10. Suppose that $f(x, \theta)$ is a positive density on the real line, which is continuous in $x$ for each $\theta$ and such that if ($X_1, X_2$) is a sample of size 2 from $f(\cdot, \theta)$, then $X_1 + X_2$ is sufficient for $\theta$. Show that $f(\cdot, \theta)$ corresponds to a one-parameter exponential family of distributions with $T(x) = x$.
Hint: There exist functions $g(t, \theta)$, $h(x_1, x_2)$ such that $\log f(x_1, \theta) + \log f(x_2, \theta) = g(x_1 + x_2, \theta) + h(x_1, x_2)$. Fix $\theta_0$ and let $r(x, \theta) = \log f(x, \theta) - \log f(x, \theta_0)$, $q(x, \theta) = g(x, \theta) - g(x, \theta_0)$. Then, $q(x_1 + x_2, \theta) = r(x_1, \theta) + r(x_2, \theta)$, and hence, $[r(x_1, \theta) - r(0, \theta)] + [r(x_2, \theta) - r(0, \theta)] = r(x_1 + x_2, \theta) - r(0, \theta)$.
