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(Abridged) Several phenomena in astrophysics generate light curves with timedelays. Among these are reverberation mapping, and lensed quasars. In somesystems, the measurement of the time-delay is complicated by the fact that thedelayed components are unresolved and that the light curves are generated froma red-noise process. We derive the likelihood function of the observationsgiven a model of either a combination of time-delayed light curves or a singlelight curve. This likelihood function is different from the auto-correlationfunction. We demonstrate that given a single-band light curve that is acombination of two (or more) time-shifted copies of an original light curve,generated from a red-noise probability distribution, we can test if thetotal-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 fluxratios between these unresolved components even when the flux ratio is about1/10. This method is useful for identifying lensed quasars and simultaneouslymeasuring their time delays, and for estimating the reverberation time scalesof active galactic nuclei. In a companion paper, we derive a method that usesthe center-of-light position (e.g., of a lensed quasar) along with the combinedflux. This allow us to identify lensed quasars and supernovae and measure theirtime delays, with higher fidelity compared to the flux-only method. Theastrometry + flux method, however, is not suitable for quasar reverberationmapping. We also comment on the commonly used method of fitting a power-lawmodel to a power spectrum, and present the proper likelihood function for sucha fit. We test the new method on simulations and provide Python and MATLABimplementations.

Measuring time delays: I. Using a flux time series that is a linear combination of time-shifted light curves