I’ve had a hard time figuring this one out for the last three years.

Say I have a model that predicts a flux, μ. When someone conducts an observation, they do not detect a source. Their background-subtracted image (say, optical) has an rms of σ. They tell me that the upper limit on the source is 3σ. I’d like to use this information (along with other available information about the source) in likelihood-based modeling. I understand that the survival function should be used here. Assuming a Gaussian error model, the appropriate quantity (from Eric Feigelson’s presentation in the 2010 Astrostatistics Summer School) is

1 + erf {(x-μ)/(1.414 σ)}

I know the value of μ: predicted flux from the model. I know the value of σ: rms of the (optical) image. What should x be? Is x = 0? Is x = 3σ ? Or is x something else? Are there any reference that discusses this?

I would be very grateful for any help in solving this vexing problem!

Thanks,

Tanmoy

Graduate student in Astronomy