  Sizes of narrow emission line regions (NLRs) of AGN could be estimated by
[O~{\sc iii}] line luminosity $L_{O3}$ through the known $R_{NLRs}-L_{O3}$
empirical relations. Unfortunately, it is not convenient to test the
$R_{NLRs}-L_{O3}$ empirical relations through structure properties of spatially
resolved NLRs of large samples of AGN. In this manuscript, a method is proposed
to test the $R_{NLRs}-L_{O3}^{\sim0.25}$ empirical relations for AGN NLRs
through SDSS Type-2 AGN having few orientation effects on NLRs sizes expected
by AGN unified model, after considering sizes $R_{fib}$ of SDSS fiber covered
regions. Comparing $R_{fib}$ and $R_{NLRs}$ estimated by $L_{O3}$, Type-2 AGN
with $R_{fib}>R_{NLRs}$ (Sample-II) and with $R_{fib}<R_{NLRs}$ (Sample-I)
should have different physical properties of NLRs. Accepted electron density
gradients in AGN NLRs, statistically higher electron densities (traced by lower
flux ratio $R_{S2}$ of [S~{\sc ii}]$\lambda6717$\AA~ to [S~{\sc
ii}]$\lambda6731$\AA) could be expected for the Type-2 AGN in the Sample-I.
Then, through the collected 1062 SDSS Type-2 AGN in the Sample-I and 3658 SDSS
Type-2 AGN in the Sample-II, statistically lower $R_{S2}$ for the Type-2 AGN in
the Sample-I can be confirmed with confidence level higher than 5$\sigma$, even
after considering necessary effects. Therefore, the results in this manuscript
can provide strong clues to support that the reported
$R_{NLRs}~\propto~L_{O3}^{0.25}$ empirical relation is preferred to estimate
NLRs sizes of SDSS AGN through SDSS fiber spectroscopic results, and also to
support the commonly expected electron density gradients in AGN NLRs.
