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.