[R] Estimated Standard Error for Theta in zeroinfl()

[Achim Zeileis]

On Sun, 14 Feb 2010, Lam, Tzeng Yih wrote:

> Dear R Users,
>
> When using zeroinfl() function to fit a Zero-Inflated Negative Binomial 
> (ZINB) model to a dataset, the summary() gives an estimate of log(theta) 
> and its standard error, z-value and Pr(>|z|) for the count component. 
> Additionally, it also provided an estimate of Theta, which I believe is 
> the exp(estimate of log(theta)).

As maximum likelihood estimation is employed, this does not matter for 
point estimation. theta is the ML estimator for theta and log(theta) is 
the ML estimator for log(theta). I don't think that there is an 
unibiasedness result for either one, but both are consistent (if the model 
is correctly specified).

What is done internally in zeroinfl() is that log(theta) is employed which 
is a standard approach for numeric optimization of positive parameters.

> However, if I would like to have an standard error of Theta itself (not 
> the SE.logtheta), how would I obtain or calculate that standard error?

You can do so by means of the delta method which is rather simple in this 
case: The standard error of theta is: theta * SE(logtheta). Thus, if obj 
is a fitted "zeroinfl" object:
   ## theta
   obj$theta
   ## associated standard error
   obj$theta * obj$SE.logtheta

Whether this is very useful is another story, of course...see also Rolf's 
remarks.
Z

> Thank you very much for your time.
>
> Best regards,
> Tzeng Yih Lam
>
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