linear combination of exponential distributions

I have an equation with linear combination of exponential distributions, i.e.
sum (c_i * exp(a_i)) on the right hand side, and on the left hand side I have something that is best fitted by cauchy or log-normal...

Now, I am looking for an estimating function for a linear combination of exponential distributions. Probably it depends on the distribution of the rates (a_i).
I was thinking about using known results of non-homogeneous poisson process, but the problem is that I don't really have pure sum of exponentials but a linear combination of them....

Please note that I am looking on the problem in the direction of estimating the right hand side and see if it fits the left. This direction is important...

Any pointer / paper / book / idea will be highly appreciated,
Thank you in advance,