Find parameters of Generalized Inverse Gaussian Distribution

Question

I have a vector of numbers and I am trying to fit the data by Generalized Inverse Gaussian Distribution. My goal is to estimate the parameters $ a,b,p $ which appears in the pdf function. As in the above wiki page, we know that the pdf function, called $f$, satisfies the following equation: $$ f(x)(x(ax-2p+2) - b)+2x^2f'(x)=0 $$ My questions are:

1. Can I use this equation to estimate the parameters $a,b,p$?

2. Does there exist other ways to approximate the parameters more accurately?

Answer 1

If you integrate out the differential equation $$\int_0^\infty \{f(x)[x(ax-2p+2) - b]+2x^2f'(x)\}\text{d}x=0$$ you get a moment equation $$\begin{align*}&\int_0^\infty [x(ax-2p+2) - b]f(x)\text{d}x+\int_0^\infty 2x^2f'(x)\}\text{d}x\\ &=\int_0^\infty [x(ax-2p+2) - b]f(x)\text{d}x-4\int_0^\infty xf(x)\}\text{d}x\\ &=\int_0^\infty [x(ax-2p+2-4) - b]f(x)\text{d} =0\end{align*}$$ which can be used in a system of moment equations, but you need two further equations.