Are expected order statistics always concave in sample size?

Question

Let $X_{1:n}\leqslant X_{2:n}\leqslant \dots \leqslant X_{n:n}$ be the $n$ (independent) order statistics of a known distribution with CDF $F(x)$. Consider the expectation of the $k$-th largest order statistics, i.e., $\mathbb{E}(X_{n-k+1:n})$, is it always concave in $n$? Formally, is the following relationship always true? $$\mathbb{E}(X_{(n+1)-k+1:(n+1)})-\mathbb{E}(X_{n-k+1:n})>\mathbb{E}(X_{(n+2)-k+1:(n+2)})-\mathbb{E}(X_{(n+1)-k+1:(n+1)})$$ where $n$ is finite. Thanks!

Answer 1

After some literature search, I realized that this claim is true for the largest expected order statistics. The following illustration uses results from David (1997). To be consistent, I use their notation. Denote $\mu_{r:n}$ as the $r$-th smallest expected order statistics out of $n$ order statistics (or equivalently, the $(n-r+1)$-th largest one), where $r\in\{1,2,...,n-1\}$, David (1997) shows that $$\mu_{r+1:n}-\mu_{r:n-1}=\binom{n-1}{r}\int_{-\infty}^{+\infty}F^r(x)[1-F(x)]^{n-r}dx$$

Let $r=n-1$, i.e., consider the largest expected order statistics, then the above relationship becomes $$\mu_{n:n}-\mu_{n-1:n-1}=\int_{-\infty}^{+\infty}F^{n-1}(x)[1-F(x)]dx$$

Similarly, increase both $r$ and $n$ by 1, we get $$\mu_{n+1:n+1}-\mu_{n:n}=\int_{-\infty}^{+\infty}F^n(x)[1-F(x)]dx$$

Because $F(x)$, as the CDF of a distribution, is always between 0 and 1, we get $$\mu_{n+1:n+1}-\mu_{n:n}<\mu_{n:n}-\mu_{n-1:n-1}$$ indicating concavity.

However, this argument does not seem to easily generalize to other expected order statistics, because of the binomial coefficients. Or maybe I'm missing something obvious. Hope this can help people obtain a more generalized result.

(Update 09/22/2017) This claim is not always true for expected order statistics that are not the largest. Here is an (numeric) counter-example:

Consider a Weibull distribution of shape parameter 0.4 and scale parameter 1. The second largest expected order statistics out of samples of 3, 4, and 5 (i.e., $n=3,4,5$) are 1.336, 2.131, and 2.931 respectively, which is convex with respect to $n$. Even more evidently, the third largest expected order statistics are 0.213, 0.541, 0.931 respectively, also convex in $n$. These numbers were calculated based on results in Lieblein (1955).

In summary, the largest expected order statistics always increase concavely with respect to sample size, irrespective of distribution. However, the same cannot be said for other expected order statistics.