1
$\begingroup$

Let $X_1,\ldots,X_n$ be an iid sample from a distribution supported on $[0, 1]$.

Question

What are some sharp concentration inequalities (i.e tail bounds) empirical statistic defined by $Z_n := \max(X_1,\ldots,X_n)$ ?

Observation

Boucheron and co-workers have results for sub-gaussian and sub-exponential distributions. These results could in principle be used to obtain rather lax tail bounds bounds for $Z_n$. I was wondering one could do better than such an approach.

Improve this question
$\endgroup$

1 Answer 1

Reset to default
2
$\begingroup$

For the tail of $Z_n$, we have the very simple exact formula $P(Z_n>t)=1-F(t)^n$ for all real $t$, where $F$ is the cdf of $X_1$. Do you want a bound on it? Of what kind?

Improve this answer
$\endgroup$
2
  • $\begingroup$ The formula you're proposing is valid for the uniform distribution on $[0,1]$. My problem is more general. The distribution is only required to supported on $[0, 1]$, not necessarily uniform thereupon. Agreed ? Concerning the remark about my use of "empirical processé", it's in fact not a very interesting process. Fixed. $\endgroup$
    – dohmatob
    Commented Dec 20, 2018 at 20:19
  • 1
    $\begingroup$ Oops, I missed it that the distribution on $[0,1]$ may arbitrary. This does not change the answer much, though. $\endgroup$
    – Iosif Pinelis
    Commented Dec 20, 2018 at 21:18

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .