Let $D(\theta)$ denote an absolutely continuous distribution on $\mathbb{R}$. (The finite dimensional vector $\theta$ collects the parameters of the distribution.) Assume that the p.d.f. of $D(\theta)$ exists and is positive on the entire real line for all $\theta$.
Let $E$ be some other absolutely continuous distribution on $\mathbb{R}$ with a positive p.d.f.. $E$ may or may not be given by $D(\theta)$ for some $\theta$.
Suppose that for all $\theta_0$ and all $l\in\mathbb{R}$:
If $X$ is a random variable drawn from $D(\theta_0)$, and $Z$ is an independent random variable drawn from $E$ (not a function of $\theta_0$), then there is some $\theta_1$ and $\theta_2$ such that:
- $X+Z$ is distributed as $D(\theta_1)$.
- $X+Z|X>l$ is distributed as $D(\theta_2)$.
What does this imply about the distribution $D$?
In particular: Are there distributions $D$ and $E$ for which (1) and (2) above hold?
A few ideas which do not work (I think!) follow:
- Shifted exponential distributions are closed under conditioning, and their sums are shifted Gamma distributed, but shifted Gamma distributions are not closed under truncation (I think). (These distributions also violate my positivity condition.)
- One-sided stable distributions are closed under linear combinations, and look superficially "truncated", but I do not think they are actually closed under truncation.
- Extended skew normal distributions embed truncated normals in a limit case, and are closed under linear combinations with the same "truncation" parameter. But the requirement that $E$ is not a function of $\theta_0$ would mean that $D(\theta_0)$ and $E$ would have different truncation parameters in general.
