# Finding the Method of Moments in two Parameters

Let $$X_1, X_2, \ldots, X_n$$ be a random sample of size $$n$$ from the following distribution $$f(x;\mu, \lambda) = \sqrt{\frac{\lambda}{2\pi x^3}}e^{\frac{-\lambda(x - \mu)^2}{2\mu^2 x}}$$

where $$x, \lambda, \mu > 0$$. Find the method of moments estimator of $$(\mu,\lambda)$$.

Answer: I tried calculating $$E[X]$$, but this is extremely complicated to solve, unless I'm missing something.

I've tried comparing the given function to the inverse Gamma PDF: $$f(x;\mu, \lambda) = \frac{\lambda^{\mu}}{\Gamma(\mu)}\Big(\frac{1}{x}\Big)^{\mu + 1}e^{-\frac{\lambda}{x}}$$

Letting $$\mu = \frac{1}{2}$$ here produces $$f\Big(x;\frac{1}{2},\lambda\Big) = \sqrt{\frac{\lambda}{\pi x^3}}e^{-\frac{\lambda}{x}} *$$

However, letting $$\mu = \frac{1}{2}$$ in the given distribution function does not yield the same function as $$*$$.

Any suggestions?

• This distribution is called an inverse Gaussian distribution. It is furthermore an exponential family, including $x$ as one of the natural statistics, therefore allows for a direct derivation of the MGF and hence of the moments. – Xi'an Nov 10 '20 at 13:41