# Re: st: estimating parameters of Weibull distribution

 From rgutierrez@stata.com (Roberto G. Gutierrez, StataCorp.) To statalist@hsphsun2.harvard.edu Subject Re: st: estimating parameters of Weibull distribution Date Fri, 21 Feb 2003 14:48:23 -0600

```Clint Thompson <cthompson@dfpm.utah.edu> asks:

> I need to estimate both parameters from the Weibull distribution (beta &
> theta) using the maximum likelihood estimation method.  I have referenced
> the reference manual and it assumes that I am using a specific data set with
> a Weibull distribution but I need the estimates using the general Weibull
> pdf.  I also want to estimate the variance-covariance matrix of the MLEs.
> Any suggestions (aside from computing it by hand)?

Very easy in Stata 8.

Clint can fit a Weibull model without covariates on his univariate data and
then use -nlcom- to remap the estimates to those from a more "typical" Weibull
pdf.  For example, if the pdf in question is

f(t) = theta/beta * t^(theta-1) * exp(-t^theta/beta)

and if one fits the Weibull model using the default proportional hazards (PH)
parameterization, then the remapping would take place as follows:

. webuse cancer

. stset studytime  /* studytime is the univariate response, no censoring */

. streg, dist(weibull) nolog nohr

(output omitted)

. nlcom (theta:exp([ln_p]_b[_cons])) (beta:exp(-_b[_cons]))

theta:  exp([ln_p]_b[_cons])
beta:  exp(-_b[_cons])

------------------------------------------------------------------------------
_t |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
theta |   1.518202   .1766894     8.59   0.000     1.171897    1.864506
beta |   74.91051   42.54283     1.76   0.078    -8.471904    158.2929
------------------------------------------------------------------------------

The estimated variance covariance matrix (VCE) of the remapping is returned
in r(V).

. mat list r(V)

symmetric r(V)[2,2]
theta       beta
theta  .03121916
beta  7.2700449  1809.8925

As for a pre-Stata 8 solution, Clint would have to use -testnl- to obtain
the derivative matrix and perform the delta method matrix calculations himself.
A bit more prone to error, hence the motivation for -nlcom-.

--Bobby
rgutierrez@stata.com
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