Model parameter distributions from Gamma distributed data

1. Jul 13, 2011

mikeclimber1

I have a set of data points {xi,yi} where each yi is a Gamma distributed variable where both the shape k and scale $\theta$ depend on i.

I then fit the data points with a power law model y=a(x)b.

I would like to know the probability distributions for the fit parameters a and b.

Is there an analytical approach for this problem? The only method I can think of is to simulate a bunch of data sets and manually build the distributions of a and b, at which point I could fit the distributions.

2. Jul 13, 2011

pmsrw3

What objective function did you use? How many points, and are they independent? There's a general theorem that says that maximum likelihood estimates based on independent variates are asymptotically normal.

Other than that, I would be surprised if there's anything substantially better than the Monte Carlo approach you mention.

3. Jul 13, 2011

mikeclimber1

By taking the log of each side ln(y)=b ln(x)+ln(a), than an ordinary least square fit can be used I think.

There are four data points and they are independent. To give you an idea of what I'm dealing with, the last point {x4,y4}, is distributed with a Gamma function with a shape parameter k=1, meaning that it's exponential, whereas for {x1,y1}, the Gamma distribution is closer to normal. I also know that shape and scale parameters vary with i in such a way that the mean scales linearly with i. This means that on average, the four data points are linear and b$\approx$1. However, I believe that the distribution of b is non-normal, given the strong asymmetry in how some of the data points are distributed. Hopefully this makes sense.

I've used the Monte Carlo approach and found that distribution of b has a mean value of $\approx$1, but it is very asymmetric. I was hoping that there was some analytical technique to determine the functional form of the distribution of b.

4. Jul 13, 2011

pmsrw3

Oh, only four points. Yeah, the asymptotic properties aren't going to be much help.

Well, I'm going to tentatively suggest, instead of a least squares fit, you look at a http://en.wikipedia.org/wiki/Generalized_linear_model" [Broken]. Common stats packages will do this, there's some analytical backing (but don't ask me about it -- I'm ignorant, you'll have to read up), and it will allow you to deal explicitly with the fact that you have gamma-distributed data.

Mucho gusto.

Last edited by a moderator: May 5, 2017
5. Jul 14, 2011

mikeclimber1

Thanks for the suggestion. I looked a little at generalized linear models because it allows for the dependent variables to be generated from any distribution of the exponential family. But I'm also ignorant, so I haven't figured out how to use it yet.

Thanks again.

6. Jul 14, 2011

pmsrw3

I actually have done GLM fits. Mathematica, R, and SPSS all have GLM fit functions -- I just used them (Mathematica to be precise). For my application, it was simple and worked very well -- much better than a least squares fit.

7. Jul 14, 2011

bpet

If using linear regression the explicit formula would be

a = exp(B-AC/D)
b = C/D

where A=E[log(X)], B=E[log(Y)], C=E[log(X)log(Y)]-AB, D=E[log(X)^2]-A^2 where the E[...] is the sample mean, so I'd be surprised if there is an explicit formula for the distribution.

Edit: if the Xj are fixed then effectively log(a) and b are linear combinations of the log(Yj), so perhaps you could try inverting the characteristic function, which can be written in terms of

$$E[\exp(itY_j)]=\theta_j^{it}\Gamma(it+k_j)/\Gamma(k_j)$$

Last edited: Jul 14, 2011