Curved exponential family

A $(p,q)$ curved exponential family, with $q<p$, restricts a $p$-parameter canonical family to a smooth rank-$q$ parameter surface, usually a nonaffine surface. For example, $N(\mu ,{\mu }^2)$ with $\mu >0$ has sufficient statistics $(Y,Y^2)$ and natural parameters $(1/\mu ,-1/(2{\mu }^2))$, constrained by ${\eta }_2=-{\eta }_1^2/2$. Canonical concavity in the ambient parameters need not imply concavity along the curved parameter surface.

This one-parameter normal distribution model has positive mean equal to its standard deviation. It is a curved exponential family: both the linear and quadratic natural parameters depend on the same scalar statistical parameter. It combines information about the center and spread, so the sample mean alone does not use all available Fisher information.

For a statistical sample of size $n$, let $A={\sum }_i{x}_i$ and $B={\sum }_i{x}_i^2$. Up to an additive constant the log-likelihood function is $\ell (\mu )=-n\log \mu -B/(2{\mu }^2)+A/\mu$. Its score function is $(B-A\mu -n{\mu }^2)/{\mu }^3$. When $B>0$, the numerator changes from positive to negative at its unique positive root, so the global maximum-likelihood estimator isThe all-zero observation vector has zero probability and is an exceptional case with an unbounded likelihood function near zero. Since $\mathbb{E}A=n\mu$ and $\mathbb{E}B=2n{\mu }^2$, the expected Fisher information is $I_n(\mu )=3n/{\mu }^2$. Thus regular asymptotic normality of a maximum likelihood estimator gives $\sqrt{n}(\hat{\mu }-\mu )\stackrel{d}{\to }N(0,{\mu }^2/3)$. The optimal asymptotic combination of a normal sample mean and standard deviation attains the same first-order precision.

For independent $N(\mu ,{\mu }^2)$ observations with $\mu >0$, the sufficient statistics are their sum and squared norm. Condition on $a=\sqrt{n}\Vert y\Vert /{\sum }_i{y}_i>0$ and write $v=\hat{\mu }=\Vert y\Vert /(q\sqrt{n})$, where $q=(\sqrt{1+4a^2}+1)/(2a)$. Polar coordinates supply radial measure $\Vert y{\Vert }^{n-1}d\Vert y\Vert$, while the conditional angular factor does not depend on $\mu$. This proves the displayed shape. Its normalization depends on $a,n$ but not $\mu$ after the substitution $v=\mu z$, and agrees with the normalized P-star approximation.