Let $Y_r:={\mathrm{\Phi }}^{-1}(d_r)$ for brevity. The original equations (5.7)–(5.9) then become

Rearranging (4) gives

Taking the square of (4) and subtracting that from (5) and then rearranging gives us

Here, we have introduced the variance, defined as

Note that (9) might result in if ${\rho }_r>0$, which is undesirable since ${\gamma }_r$ was defined as a variance when the original system of equations was set up. Negative values of ${\rho }_r$ could therefore be considered, which would require an extensive modification of (1)–(3), using $\sqrt{-{\rho }_r}$ and changing signs in several places.

Taking the third power of (4) and subtracting this from (6) gives

Inserting (7) and (9) into the inner square brackets on the right-hand side yields, after rearranging, an expression that is solvable for ${\rho }_r$ alone:

Here, we have defined

which is the nonnormalized skewness (to obtain normalized skewness, multiply $?[X]$ by $?{[X]}^{-3/2}$).

After evaluating ${\rho }_r$ from (10), use (9) and (7) to obtain values for ${\gamma }_r$ and ${\mu }_r$, respectively.

Note that if the distribution of $\widehat{Z}$ is symmetrical, ie, $?[\widehat{Z}]=0$, then (10) has no solution if $?[Y_r]\ne 0$. There is no unique solution if both $?[\widehat{Z}]=0$ and $?[Y_r]=0$. For the limiting case ${\rho }_r=1$, the whole system of equations (4)–(6) would be invalid.