I need to estimate a truncated gamma distribution parameters (shape , scale).

But, I only know the data mean and std. dev. I do not know the data set.

Given the mean and std. dev. of a data set from a truncated gamma distribution, how to find shape and scale of the distribution parameters ?

I know MLE may be useful for solving this problem. But, they depend on knowing the whole data set.

Any help would be appreciated.


I have used the Method of moments here http://en.wikipedia.org/wiki/Method_of_moments_%28statistics%29#Example

But, the QQ-plot from the estimated parameters (got by the above method) is very bad.

I think this is because the method of moments do not consider the truncation points.

  • 1
    $\begingroup$ Does this paper help at all? $\endgroup$
    – Peter Flom
    Mar 17, 2014 at 21:27
  • 1
    $\begingroup$ If you know the truncation point, this is possible - e.g. by method of moments. If you don't know the truncation point, it must be estimated, in which case you have three parameters to estimate from only two quantities. So, crucially do you know the truncation point? $\endgroup$
    – Glen_b
    Mar 17, 2014 at 21:33
  • $\begingroup$ @Glen_b, yes, we know the truncation point. Thanks ! $\endgroup$ Mar 17, 2014 at 21:41
  • $\begingroup$ @Glen_b, would you please explain about "method of moments", thanks ! $\endgroup$ Mar 17, 2014 at 21:55

1 Answer 1


Using the method of moments to estimate parameters is simply this:

1) Write the parameters in terms of the first few population moments.

2) equate population moments and sample moments

3) solve for the parameters

the result is parameter estimates.

The justification for step 2 of this approach is the law of large numbers.

There's an example of this approach being used on an (untruncated) gamma density here.

So how do you apply it on your problem?

If you can write the moments of the truncated gamma explicitly in terms of the parameters and the known truncation point, you could attempt to solve the equations for the parameters (via methods for solving nonlinear systems of equations).

Failing that, since for a given set of parameter values (and truncation point) you can always find the corresponding moments, you could just quantify the discrepancy between sample and population moments at any given set of parameter values (say via sum of squares of deviations) and use optimization routines (nonlinear least squares should be possible) to minimize the discrepancy (which should go to zero, if that's possible with the specific data for that truncation point, but in any case would be finding a 'best available' in a least-squares-closest-matching-moment sense). The resulting parameter estimates will be method of moments estimators.

(There are functions in the R packages I mentioned in my answer to your other question can evaluate the mean and variance for a given truncation point for many distributions, including the gamma.)

  • $\begingroup$ the population and sample are unknown. We only know mean and std.dev. , which are two numerical values. And, we know the truncate. We need to find the truncated gamma distribution's parameters from them. Thanks ! $\endgroup$ Mar 18, 2014 at 4:19
  • $\begingroup$ I'm not sure what you're trying to tell me that's not already covered by my answer. I suspect you've misunderstood something about what I wrote. I KNOW you only know the sample mean and standard deviation (plus the truncation point). My entire answer is based on that. If at any point in my answer you think it's saying otherwise, it's either (i) a typographical error, or (ii) you misunderstood what you were reading. What made you think I didn't already understand the information in your comment? $\endgroup$
    – Glen_b
    Mar 18, 2014 at 4:51
  • $\begingroup$ I really appreciate your help. I have tried the method of moment. Please see the updates in my OP. I cannot find the functions in the R packages that you mentioned in your answer to my other question can evaluate the mean and variance for a given truncation point for gamma distributions. Any help would be appreciated ! $\endgroup$ Mar 18, 2014 at 14:04
  • $\begingroup$ Would you please tell me what functions in R they are ? Thanks ! $\endgroup$ Mar 18, 2014 at 14:11

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