Suppose $X_1,\ldots,X_n$ are each unimodal random variables with mode around zero. In particular, for each $i$, $X_i=X_i'-X'$ where $X'_i$ and $X'$ are i.i.d. unimodal random ($X_i'$ and $X_j'$ are ind. as well) variables so that $X_i$ is indeed unimodal. Is it the case that joint distribution of $X_i$ has the form, say $F(\cdot)$ satisfies the following:
Given $x_2,\ldots,x_n$ $F(u,x_2,\ldots,x_n)$ is convex for $u\in (-\infty,a)$ and concave for $u \in (a,\infty)$. In particular, the conditional (on truncating the other variables) distribution is unimodal. If so, how would one go about proving such a claim?
Thanks in advance