Introduction. Assume that two populations $A$ and $B$ are distributed with normal distributions $N(\mu_A, \sigma_A^2)$ and $N(\mu_B, \sigma_B^2)$.
This is a general problem, but as an example, I will consider height distributions of men and women. Assume that the heights of men and women are normally distributed with the following means and standard deviations:
$$\begin{align} & \textbf{Mean} & \textbf{St.d.}\\ \textbf{Female}\quad & 161\,\text{cm} & 7.1\,\text{cm}\\ \textbf{Male}\quad& 175\, \text{cm} & 8.5\,\text{cm} \end{align}$$
Importantly, I don't know these parameter values, but I wish to infer/estimate them (Read on). Instead, what I have is a table of the proportion of men and women exceeding some (unknown) threshold. For example, I have manually created some thresholds. The following table contains the proportion of women and men, respectively, who exceed a certain threshold of height:
$$\begin{align} \textbf{Threshold} &\quad \textbf{Female} & \textbf{Male}\\ \geq158\,\text{cm} &\quad0.6637 & 0.9772\\ \geq 162\,\text{cm} & \quad 0.4440 & 0.9369\\ \geq 166\,\text{cm} & \quad 0.2406 & 0.8552 \\ \geq 170\,\text{cm} & \quad 0.1025 & 0.7218\\ \geq 174\,\text{cm} & \quad 0.0336 & 0.5468\\ \geq 178\,\text{cm} & \quad 0.0083 & 0.3621\end{align}$$
I only have the actual proportion values (0.6636, 0.9772, ... etc) but I don't have the threshold values (158 cm, 162 cm, etc) as they are unknown.
Question. Given such a threshold table, how do I find the group mean difference (in SD units) and variance ratio? (Assuming that they are normally distributed)
Results. Given I have manually constructed the data, I can say what the results should approximately be. If we standardize the female distribution, then the mean difference is
$$\Delta := \frac{\mu_M-\mu_F}{\sigma_F} = \frac{175-161}{7.1} \approx 1.972 \quad\text{female standard deviation units} $$
The variance ratio is $$\rho^2 := \frac{\sigma_M^2}{\sigma_F^2} = \frac{8.5^2}{7.1^2} \approx 1.433$$
Which are the two values that I'm looking for (mean difference and variance ratio).
Thoughts. My thinking is to first reduce the problem from four unknown variables ($\mu_A$, $\mu_B$, $\sigma_A^2$, $\sigma_B^2$) to a problem of two unknowns. This can be done by standardizing one group to be distributed with mean 0 and variance 1. Then I am simply looking for the mean difference (say, $\Delta$) and the variance ratio.
Then I probably have to calculate some density function or cumulative density function for the difference, and make a least squares optimization problem with the table (There can be noise in the data, hence I am thinking a least squares minimization). But I'm unsure exactly how.