Exponential decay parameter estimation

I was asked today how to estimate the parameters of an exponential decay model based on sample data. I said sure I'll have a look remembering the MLE method for estimating lambda from the exponential model. However I realise now that the decay model looks slightly different. F(x)=F0exp(-lambda*t) as opposed to F(x)=lambda*exp(-lambda*t). Does anyone know where I could start looking for a solution to this one - thanks.

 

It's the same thing - you estimate lambda via the MLE (which is just the inverse of the mean if you have no censored values) and F_0 is just your initial constant. The extra lambda in the PDF simply normalises the area under the curve to 1.

 

Right. I didn't get the significance of the second lambda but that makes sense. So the inverse mean gives you the rate of decline effectively which is the same regardless of anything else. So by initial constant, do you mean like this - my object is 2.5 now (F0=2.5) and I want to know what it will decay to in 4 years (t=4)?

Thanks.

 

So another question if I may - so the data (3 data points) is at time 0, 7 and 13 with values 17.2, 0.58, 0.32. Given they are at different time intervals, how do I calculate the average? Is it simply (17.2+0.58+0.32)/(0+7+13)?

 

Oh, I see. I was under the impression you had survival type data, rather than quantity versus time.

You have \[F=F_0e^{\lambda t}\]

with certain values of F at times t.

One approach is to transform the equation

\[\ln F = \ln F_0 + e^{\lambda t}\]

and then fit a least squares straight line.

What I am not sure of is how to estimate the bias/variance under this approach - I think it may be invariant under the ML principle but I would take advice on this!

 

Thanks. After getting some more information, it seems the method they have been using is non-linear regression and solving for the parameters via a convergence theorem. So pretty much as you have said. Also I have other questions but a aware this is a CT subject forum and don't want to push the friendship. Is there another forum I can use for this sort of thing?