  (Abridged) Several phenomena in astrophysics generate light curves with time
delays. Among these are reverberation mapping, and lensed quasars. In some
systems, the measurement of the time-delay is complicated by the fact that the
delayed components are unresolved and that the light curves are generated from
a red-noise process. We derive the likelihood function of the observations
given a model of either a combination of time-delayed light curves or a single
light curve. This likelihood function is different from the auto-correlation
function. We demonstrate that given a single-band light curve that is a
combination of two (or more) time-shifted copies of an original light curve,
generated from a red-noise probability distribution, we can test if the
total-flux light curve is a composition of time-delayed copies or,
alternatively, is consistent with being the original light curve. Furthermorew,
in some realistic cases, it is possible to measure the time delays and flux
ratios between these unresolved components even when the flux ratio is about
1/10. This method is useful for identifying lensed quasars and simultaneously
measuring their time delays, and for estimating the reverberation time scales
of active galactic nuclei. In a companion paper, we derive a method that uses
the center-of-light position (e.g., of a lensed quasar) along with the combined
flux. This allow us to identify lensed quasars and supernovae and measure their
time delays, with higher fidelity compared to the flux-only method. The
astrometry + flux method, however, is not suitable for quasar reverberation
mapping. We also comment on the commonly used method of fitting a power-law
model to a power spectrum, and present the proper likelihood function for such
a fit. We test the new method on simulations and provide Python and MATLAB
implementations.
