TITLE: Probability of one sample mean being maximal among a set of other sample means
QUESTION [5 upvotes]: Let $\mathcal{D}$ be a distribution, and consider a finite set of sample means $x_i$ of $k_i$ draws from $\mathcal{D}$ ($k_i$ can be different for each mean). 
I want to show that if $x_i$ and $x_j$ have $k_i\leq k_j$ draws, $\mathbb{P}(x_i =\max_\ell x_\ell)\geq \mathbb{P}(x_j =\max_\ell x_\ell)$. I also suspect that if $\mathcal{D}$ meets some kind of regularity condition, then if the first inequality is strict, so is the second.
I know that sample variance decreases as $\dfrac{var(\mathcal{D})}{k_i}$, but I'm not sure how to prove the full result.
EDIT: Not true in general. Is it true for the uniform distribution on $[0,1]$? If so, why?
Here's the distribution of sample means for uniform $\mathcal{D}$: 

For $k_1=1,k_2=3,k_3=10$, we have $x_1$ maximal about 42% of the time, $x_2$ maximal about 31% of the time, and $x_3$ maximal about 27% of the time.
So far, numerical evidence supports the conjecture for the following distributions: 

uniform
normal
Bernoulli($.5$)
geometric($.5$), geometric($.1$)
beta($.5,.5$), beta($2,5$)

Numerical evidence contradicts the conjecture for: 

Bernoulli($.2 $)
exponential (the conjecture holds for $k=[1,5,10,27]$ but not for $k=[1,5,10]$ or $k=[1,10]$)

library(ggplot2)
n = 50000 # number of draws for histogram

k = c(1,3,10)

out <- c()
for (i in 1:length(k)) {
  out[[i]] = replicate( n, mean( runif(k[[i]]) ) )
}

df <- data.frame(
  samples=factor(rep(k, each=n)),
  value=c(out, recursive=TRUE)
  )
p<-ggplot(df, aes(x=value, fill=samples, color=samples))+
  geom_histogram(position="identity", alpha=0.5, binwidth=.01)
p

# Count how often each sample mean is optimal among x1,...,x(length(k))
tallies = c(1:length(k))*0
for (i in 1:n) {
    samples <- c()
    for (j in 1:length(k)) {
      samples[[j]] = sample(out[[j]],1)
   }
   tallies[[which.max(samples)]] = tallies[[which.max(samples)]] + 1
}
tallies = tallies/n

print(k)
print(tallies)

REPLY [3 votes]: In general, this is not true.  Consider a distribution where $P(X=1)=p$ and $P(X=0)=1-p$, and $p$ is much much smaller than $\frac{1}{\sum k_i}$
Then typically all the sample means will be $0$.  The remaining cases are dominated by the situation where only one draw is equal to $1$, and the ones with more draws are more likely to be positive.