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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?

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    $\begingroup$ 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. $\endgroup$ – Xi'an Nov 10 '20 at 13:41

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