= Inverse Gaussian mean estimation
{c}
{title2=$\widehat\mu=\bar Y,\quad I_1(\mu)=\lambda/\mu^3$}
For an independent <statistical sample> from an <Inverse Gaussian distribution> with mean $\mu>0$ and known shape $\lambda>0$, the <maximum-likelihood estimator> of $\mu$ is the <sample mean>. To prove this, expand the <log-likelihood function> to get $\ell(\mu)=C-\lambda\sum_i y_i/(2\mu^2)+n\lambda/\mu$. Its <score function> is $\lambda(\sum_i y_i-n\mu)/\mu^3$, positive below $\bar y$ and negative above it. Taking the negative expected second derivative gives one-observation <Fisher information> $I_1(\mu)=\lambda/\mu^3$. Thus the regular <asymptotic normality of a maximum likelihood estimator> yields $\sqrt n(\bar Y-\mu)\xrightarrow d N(0,\mu^3/\lambda)$. The same conclusion follows from the <central limit theorem> and the distribution's <variance> $\mu^3/\lambda$.
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