the standard candles used. In this
paper, we study the properties of a large sample of 156 sources (25 high-$z$
HII galaxies, 107 local HII galaxies, and 24 giant extragalactic HII regions)
compiled by Terlevich et al.(2015). Using the the cosmological distances
reconstructed through two new cosmology-independent methods, we investigate the
correlation between the H$\beta$ emission-line luminosity $L$ and ionized-gas
velocity dispersion $\sigma$. The method is based on non-parametric
reconstruction using the measurements of Hubble parameters from cosmic clocks,
as well as the simulated data of gravitational waves from the third-generation
gravitational wave detector (the Einstein Telescope, ET), which can be
considered as standard sirens. Assuming the emission-line luminosity versus
ionized gas velocity dispersion relation, $\log L ($H$\beta) = \alpha \log
\sigma($H$\beta)+\kappa$, we find the full sample provides a tight constraint
on the correlation parameters. However, similar analysis done on three
different sub-samples seems to support the scheme of treating HII galaxies and
giant extragalactic HII regions with distinct strategies. Using the corrected
"$L$--$\sigma$" relation for the HII observational sample beyond the current
reach of Type Ia supernovae, we obtain a value of the matter density parameter,
$\Omega_{m}=0.314\pm0.054$ (calibrated with standard clocks) and
$\Omega_{m}=0.311\pm0.049$ (calibrated with standard sirens), in the spatially
flat $\Lambda$CDM cosmology.
