= Three-cumulant shifted gamma approximation
{title2=$\nu=2m_2/m_3,\quad\alpha=4\lambda m_2^3/m_3^2,\quad k=\lambda m_1-2\lambda m_2^2/m_3$}
Approximate a <compound Poisson distribution> of positive claims by a <shifted gamma distribution> whose first three <cumulants> agree. Here $m_j=\mathbb E X^j$ are raw severity <moments>, while the compound <cumulants> are $\lambda m_j$. Solve $k+\alpha/\nu=\lambda m_1$, $\alpha/\nu^2=\lambda m_2$, and $2\alpha/\nu^3=\lambda m_3$ to obtain the displayed parameters. The approximation captures positive <skewness>, but not the atom $e^{-\lambda}$ at zero. Its shift can be negative; with exponential claims of mean $\mu$ it is $-\lambda\mu/3$. Matching three <cumulants> alone supplies no general tail-error bound.
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