Bounds for the expectation of the max

Question

Consider the following expected value
$$
E(max{a_1+epsilon_1, a_2+epsilon_2,...,a_n+epsilon_n})
$$
where the expectation is taken with respect to the random variables $(epsilon_1,...,epsilon_n)$ and $a_1,...,a_n$ are real numbers.
Can we say whether $E(max{a_1+epsilon_1, a_2+epsilon_2,...,a_n+epsilon_n})$ is greater or smaller than $max{a_1,...,a_n}$?
Can we characterise some upper or lower bound of  $E(max{a_1+epsilon_1, a_2+epsilon_2,...,a_n+epsilon_n})$ that does not depend on the distribution of $(epsilon_1,...,epsilon_n)$?

gunes · Answer

No, consider the easiest case $n=1$. $E[a+epsilon]=a+E[epsilon]$ and it can be either greater or smaller than $a$ based on the distribution of $epsilon$.

Bounds for the expectation of the max

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