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.