User:DancesWithChances/sandbox

Note that the a-subscripts for the μ expansion are all odd except for the second, which is a_4, and the a-subscripts for the s expansion are all even except for the second which is a3. These coefficients were chosen so that k=2 yields non-zero mean and variance, but zero skewness and kurtosis, k=3 yields non-zero mean, variance and skewness, but zero kurtosis, and k=4 yields non-zero mean, variance, skewness and kurtosis.[10]: p.252

Note that the $a$-coefficients are numbered such that the first coefficient $a_{1}$ is from the $\mu$ expansion; the second and third coefficients, $a_{2}$and $a_{3}$, are from the $s$ expansion; the fourth and fifth are from the expansion; and they alternate there after.

Note that the subscripts of the $a$-coefficients are such that $a_{1}$ and $a_{4}$ are in the $\mu$ expansion, $a_{2}$ and $a_{3}$ are in the $s$ expansion, and subscripts alternate thereafter. This ordering was chosen so that the first two terms in the resulting metalog quantile function correspond to the logistic distribution exactly; adding a third term with $a_{3}\neq 0$ adjusts skewness; adding a fourth term with $a_{4}\neq 0$ adjusts kurtosis primarily; and adding subsequent non-zero terms yields more nuanced shape refinements.