[R-sig-ME] question on unit variance function

[HATICE T KUBRA AKDUR]

Dear Group Members,

I hope you are keeping well during this difficult time.
Thank you in advance.
I have some questions below:

In the beta regression model, it is stated V(\mu)= \mu(1-\mu).

It is known that even though beta distribution is in the exponential
family, logit link is not canonical link for beta regression.

And, V(\mu)= \mu(1-\mu) is unit variance function for binomial
distribution, logit link is canonical for binomial distribution.

 I am not sure,  V(mu)=mu(1-mu), is it still unit variance function for
beta regression model with logit link.  Even though, it is not in natural
exponential family. OR we can say that V(\mu)= \mu(1-\mu) is variance
function for beta regression with logit link.

1/g'(mu) (g(mu) is link function) can be used to obtain unit variance
function for not natural exponential family distribution? or it is only
true for natural exponential family?

For example, unit variance function of Simplex distribution is
V(mu)=mu_3(1-mu)_3
because simplex distribution is in dispersion family. So, to obtain unit
variance function, the second order derivative of the deviance function is
used.

I will be very glad, if you enlighten me.

Best,

References:

https://onlinelibrary.wiley.com/doi/abs/10.1111/j.0006-341X.2000.00496.x

https://www.tandfonline.com/doi/pdf/10.1080/0266476042000214501

On Thu, Apr 30, 2020 at 11:10 AM HATICE T KUBRA AKDUR <hkubrasenol using gmail.com>
wrote:

> Dear Group Members,
>
> I hope you are keeping well during this difficult time.
> Thank you in advance.
> I have some questions below:
>
> In the beta regression model, it is stated V(\mu)= \mu(1-\mu).
>
> It is known that even though beta distribution is in the exponential
> family, logit link is not canonical link for beta regression.
>
> And, V(\mu)= \mu(1-\mu) is unit variance function for binomial
> distribution, logit link is canonical for binomial distribution.
>
>  I am not sure,  V(mu)=mu(1-mu), is it still unit variance function for
> beta regression model with logit link.  Even though, it is not in natural
> exponential family. OR we can say that V(\mu)= \mu(1-\mu) is variance
> function for beta regression with logit link.
>
> 1/g'(mu) (g(mu) is link function) can be used to obtain unit variance
> function for not natural exponential family distribution? or it is only
> true for natural exponential family?
>
> For example, unit variance function of Simplex distribution is V(mu)=mu_3(1-mu)_3
> because simplex distribution is in dispersion family. So, to obtain unit
> variance function, the second order derivative of the deviance function
> is used.
>
> I will be very glad, if you enlighten me.
> I attached beta regression and marginal simplex model papers FYI.
>
> Best Regards,
> --
> Asst. Prof. Dr. Hatice T. Kubra AKDUR
> Department of Statistics, Faculty of Science
> Gazi University
> 06500 Teknikokullar ANKARA, TURKEY
> Phone: +90 553 324 5380
> Email: hatice_senol using wsu.edu
> Homepage: http://websitem.gazi.edu.tr/site/haticesenol
>


-- 
Asst. Prof. Dr. Hatice T. Kubra AKDUR
Department of Statistics, Faculty of Science
Gazi University
06500 Teknikokullar ANKARA, TURKEY
Phone: +90 553 324 5380
Email: hatice_senol using wsu.edu
Homepage: http://websitem.gazi.edu.tr/site/haticesenol

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