Let $X\_{(k)}$ be the $k$th order statistic out of $n$ uniform $[0,1]$ random variables. Let $q$ be the location of the $p$ quantile of $X\_{(k)}$, i.e. $\Pr[X\_{(k)}\leq q] = p$. For small $p$, Is it true that $q = O(p^{1/k} \frac{k}{n})$?



The behavior of a uniform order statistic near zero
Your conjecture is true. More specifically, 
\begin{equation\*}
 q \lesssim 2C p^{1/k} \frac kn \tag{1}
\end{equation\*}
uniformly as 
\begin{equation\*}
p\to0,\quad C\_1k\ge\ln n, \quad n-k\to\infty, \tag{1a}
\end{equation\*}
where $C\_1$ is any positive real constant and $C$ is any positive real constant such that 
\begin{equation}
 C>C\_2:=e^{C\_1/2}. \tag{2} 
\end{equation}


Indeed, $X\_k:=X\_{(k)}$ has the beta distribution with parameters $k,n-k+1$. So, letting  

\begin{equation\*}
 c:=k/n,
\end{equation\*}
for the [mean](https://en.wikipedia.org/wiki/Beta_distribution#Mean) and the [variance](https://en.wikipedia.org/wiki/Beta_distribution#Variance) of $X\_k$ we have 
\begin{equation\*}
 EX\_k=\frac k{n+1}\le c,\quad Var\,X\_k\le\frac{c(1-c)}n\le\frac cn. 
\end{equation\*}
So, letting 
\begin{equation\*}
 F(q):=P(X\_k\le q)
\end{equation\*}
and using Chebyshev's inequality, we have 
\begin{equation\*}
F(2c)\ge1-\frac{c/n}{c^2}=1-\frac1k\to1. 
\end{equation\*}
So, eventually $q\le2c$, which implies (1) unless 
\begin{equation\*}
 Cp^{1/k}\le1, \tag{3}
\end{equation\*}
which may and will be henceforth assumed. 


Let 
\begin{equation\*}
 q\_\*:=Ccp^{1/k},
\end{equation\*}
where $C$ is as in (2), so that $q\_\*\le c$. 
So, in view of Stirling's formula, (1a), and (2), 
\begin{align\*}
 F(q\_\*)&=k\binom nk\int\_0^{q\_\*} x^{k-1}(1-x)^{n-k}\,dx \\ 
 &\ge k\binom nk\frac{q\_\*^k}k(1-q\_\*)^{n-k} \\ 
 &\gtrsim\frac1{\sqrt n}\, \frac{n^n}{k^k(n-k)^{n-k}}\,q\_\*^k(1-q\_\*)^{n-k} \\ 
 &=\frac1{\sqrt n}\,\Big(\frac{q\_\*}c\Big)^k\Big(\frac{1-q\_\*}{1-c}\Big)^{n-k} \\ 
 &\ge\frac1{\sqrt n}\, \Big(\frac{q\_\*}c\Big)^k
 \ge \Big(\frac{q\_\*}{C\_2c}\Big)^k=\Big(\frac C{C\_2}\Big)^k p,
\end{align\*}
so that $F(q\_\*)>p$ eventually, 
whence (1) follows as well in the case when (3) holds. 


