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# Re: st: Estimating the parameters of the GB2 distribution with incomplete knowledge of sample statistics

 From Austin Nichols To statalist@hsphsun2.harvard.edu Subject Re: st: Estimating the parameters of the GB2 distribution with incomplete knowledge of sample statistics Date Sun, 26 Sep 2010 22:25:57 -0400

```Julio Estevez <je_123@hotmail.com>:
-gbgfit- on SSC will give the MLE based only on the quantiles; you can
write the MLE incorporating knowledge of the mean.
Or you can use -nl- or Mata's optimize() of which there are examples
in the archives.

On Sun, Sep 26, 2010 at 4:32 PM, Julio Estevez <je_123@hotmail.com> wrote:
> Hi
> I am interested in deducing the parameters of a GB2 distribution given that I know just a couple of  quantiles and sample statistics of the population I am interested in.
>
> I know the mean  of the population of interest. I also know some key quantiles from the left side of the distribution. That is  I know x1, x2, x3  such that F(x1) = 0.1, F(x2) = 0.2, F(x3) = 0.3 been F() the c.d.f. Although I do not know much about the upper tail, from previous research I beleive that the GB2 distribution may be good approximation of the distribution I am looking at.
>
> I beleive I  can use the fact that  GB2 distribution has distribution function (c.d.f.)  F(x) = ibeta(p, q, (x/b)^a/(1+(x/b)^a) ) to estimate the parameters of the distribution function (a,b,p,q)  given my knowledge about the mean and the first qualtiles. Then I can explore the upper tail under the assumption that my data is distributed as a GB2
>
> Can somebody help me to  set up the program to estimate these parameters?  I believe it is "just" a matter of solving a set of 4 nonlinear equations with 4 unknowns, but my programming skills are not that good.
>
> The four equations will look like
> mean = b*G(p+1/a)*G(q-1/a)/[G(p)G(q)]F(x1) = 0.1 = ibeta(p, q, (x1/b)^a/(1+(x1/b)^a)F(x2) = 0.2 = ibeta(p, q, (x2/b)^a/(1+(x2/b)^a)F(x3) = 0.3 = ibeta(p, q, (x3/b)^a/(1+(x3/b)^a)
> from which the unknowns, a,b, p and q can be retrieved.
> with G(.) been the gamma function
>
> Any help will be greatly appreciated!
> Thanks
>
> Julio
>
> *

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