  The validity of the emission line luminosity vs. ionised gas velocity
dispersion ($L - \sigma$) correlation for HII galaxies (HIIGx), and its
potential as an accurate distance estimator are assessed. For a sample of 128
local ($0.02\lesssim z\lesssim 0.2$) compact HIIGx with high equivalent widths
of their Balmer emission lines we obtained ionised gas velocity dispersion from
high S/N high-dispersion spectroscopy (Subaru-HDS and ESO VLT-UVES) and
integrated H$\beta$ fluxes from low dispersion wide aperture spectrophotometry.
We find that the $L - \sigma$ relation is strong and stable against
restrictions in the sample (mostly based on the emission line profiles). The
`gaussianity' of the profile is important for reducing the rms uncertainty of
the distance indicator, but at the expense of substantially reducing the
sample. By fitting other physical parameters into the correlation we are able
to significantly decrease the scatter without reducing the sample. The size of
the starforming region is an important second parameter, while adding the
emission line equivalent width or the continuum colour and metallicity,
produces the solution with the smallest rms scatter=$\delta \log
L(\mathrm{H}\beta) = 0.233$. The derived coefficients in the best $L - \sigma$
relation are very close to what is expected from virialized ionising clusters,
while the derived sum of the stellar and ionised gas masses are similar to the
dynamical mass estimated using the HST corrected Petrosian radius. These
results are compatible with gravity being the main mechanism causing the
broadening of the emission lines in these very young and massive clusters. The
derived masses range from about 2 $\times10^6$ $\mathrm{M}_{\odot}$ to $10^9$
$\mathrm{M}_{\odot}$ and their `corrected' Petrosian radius, from a few tens to
a few hundred parsecs.
