Are the probability distributions of order statistics log concave? If some distribution is log concave, then would the distribution of order statistics sampled from that distribution also be log concave? Specifically, I'm thinking of a case where I have a random vector $X = [X_1, X_2, \dots, X_n]$ and I set everything but the $k$ highest magnitude entries to 0. If $X$ were drawn from a log concave distribution, could I say anything about the resulting vector?
I have heard that the extreme value distribution is log concave, and also that the distribution of a log concave rv is also log concave (Section 2.2.5). These facts seem relevant, though not exactly what I'm looking for.

Yes. Let $X_{(1)}<…<X_{(n)}$ denote the order statistic of $(X_1,…,X_n)$, where the $(X_i)_{i=1…n}$ are i.i.d. with density $f$ and cdf $F$. Let $f_k$ denote the density of $X_{(k)}$. Then (see here):
$$f_k(x)=C_{n,k} F^{k-1}(x)(1-F(x))^{n-k}f(x),$$
for some constant $C_{n,k}$. Thus,
$$\ln f_k(x)=\ln C_{n,k} + (k-1) \ln F(x)+ (n-k)\ln(1-F(x))+\ln f(x).$$
Now, remark that if $\ln f$ is concave, so are $\ln F$ and $\ln(1-F)$ (see here). This implies that $\ln f_k$ is also concave.
