Estimating Expected Order Statistics

Question

I have a fairly basic question that I'm looking for a reference for.

First, a couple definitions. Let's say $X_1,\ldots,X_n$ are IID samples from a distribution $F$ over $[0,1]$. For any $k\in\{1,\ldots,n\}$, we can define the $k$th of $n$ order statistic $X_{k:n}$ to be the $k$th highest of the $n$ samples. Let $\mu_{k:n}$ denote the expected value of $X_{k:n}$.

For any given $k$ and $n$, I would like to estimate $\mu_{k:n}$ for $F$. Specifically, I would like find an estimator $\hat \mu_{k:n}$ which takes a profile of $m$ IID samples and minimizes the mean absolute error $ E[|\hat \mu_{k:n}-\mu_{k:n}|]$ in the worst case over all $F$ (again, distributed over $[0,1]$). (With $m$ generally being distinct from and larger than $n$.)

What kinds of guarantees are known for this problem (in terms of $k$, $n$, $m$, and maybe $\mu_{k:n}$)? Is there anything that does significantly better than just dividing $m$ into blocks of $n$ samples and computing an empirical mean of the order statistics?

Answer 1

I would like find an estimator $\hat{μ}_{k:n}$ which takes a profile of $m$ iid samples and minimizes the mean absolute error $\mathbb{E}[|\hat{μ}_{k:n}−μ_{k:n}|]$ in the worst case over all $F$

Minimising an error over all possible distributions is impossible since all distributions include Dirac masses at an arbitrary $a\in (0,1)$ for which the minimiser is $\hat{μ̂}_{k:n}=a$.

If no constraint is imposed on $F$, I think the solution has to rely on an empirical cdf $\hat{F}_m$ based on the sample of size m, from which an estimate of $μ_{k:n}$ can be derived by simulation (i.e., bootstrap in this case).