I have the following question, any help is greatly appreciated. Thanks!
Let $Q(y)=E[X\mid X>y]$, the truncated expected value of a given distribution.
Is it possible to characterize the family of distributions where $\dfrac{dQ}{dy}\geq1$?
I know that $$\frac{dQ}{dy}=h(y)(Q(y)-y)$$ where $h(y)$ is the hazard rate at $y$. But I do not know how to proceed from here.
Thanks!
For simplicity, I am going to assume that you have a continuous differentiable survival function, so that the standard rules of survival analysis apply. To characterise this rule it is best to translate the rule so that it is expressed solely in terms of the hazard function $h$ for the distributional family - this will give a partial solution to the problem.
The conditional-survival function can be written in terms of the hazard rate as:
$$R(x | y) \equiv \mathbb{P}(X > x | X > y) = \exp \Bigg( - \int \limits_y^x h(r) dr \Bigg) \quad \quad \text{for all }x \geqslant y.$$
So your conditional expectation can be written as:
$$Q(y) \equiv \mathbb{E}(X|X > y) = y + \int \limits_y^\infty R(x |y) dx = y + \int \limits_y^\infty \exp \Bigg( - \int \limits_y^x h(r) dr \Bigg) dx. $$
Differentiating this expression and applying Leibniz integral rule yields:
$$\begin{equation} \begin{aligned} \frac{dQ}{dy}(y) &= 1 - \exp \Bigg( - \int \limits_y^y h(r) dr \Bigg) + \int \limits_y^\infty \frac{\partial}{\partial y} \exp \Bigg( - \int \limits_y^x h(r) dr \Bigg) dx \\[8pt] &= 1 - \exp(0) - \int \limits_y^\infty \Bigg( \frac{\partial}{\partial y} \int \limits_y^x h(r) dr \Bigg) \exp \Bigg( - \int \limits_y^x h(r) dr \Bigg) dx \\[8pt] &= 1 - 1 + \int \limits_y^\infty h(y) \exp \Bigg( - \int \limits_y^x h(r) dr \Bigg) dx \\[8pt] &= h(y) \int \limits_y^\infty \exp \Bigg( - \int \limits_y^x h(r) dr \Bigg) dx. \\[8pt] \end{aligned} \end{equation}$$
Hence, your condition can therefore be written as:
$$\int \limits_y^\infty \exp \Bigg( - \int \limits_y^x h(r) dr \Bigg) dx \geqslant \frac{1}{h(y)}.$$
This gives you a rule that you can use to characterise the required hazard function, which then characterises the allowable distributional families. I have been unable to make any further progress, and honestly, I'm not sure if it is actually possible for any hazard function to obey this rule for all $y$. So although I don't have a full solution, this constitutes a characterisation where we have got things down to a single functional object representing the underlying distribution.