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I am trying to better understand how model parameters are estimated in Dispersion Models. In general, Dispersion Models can be stated as follows:

enter image description here

A specific example of a probability distribution belonging to the family of Dispersion Models is the Tweedie Distribution. The Tweedie Distribution is defined below, with model parameters "lambda", "alpha" and "beta":

enter image description here

Apparently, "the parameters ϕ and p are estimated by numerically maximizing the profile likelihood".

My Questions:

  • Are there any references which provide the full Maximum Likelihood Equations for "lambda-hat", "alpha-hat" and "beta-hat"? As I understand, these Maximum Likelihood Equations are then solved using numerical methods (i.e the final value of "lambda-hat" is selected by optimizing the the Maximum Likelihood Equations using numerical methods).

  • In the above example (for the Tweedie Distribution), the random variable "Y" is of interest. I have been trying to find a reference that states the equations needed to calculate the "expected value of Y" (i.e. E(Y) ) and the "variance of Y" (i.e. Var(Y) ). Does anyone know of any references where they clearly demonstrate how to set up the equations for determining the "mean of Z" and the "variance of Y"?

In the above picture, the mean of of "Y" is E(Y) = mu = k'(thetha) and the variance of "Y" is var(Y) = k''(thetha)*phi. I have seen the function "k" referred to as the "Cumulant Function" and expressed as follows (for the Tweedie Distribution):

enter image description here

Thus, the first derivative and the second derivative of the "k" function appear to be required for calculating the mean and variance of "Y". But then it would also appear that these derivatives require the Maximum Likelihood Estimates of the original model parameters "alpha" and "beta" - thus, the mean and variance of "Y" can not be calculated until all the model parameters of the Tweedie Distribution are first estimated . However, it still remains unclear as to how "thetha" should be defined in the "k(thetha)" equation : should "thetha" be set to 0 in the "k(tetha)" equation as apparently this is customary in the cumulant function? Or does "thetha" itself also need to be estimated from the data using MLE/numerical methods?

Is this correct?

Thanks!

References:

Note:

I tried to manually evaluate the first derivative of "k(thetha)", e.g. for when p "not equal" to 1,2 :

k'(thetha) = ((alpha-1)/alpha) * (1/(alpha_1))^alpha * alpha * thetha ^ (alpha - 1).

If you set "thetha = 0", then k'(thetha) = 0, implying that the expected mean of "Y" for any value of "alpha" (estimated from MLE and numerical methods) will always be 0 ... and I don't think that this is correct.

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