  We present a possible Cepheid-like luminosity estimator for the long
gamma-ray bursts based on the variability of their light curves. To construct
the luminosity estimator, we use CGRO/BATSE data for 13 bursts, Wind/KONUS data
for 5 bursts, Ulysses/GRB data for 1 burst, and NEAR/XGRS data for 1 burst.
Spectroscopic redshifts, peak fluxes, and high resolution light curves are
available for 11 of these bursts; partial information is available for the
remaining 9 bursts. We find that the isotropic-equivalent luminosities L of
these bursts positively correlate with a rigorously-constructed measure V of
the variability of their light curves. We fit a model to these data that
accommodates both intrinsic scatter (statistical variance) and extrinsic
scatter (sample variance). If one excludes GRB 980425 from the fit on the
grounds that its association with SN 1998bw at a redshift of z = 0.0085 is not
secure, the luminosity estimator spans approx. 2.5 orders of magnitude in L,
and the slope of the correlation between L and V is positive with a probability
of 1 - 1.4 x 10^-4 (3.8 sigma). Although GRB 980425 is excluded from this fit,
its L and V values are consistent with the fitted model, which suggests that
GRB 980425 may well be associated with SN 1998bw, and that GRB 980425 and the
cosmological bursts may share a common physical origin. If one includes GRB
980425 in the fit, the luminosity estimator spans approx. 6.3 orders of
magnitude in L, and the slope of the correlation is positive with a probability
of 1 - 9.3 x 10^-7 (4.9 sigma). Independently of whether or not GRB 980425
should be included in the fit, its light curve is unique in that it is much
less variable than the other approx. 17 light curves in our sample for which
the signal-to-noise is reasonably good.
