Nearest neighbour distance (${\rho }_{(k)}(x)$)

For observations $X_1,\dots ,X_n$, the nearest neighbour distance of order $k$ at $x$ is the $k$th order statistic of the distances $|X_i-x|$. An atomless distance distribution function ensures ties occur with probability zero.

For independent and identically distributed random variables with continuous distance distribution function $p_x$, the probability integral transform makes the probability content of the order-$k$ nearest ball a uniform order statistic, with the Beta distribution $\mathrm{Beta}(k,n+1-k)$.

For a strictly positive probability density function with Lipschitz bound $L$, $p_x(r)={\int }_{x-r}^{x+r}f(y)dy$ satisfies $|p_x(r)-2rf(x)|\le L{r}^2$. If $0<s<1$ and $s\le f(x{)}^2/L$, monotonicity gives $p_x^{-1}(s)\le s/f(x)$ and hence $|2f(x)p_x^{-1}(s)-s|\le L{s}^2/f(x{)}^2$. The Lipschitz density height bound prevents extending this local argument to every $s\in (0,1)$ by assuming $f(x)\ge \sqrt{L}$ for a strictly positive density.