  We propose a redshift estimator for the long ($T_{90} > 20$) gamma-ray bursts
(GRBs) observed by the BATSE. It is based on an empirical relation between the
redshift and the power-law index of power density spectra (PDSs) of the
observed GRBs. This relation is constructed by using the fact that the
power-law index is dependent upon a characteristic timescale of GRB light
curves which are inevitably stretched by cosmological time dilation. We
construct the empirical relation using both individual PDSs and averaged PDSs.
An error estimates of $z$ are 1.09 and 1.11 for the empirical relation by
individual PDS fits, 1.72 and 1.56 by averaged PDS fits, for the least squares
fit and the maximum likelihood fit, respectively. We attempt to determine the
spatial distribution of the GRBs observed by the BATSE as a function of
redshifts on the basis of the resulting redshift estimator. We find that the
obtained spatial distribution of the observed GRBs seems consistent with that
of the GRBs whose redshifts are reported, even though the estimated errors are
not very accurate. The GRBs observed by the BATSE seem distributed within $z
\sim 5-6$. This result has implications on theoretical calculations of stellar
formations at high redshifts and beaming geometry via a statistical study of
the observed GRBs involving beaming-induced luminosity functions. We discuss
such implications, and possible uncertainties of the suggested method.
