Problem:

You are given a sample of size $m$ from $n$ independent normally distributed random variables. Expectations and standard deviations of the random variables are unknown. Estimate, which random variable has the largest expectation.

Discussion:

The interesting case is when $n > 2$ and the expectations of the random variables are close to each other compared to the standard deviations divided by $\sqrt{m}$. In this case just comparing the sample means may not give the best results. For example, with 3 random variables $X_1, X_2, X_3$, with $X_i \sim N(\mu_i, \sigma_i^2)$, with $$ \mu_1 = 0, \; \mu_2, \mu_3 \in [-1, 1] $$ $$ \sigma_1 = \sqrt{m}, \; \sigma_2, \sigma_3 = 10 \sqrt{m} $$ there is only a roughly $1/4$ chance for $X_1$ to have the largest sample mean, whether any of $\mu_2, \mu_3$ are positive or not. For a good test I would expect this probability to be around $1/3$. Hence the question.

Thanks.