  We conduct a first comprehensive study of the Luminosity Function (LF) using
a non-parametric approach. We use Gaussian Process to fit available luminosity
data between redshifts $z \sim 2-8$. Our free-form LF in the non-parametric
approach rules out the conventional Schechter function model to describe the
abundance-magnitude relation at redshifts $z=3$ and $4$. Hints of deviation
from the Schechter function are also noticed at redshifts 2, 7 and 8 at lower
statistical significance. Significant deviation starts for brighter ionizing
sources at $M_{\rm UV} \lesssim -21$. The UV luminosity density data at
different redshifts are then derived by integrating the LFs obtained from both
methods with a truncation magnitude of $-17$. In our analysis, we also include
the first 90 arcmin$^2$ JWST/NIRCam data at $z \sim 9-12$. Since at larger
magnitudes, we do not find major deviations from the Schechter function, the
integrated luminosity density differs marginally between the two methods.
Finally, we obtain the history of reionization from a joint analysis of UV
luminosity density data along with the ionization fraction data and Planck
observation of Cosmic Microwave Background. The history of reionization is not
affected by the deviation of LFs from Schechter function at lower magnitudes.
We derive reionization optical depth to be $\tau_{\rm
re}=0.0494^{+0.0007}_{-0.0006}$ and the duration between 10$\%$ and 90$\%
$completion of ionization process is found to be $\Delta z\sim
1.627^{+0.059}_{-0.071}$.
