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How do you fit non-negative exponential decay that is biased with non-uniform noise over time?

 We often make assumptions when we model exponential decays.

 
The most common assumptions that I know of are, 1) the signal is contaminated with zero-mean, gaussian distributed noise and 2) the noise distribution is uniform across time with the decay.
 
What do we do when we know these assumptions are both false. In my particular case, the signal is biased by a positive, non-zero mean, noise distribution that is dependent on the signal intensity, and thus the noise distribution changes with time as the signal decays.

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Ok, so postulate some sort of behavior for that distribution as a function of time. Use a couple of parameters to model the behavior, choosing some logical candidate distribution. It need not be perfect, just as good as you can do.
 
Then use maximum likelihood estimation to find the parameters for the noise model, as well as the exponential decay model. This is just an optimization, although you will probably need constraints on the parameters. And most of the time, you will need to optimize the log of the likelihood function to make things numerically tractable.
 
So, for example, you might postulate some sort of gamma distribution for the noise model, where the gamma parameters are a function of time. Or, pick some other distribution, anything from lognormal, to beta, even uniform. Whatever makes sense in context.
 
Sorry, I won't/can't write it for you, since you need to do the work up front to choose these models. But MLE is simple in concept. 

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