  This paper presents results from photometric and statistical-parallax
analysis of a sample of 850 field RR Lyrae (RRL) variables. The photometric and
spectroscopic data for sample RRLs are obtained from (1) our new spectroscopic
observations (for 448 RRLs) carried out with the Southern African Large
Telescope (SALT); (2) our photometric observations using the 1.0-m telescope of
the South African Astronomical Observatory (SAAO), and (3) literature. These
are combined with accurate proper motion data from the second release of
\textit{Gaia} mission (DR2). This study primarily determines the velocity
distribution of solar neighborhood RRLs, and it also calibrates the zero points
of the RRLs visual V-band luminosity-metallicity (LZ or $M_V-$[\text{Fe/H}])
relation and their period-luminosity-metallicity (PLZ) relations in the
\textit{WISE} $W_1-$ and \textit{2MASS} $Ks-$band. The calibrated PLZ and LZ
relations are used to estimate the Galactic Center distance and the distance
modulus of the Large Magellanic Cloud (LMC), which are found to be
7.99$\pm$0.49\,kpc and 18.46$\pm$0.09 \,mag, respectively. All our results are
in excellent agreement with available literature based on statistical parallax
analysis, but are considerably more accurate and precise. Moreover, the
zero-points of our calibrated PLZ and LZ relations are quite consistent with
current results found by other techniques and yield the LMC distance modulus
that is within 0.04\,mag of the current most precise estimate.
