Best way to test that one mean is greater than all the others? Suppose I have $k$ samples of different sizes, each of a different univariate variable. I want to test the significance of the hypothesis that the population of $k_0$ has a mean greater than all of the other means.
I know of one-way ANOVA, but my understanding is that it only checks for significant differences across the groups.
Is there a statistical test that establishes significance for which mean is greatest? Feel free to give Bayesian or Frequentist solutions (preferably both).
Here is my example:
I have to compare the heights of 5 different people; however, my tape measure operator has a very unsteady hand, so there is significant variance on measurements. I can measure as many times as I like. Obviously I do not know beforehand which person is tallest, but I would like to give a confidence that a particular person is tallest.
Would I perform 5 one-versus-all two-sample $t$-tests, and quintuple their $p$-values?
What would be the Bayesian approach?
 A: 
Would I perform 5 one-versus-all two-sample t-tests, and quintuple their p-values?

Intuitively, I would think that it's less powerful than a permutation test. I hope you will receive others answer to this question as I would like to know what the alternatives are.
My suggestion is to compute how unlikely it is that the group with the biggest mean is that far or farther from the group with the second biggest mean. To measure how unlikely it is, you do a random permutation mixing which individual is in which group. At each permutation you look if the difference between the two biggest mean is bigger or equal to the two biggest mean you currently have. If it's a very rare event, you prove your point that the apparent gap between the biggest mean and the others cannot be caused by chance. 
In more algorithmic way :
1-Compute $d=mean(k_0)-max(mean(k_1),mean(k_2)...mean(k_n))$
2-Permute the label indicating which points belongs to which group.
For each of these permuted dataset :
3-Compute $d'$ with taking care to pick as $k_{0}$ the group which has the biggest mean (because you mentioned that $k_{0}$ is selected after the experiment).   
4- Count the ratio of times when $d'\geqslant d$, this is your p-value.
