Let $X_1$ and $X_2$ be two i.i.d. Uniform variate on $[\alpha,\beta]$. Does there exist a sufficient statistic for $X_1+X_2$?
Let us divide the problem in two parts. First part obviously consists to find the distribution that $X_1+X_2$ follows. To serve this purpose, let us take $x_1+x_2=u$ and $x_1-x_2=v$. Skipping some multiple integration steps and finding the distribution for $u$, we have the corresponding distribution function as below : $$g(u)=\begin{cases}\frac{u-2\alpha}{(\beta-\alpha)^2} \ \ \ \text{if} \ \ \ 2\alpha<u<\alpha+\beta \\ \frac{2\beta-u}{(\beta-\alpha)^2} \ \ \ \text{if} \ \ \ \alpha+\beta<u<2\beta \end{cases}$$ This completes our first part, i.e. finding distribution of $X_1+X_2=U$. Now since this is a step function, I don't know whether any method of finding sufficient statistic for step functions exists, or to check whether the sufficient statistic even exists.
What will be the approach to check for the sufficient statistic for $X_1+X_2$ afterwards? Any help is appreciated.