Abstract:
Our aim in the present paper is to give a new representation of the previously known estimates and further investigate upper and lower bounds for expected maxima of $n$ independent and identically distributed (i.i.d.) standardized random variables (r.v.'s)
from known expected maxima of $m$ and $p$ r.v.'s with the same distribution, where $1<m<p<n$.
A new representation is obtained from expansion of the inverse distribution function in a system of orthonormal functions on the unit interval.
A criterion for attainability of the resulting bounds is put forward. We also obtain asymptotic properties of normed bounds for maxima expectations and normed maxima of r.v.'s with distribution where a criterion for attainability of these bound holds.
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