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Model parameter distributions from Gamma distributed data

  1. Jul 13, 2011 #1
    I have a set of data points {xi,yi} where each yi is a Gamma distributed variable where both the shape k and scale [itex]\theta[/itex] 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. jcsd
  3. Jul 13, 2011 #2
    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.
  4. Jul 13, 2011 #3
    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[itex]\approx[/itex]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 [itex]\approx[/itex]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.
  5. Jul 13, 2011 #4
    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
  6. Jul 14, 2011 #5
    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.
  7. Jul 14, 2011 #6
    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.
  8. Jul 14, 2011 #7
    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

    Last edited: Jul 14, 2011
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