Understanding the Tweedie Distribution I am trying to better understand the basic details and properties about the Tweedie Probability Distribution.
The Tweedie Probability Distribution is a bimodal probability distribution function which is useful for capturing shapes and patterns in datasets having a high level of "over dispersion" (i.e. many zeros, also called zero-inflated). Here are some examples of the Tweedie Probability Distribution Function:

I am trying to better understand the general form of the Tweedie Probability Distribution Function. In general, a Tweedie Distribution is said to be a sum of an "M" number of Gamma Distributions - where "M" itself is a random variable from a Poisson Distribution. According to Wikipedia, the Tweedie Probability Distribution Function (for a random variable "Y") can be written as follows:

From the above statement, it would appear that the Tweedie Probability Distribution Function is parametrized by "alpha", "p" and "sigma^2" (i.e. since "thetha" is a function of "p" and "sigma^2"). I spent some time trying to understand this, but I still have the following questions:
1) I noticed that there is a "z" term in the Tweedie Probability Distribution Function - I assume that this actually refers to the original random variable "Y", and that replacing "Y" with "z" is just a common tradition when writing integrals. Is this correct? (e.g. "dz" is effectively "dy"?)
2) In the Tweedie Probability Distribution Function, what exactly is "v-lambda"? The Wikipedia page states that "v-lambda" refers to a "sigma finite measure" - but I am not sure what this means. In practice (e.g. when modelling real world data using the Tweedie Distribution), does this "v-lambda" term "drop out" of the Probability Distribution Function? Is writing "v-lambda" simply a mathematical "formality" ? Or does "v-lambda" actually refer to some physical term?
3) If you have some observations (e.g. heights of basketball players) X : X1, X2...XN, and you decide that these observations likely originated from a Tweedie Probability Distribution Function.  Given these observations, how do you calculate "alpha", "p" and "sigma^2" (e.g. are there formulas for the Maximum Likelihood Estimates of these parameters) ? If you assume that X: X1, X2...XN come from a Tweedie Probability Distribution Function - how do you calculate the "mean" ( E(X) ) and the "variance" ( Var(X) ) of X? (I have a feeling that you obviously just can't use the same formulas for mean and variance based on a normal probability distribution)
4) Are there any standard methods to (numerically) integrate the Tweedie Probability Distribution? For example, the main R package for the Tweedie Distribution (https://cran.r-project.org/web/packages/tweedie/tweedie.pdf) uses the "saddle point approximation" method to compute densities :
> library(tweedie)

> power <- 2.5
> mu <- 1
> phi <- 1
> y <- seq(0, 6, length = 500)
> fy <- dtweedie(y = y, power = power, mu = mu, phi = phi)
> plot(y, fy, type = "l", lwd = 2, ylab = "Density")
> 
> f.saddle <- dtweedie.saddle( y = y, power = power, mu = mu, phi = phi)
> lines( y, f.saddle, col = 2 )
> legend("topright", col = c(1, 2), lwd = c(2, 1),
       legend = c("Actual", "Saddlepoint") )
> 
> hist( rtweedie( 1000, power = 1.2, mu = 1, phi = 1) )


In practice, could MCMC be used to evaluate samples from a Tweedie Probability Distribution Function?
Can someone please help me clarify these points?
Thanks!
References:

*

*https://en.wikipedia.org/wiki/Tweedie_distribution

*https://converged.yt/papers/dsm-paper-AppendixD.pdf
Note: I have also seen the Tweedie Probability Distribution expressed as a sum of infinite series:

And this sum of the infinite series itself is approximated using Stirling's Formula and Fourier Inversion (e.g. https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.449.2161&rep=rep1&type=pdf).
 A: *

*The $z$ in the probability function is just an arbitrary integration variable. There is no standard or convention about this, it is just arbitrary notation. You could just as well change the $z$ to $y$ with no change of meaning. The author of the Wikipedia page seems more concerned with mathematical formality than with clarity unfortunately.


*The notation in the Wikipedia article really is terrible. The $\lambda$ parameter isn't even defined. Even if the notation was properly defined, the use $\sigma$-algebras for this type of article is totally unnecessary and unhelpful. Anyway, $\nu_\lambda()$ is just a mathematical formality to make sure the density integrates to 1 over the whole distribution. In my opinion, you would be better off ignoring the Wikipedia page and read instead the much better descriptions of the Tweedie distributions in the cited papers.


*It is much better to estimate $\mu$ instead of $\alpha$. See the Dunn & Smyth (2018) book for detailed advice and examples. The power parameter $p$ is often pre-specified, in which case estimating $\mu$ and $\sigma^2$ becomes relatively easy. See also Tweedie Dispersion Parameter Estimation Methods.


*No there aren't. If there were good methods they would be implemented in the tweedie package. Type help("dtweedie") and consult the references cited to understand the challenges.


*It is difficult to imagine how MCMC could possibly be of any help to you. MCMC is not a tool for evaluating goodness of it, if that is what you have in mind.
