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

[Lam, Tzeng Yih]

Dear Dr. Zeileis,

Thank you for pointing out on the maximum likelihood estimator property as well as the delta method to obtain the standard error of estimated Theta.

I agree with you in that whether getting the standard error of estimated Theta is useful or not. I will think about this further.

Best regards,
Tzeng Yih Lam
------------------------------------------------------------------------------
PhD Candidate
Department of Forest Engineering, Resources and Management
College of Forestry
Oregon State University
321 Richardson Hall
Corvallis OR 97330 USA
Phone: +1.541.713.7504
Fax: +1.541.713.7504
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________________________________________
From: Achim Zeileis [Achim.Zeileis at uibk.ac.at]
Sent: Monday, February 15, 2010 2:03 AM
To: Lam, Tzeng Yih
Cc: r-help at r-project.org
Subject: Re: [R] Estimated Standard Error for Theta in zeroinfl()

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
>
> ------------------------------------------------------------------------------
> PhD Candidate
> Department of Forest Engineering, Resources and Management
> College of Forestry
> Oregon State University
> 321 Richardson Hall
> Corvallis OR 97330 USA
> Phone: +1.541.713.7504
> Fax: +1.541.713.7504
> ------------------------------------------------------------------------------
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