Estimating $|\mu_1 - \mu_2|$ in a mixture of two Gaussians with known variance. Consider $n$ samples denoted with $X^n = X_1, \ldots, X_n$ from mixture of two Gaussians with equal variance and equal mixture proportions:
$X^n \sim 1/2 \mathcal{N}(\mu_1, 1) + 1/2 \mathcal{N}(\mu_2, 1)$.
Now the goal is to estimate the difference in means $|\mu_1 - \mu_2|$ from this $n$ samples. One idea is to rank them from lowest to highest, and take the difference between the 1/4th highest and lowest sample point. But how accurate is this estimate, and is there a better way?
 A: The first two moments of this mixture are:
$$\mathbb{E}[X] = \frac{\mu_1+\mu_2}{2}$$
$$\mathbb{E}[X^2] = \frac{\mu_1^2+\mu_2^2}{2}+1$$
Substituting the true expectation by their respective means:
$$S_1=\frac{1}{n}\sum_{k=1}^nX_k$$
$$S_2=\frac{1}{n}\sum_{k=1}^nX_k^2$$
We can transform the system of equations:
$$
\begin{cases}
S_1 = \frac{\mu_1+\mu_2}{2} \\
S_2 = \frac{\mu_1^2+\mu_2^2}{2}+1
\end{cases}
$$
To get:
$$
\begin{cases}
\mu_1+\mu_2 = 2S_1 \\
\mu_1^2+\mu_2^2 = 2S_2 - 1
\end{cases}
$$
Now, knowing that $(x+y)^2+(x-y)^2=2(x^2+y^2)$ or equivalently $(x-y)^2=2(x^2+y^2) - (x+y)^2$ we can obtain the value of:
$$(\mu_1-\mu_2)^2 = 4(S_2-S_1^2-1)$$
And so:
$$|\mu_1-\mu_2| = 2\sqrt{S_2-S_1^2-1}$$
This estimator is at least consistent. Moreover, both $S_1$ and $S_2$ will be asymptotically normal. They are definitely not independent though and I don't even know about their covariance (though I suspect that their asymptotic distribution is jointly normal so the covariance is the only factor deciding their dependence). It is therefore not easy (at least not for me) to compute the bounds for the error of this estimator other than the MC simulation.
