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# RE: st: estimating parameters of Weibull distribution

 From "Nick Cox" To Subject RE: st: estimating parameters of Weibull distribution Date Fri, 21 Feb 2003 21:02:00 -0000

```Roberto G. Gutierrez, StataCorp.
>
> 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-.
>

As a footnote to Bobby's posting:

if you have not yet upgraded to Stata 8,
then -wbull- on SSC does some of this -- but
not the variance-covariance matrix you will want.
Blowing the dust off this ancient code, I see
that it requires Stata 5, so should work on 5, 6 and 7.

The graphics routines -qweibull- and -pweibull-
are of the same age and also are on SSC.

Nick
n.j.cox@durham.ac.uk
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```

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