Approximation for expected value of largest order statistic

Question

Given $X_1,\dots,X_n \stackrel{iid}{\sim} F$ how can one obtain the following approximation for the largest order statistic: $\mathbb{E}{X_{(n)}} \approx F^{-1}{(\frac{n}{n+1})}$ , if $\mathbb{E}{X_{(n)}}$ exists. My professor made a reference about Q-Q plots but i don't understand how they are linked to this.

Answer 1

Since you do not mention anything about continuity of $F$, I take $F^{-1}$ to be the quantile function $$F^{-1}(y)=\inf\{x:F(x)\ge y\}$$

Now it is a well-known theorem (discussed here for example) that if $U$ has a uniform distribution on $(0,1)$ and $F$ is any distribution function, then $F^{-1}(U)$ has distribution function $F$. So if $U_1,\ldots,U_n$ are i.i.d $\operatorname{Uniform}(0,1)$ and $X_1,\ldots,X_n$ are i.i.d with common distribution function $F$, then we have this equality in distribution:

$$(X_{(1)},\ldots,X_{(n)})\stackrel{d}= (F^{-1}(U_{(1)}),\ldots,F^{-1}(U_{(n)}))$$

In particular, $$X_{(n)}\stackrel{d}=F^{-1}(U_{(n)})$$

Expanding $F^{-1}(U_{(n)})$ in a Taylor series about $E\left[U_{(n)}\right]$, we have the first order approximation

$$F^{-1}(U_{(n)}) \approx F^{-1}\left(E\left[U_{(n)}\right]\right)$$

So for sufficiently large $n$,

$$E\left[X_{(n)}\right]=E\left[F^{-1}(U_{(n)})\right] \approx F^{-1}\left(E\left[U_{(n)}\right]\right)=F^{-1}\left(\frac{n}{n+1}\right)$$