Claim about Empirical Quantile Function: $\hat{F}_n^{-1}\left(\frac{i}{n+1}\right) = X_{(i)}$

Question

The lecturer in a class I'm taking defined the empirical quantile function for a sample of $n$ random variables $\{X_i\}_{i = 1}^n$ as follows: $$ \hat{F}_n^{-1}(p) = \left\{\begin{aligned} &X_{(np)} &&, np \in \mathbb{N}\\ &X_{(\lfloor np+1 \rfloor)} &&, np \notin \mathbb{N}, \end{aligned} \right.$$

where $X_{(i)}$ represents the $i^{\text{th}}$ order statistic of the sample.

Based on this definition, I'm trying to understand the following claim: $$ \hat{F}_n^{-1}\left(\frac{i}{n+1}\right) = X_{(i)}. $$

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My progress to this point:

• I was able to show that: $n \in \mathbb{N}$ and $i \in [0,n] \cap \mathbb{N} \implies n+1 \nmid ni$. Hence, $\frac{ni}{n+1} \notin \mathbb{N}$ in this situation.
• What then remains to show is that $i \leq \frac{ni}{n+1} + 1 < i+1$. The second inquality is clear, since $\frac{n}{n+1} < 1$.
• Thus, showing this boils down to showing: $i \leq \frac{n}{n+1}\cdot i + 1$ when $n \in \mathbb{N}$, $i \in [1,n]$.

Answer 1

To prove the third bulletpoint, suppose for purpose of establishing a contradiction that for $k \in \{1,...,n\}$:

\begin{equation} \begin{split} k & > \frac{nk}{n+1} + 1\\ \frac{n+1}{n+1}k & > \frac{nk}{n+1} + \frac{n+1}{n+1}\\ nk + k & > nk + n + 1\\ k & > n + 1 \end{split} \end{equation}

Which by assumption is not true.

Answer 2

Maybe an example in R will help you visualize the ECDF function.

Here is a sorted normal sample rounded to two places:

x = sort( round(rnorm(10, 20, 3),2) );  x
[1] 16.38 18.07 18.23 18.37 19.31 20.29 20.55 20.68 22.20 26.29

The fifth order statistic is 19.31:

x[5]
[1] 19.31

The x-values of the ECDF are the order statistics. Its y-values are shown below:

F = (1:10)/11;  F
 [1] 0.09090909 0.18181818 0.27272727 0.36363636 0.45454545 0.54545455
 [7] 0.63636364 0.72727273 0.81818182 0.90909091

The fifth y-value of the ECDF is $5/11.$

F[5]; 5/11
[1] 0.4545455
[1] 0.4545455

Here is a plot of this ECDF, in which the x- and y-values mentioned above are emphasized:

plot(x, F, type="s", lwd=2, ylim=c(0,1))
  abline(h=0:1, col="green2")
  points(x, F, pch=19)
  abline(v = x[5], col="red", lty="dotted")
  abline(h = F[5], col="blue", lty="dotted")

Some authors say that the ECDF function consists only of the heavy dots, and some say that the horizontal lines are also part of the function. (If included in a plot, the vertical lines at 'jump points' or 'knots' are just to help the eye follow the function; sometimes they are dotted lines.)

You should know that different texts and statistical software programs (R among them) define the ECDF in various slightly (but fundamentally) different ways. Here is a plot of the ECDF from R.

plot(ecdf(x))