I need to fit a gamma distribution that is shifted to the left and truncated at zero (so that for example, my data may only come from the right tail of the full distribution, and I don't have any observations less than zero).

I can find the alpha and beta parameters to fit a regular old two parameter gamma no problem by MLE, but I can't find a good reference on how to fit the parameters for the shifted (and truncated) gamma case. In the shifted-truncated Gamma case I have now three parameters: alpha, beta, and lambda, where lambda is the shifting parameter.

Does anyone knows of a paper that shows how to fit a shifted gamma using MLE? I'd also be open to hear about other methods for fitting such distribution ...

  • $\begingroup$ get your confusion. What i meant is that when you translate it to the left you will have negative values, but you want to keep your same bound of zero as in the original gamma. $\endgroup$
    – Dnaiel
    Dec 12, 2012 at 23:35
  • $\begingroup$ To add my own attempt at clarification - is this a Gamma that is shifted left, then truncated at zero, so you don't know the magnitude of the shift? If so, do you know how many data points fell below zero, or just have the > 0 part of the sample? $\endgroup$
    – jbowman
    Dec 12, 2012 at 23:46
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    $\begingroup$ You won't get a closed form answer, so MLE will have to be carried out numerically. That seems straightforward to me; it wouldn't be any different than fitting any other three-parameter distribution with MLE. What prevents you from doing this? $\endgroup$
    – whuber
    Dec 12, 2012 at 23:58
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    $\begingroup$ @Proc I read the question as stating that the PDF of $x$, $x\ge 0$, is proportional to $\frac{x+\lambda}{\beta}\exp{(-(x+\lambda)/\beta)} \frac{dx}{x+\lambda}$. $\endgroup$
    – whuber
    Dec 13, 2012 at 0:06
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    $\begingroup$ @whuber, I am using C to do the optimization so I'd have to program most of the optimization procedure. If you have some suggestions that'd be great! $\endgroup$
    – Dnaiel
    Dec 13, 2012 at 20:50


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