  Originating from neutron star-neutron star (NS-NS) or neutron star-black hole
(NS-BH) mergers, short gamma-ray bursts (SGRBs) are the first electromagnetic
emitters associated with gravitational waves. This association makes the
determination of SGRB formation rate (FR) a critical issue. We determine the
true SGRB FR and its relation to the cosmic star formation rate (SFR). This can
help in determining the expected Gravitation Wave (GW) rate involving small
mass mergers. We present non-parametric methods for the determination of the
evolutions of the luminosity function (LF) and the FR using SGRBs observed by
{\it Swift}, without any assumptions. These are powerful tools for small
samples, such as our sample of 68 SGRBs. We combine SGRBs with and without
extended emission (SEE), assuming that both descend from the same progenitor.
To overcome the incompleteness introduced by redshift measurements we use the
Kolmogorov-Smirnov (KS) test to find flux thresholds yielding a sample of
sources with a redshift drawn from the parent sample including all sources.
Using two subsamples of SGRBs with flux limits of $4.57 \times 10^{-7}$ and
$2.15 \times 10^{-7}$ erg cm$^{-2}$ s$^{-1}$ with respective KS {\it p=(1,
0.9)}, we find a 3 $\sigma$ evidence for luminosity evolution (LE), a broken
power-law LF with significant steepening at $L\sim 10^{50}$ erg s$^{-1}$, and a
FR evolution that decreases monotonically with redshift (independent of LE and
the thresholds). Thus, SGRBs may have been more luminous in the past with a FR
delayed relative to the SFR as expected in the merger scenario.
