Regularity conditions for the approximation of the mean of order statistics (i.e. when is $E[X_{(pn)}] \approx F^{-1}(p) $)

Question

Let $X_{(1)} \le X_{(2)}\le ... \le X_{(n)}$ be a sequence of order statistics generate from an i.i.d. sequence $X_1, X_2, ... , X_{n}$. We are interested in expected value of $X_{(pn)}$, for $p \in (0,1)$, that is \begin{align} E[X_{(pn)}], \end{align} we assume $pn$ is always an integer or just consider $E[X_{( \lceil pn \rceil)}]$.

A commonly used, large $n$, approximation of expectation is \begin{align} E[X_{(pn)}] \approx F^{-1}(p) \end{align} where $F$ is cdf of the original distribution and $F^{-1}$ the quantile function (inverse cdf).

Question: Under what regularity conditions is this approximation valid? In other words, under what regularity conditions do we have that \begin{align} \lim_{n \to \infty} | E[X_{(pn)}] - F^{-1}(p) |=0? \end{align}

What Have I done: I have looked at books like "First Course on Order Statistics" by Bary Arnold and other references. While these approximations appear there and in other sources, I was not able to find conditions under which this approximation is valid.

The proof that I was able to find goes as follows: \begin{align} E[X_{(pn)}]=&E[F^{-1}(U_{(pn)})] \text{ where $U_{(pn)}$ is order statistics of uniform distribution}\\ &=E[F^{-1}(p) -\frac{d}{du}F^{-1}(u)|_{u=D} (p-U_{(pn)})] \end{align} where in the above we used Taylor's reminder theorm and where the random variable $D$ is between $p$ and $U_{(pn)}$. This leads to

\begin{align} E[X_{(pn)}] &=F^{-1}(p) - E \left[\frac{1}{f(F^{-1}(D))} (p-U_{(pn)})\right] \end{align} where we used that $\frac{d}{du}F^{-1}(u)=\frac{1}{f(F^{-1}(u))}$ and where $f$ is pdf of $F$.

At this point we need to show that \begin{align} \lim_{n \to \infty} E \left[\frac{1}{f(F^{-1}(D))} (p-U_{(pn)})\right]=0. \end{align} However, I was not able to find conditions that guarantee this. The key is to flip the expectation and limit, but I was not able to find a dominating random variable for this.

Answer 1

1. From the consistency of estimator of quantile(c.f. A.W. van der Vaart, Asymptotic Statistics, Cambridge University Press 1998. p.43, p.304, \$21.1), if $F$ is strictly increasing and continuous in a neighbourhood $(F^{-1}(p)-\delta, F^{-1}(p)+\delta)$, then \begin{equation*} X_{n,([pn])}\overset{pr}{\longrightarrow}F^{-1}(p),\qquad \text{as }n\to\infty.\tag{1} \end{equation*}

2. If (1) is true and $\{ X_{n,([pn])},n\ge 1 \}$ is uniformly integrable, then \begin{equation*} \lim_{n\to\infty}\mathsf{E}[X_{n,([pn])}]=F^{-1}(p). \tag{2} \end{equation*}

3. Suppose that (i) $F$ is strictly increasing and continuous in a neighbourhood $(F^{-1}(p)-\delta, F^{-1}(p)+\delta)$, (ii) \begin{equation*} -\infty<a=\inf\{x:F(x)>0\}\le \sup\{x:F(x)<1\}=b<\infty, \tag{3} \end{equation*} then (2) is true.

Proof: From (3), $a\le X_{n,([pn])}\le b $.