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Let $f(t)$ be the density of a non-negative, absolutely continuous random variable $T$. While $f(t)$ has no closed form representation, suppose its Laplace transform $$ \int_0^\infty e^{-st}f(t)\,\text{d}t = F_T(s)$$ does have one, and is continuously differentiable. Given $z \in \mathbb{R}$ and $z = n - \alpha$ with $\alpha > 0$, this potentially allows to calculate moments in closed form via \begin{equation} \label{eq:rkl} \mathbb{E}[T^z] = \frac{(-1)^n}{\Gamma(\alpha)} \int_0^\infty F_T^{(n)}(s) s^{\alpha-1}\,\text{d}s. \end{equation} My questions:

  1. Suppose $\phi:\mathbb{R}_+ \to \mathbb{R}_+$ is smooth, strictly increasing, and strictly concave. Given the knowledge above, what, if anything, can I infer about $\mathbb{E}[\phi(T^z)]$? I was hoping for an analog of the LOTUS in Laplace space.
  2. Is the case $\phi = W_0$ of the Lambert function's principal branch special by any chance?
  3. Under which circumstances, if any, does knowledge about $F_T(s)$ help to find $F_{\phi(T)}(s)$?
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    $\begingroup$ TIL people use LOTUS for the Law of the Unconscious Statistician. $\endgroup$ Jun 8, 2021 at 0:21
  • $\begingroup$ Thanks for the clarification. What does "TIL" stand for? $\endgroup$
    – bodhi
    Jun 8, 2021 at 12:17
  • $\begingroup$ "Today I learned," from the subreddit of the same name. Regarding your question, I don't know if (1) has a dual - i.e., inference about expectation of a well-behaved function. $\endgroup$ Jun 8, 2021 at 15:50

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