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