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Cosmological applications of HII galaxies (HIIGx) and giant extragalactic HIIregions (GEHR) to construct the Hubble diagram at higher redshifts requireknowledge of the "$L$--$\sigma$" relation of the standard candles used. In thispaper, 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 distancesreconstructed through two new cosmology-independent methods, we investigate thecorrelation between the H$\beta$ emission-line luminosity $L$ and ionized-gasvelocity dispersion $\sigma$. The method is based on non-parametricreconstruction using the measurements of Hubble parameters from cosmic clocks,as well as the simulated data of gravitational waves from the third-generationgravitational wave detector (the Einstein Telescope, ET), which can beconsidered as standard sirens. Assuming the emission-line luminosity versusionized gas velocity dispersion relation, $\log L ($H$\beta) = \alpha \log\sigma($H$\beta)+\kappa$, we find the full sample provides a tight constrainton the correlation parameters. However, similar analysis done on threedifferent sub-samples seems to support the scheme of treating HII galaxies andgiant extragalactic HII regions with distinct strategies. Using the corrected"$L$--$\sigma$" relation for the HII observational sample beyond the currentreach 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 spatiallyflat $\Lambda$CDM cosmology.

Exploring the "$L$--$\sigma$" relation of HII galaxies and giant extragalactic HII regions acting as standard candles