Tweedie generalized linear models assume a mean-variance relationship with variance power $p$, defined by
$$E(y_i)=\mu_i$$
and
$${\rm var}(y_i)=\phi \mu_i^p$$
where $y_i$ is the $i$th observation, $\mu_i$ is the expected value, $\phi$ is the dispersion and $p$ is the mean-variance power parameter, also called the Tweedie index parameter.
The purpose of the tweedie.profile
function is to estimate $p$ by profile maximum likelihood.
The dispersion $\phi$ is a nuisance parameter in the profile likelihood, as are the glm regression coefficients.
It is a requirement of the profile likelihood method in statistics that all nuisance parameters be set to their maximum likelihood estimates.
It would be invalid to use the Pearson estimator for $\phi$ is this context because then the resulting likelihood for $p$ would not be a profile likelihood and differences in log-likelihoods would not have their usual meaning.
We recommend that you use tweedie.profile
to guide your choice of the Tweedie index parameter. Then you can fit a generalized linear model treating $p$ as a known parameter, for example by:
library(statmod)
fit <- glm(y~x, family=tweedie(var.power=p, link.power=0))
summary(fit)
In the above code, the summary
function in R will by default use the Pearson estimator for $\phi$.
The use of the Pearson estimator is not specific to Tweedie distributions, but is a generic property of the summary.glm
function in R.
The fact that tweedie.profile
and summary.glm
use different estimators for $\phi$ does not cause any problems for the process.
As hinted at in Section 6.8 of our book (Dunn & Smyth, 2018), I would ideally prefer to use modified profile likelihood for both $p$ and $\phi$, but that is difficult to implement. The above process is much simpler and will still give good results.
I see that I answered a related question from another poster 3 years ago:
GLM Tweedie dispersion parameter.
Reference
Dunn PK, Smyth GK (2018). Generalized linear models with examples in R. Springer, New York, NY.
tweedie
package probably does not assume that $p$ is given. If $p$ is given/fixed then you might use glm instead (with a custom family function). $\endgroup$