The limit distribution of the $L_{\infty}$-error of Grenander-type estimators

Cécile Durot, Vladimir N. Kulikov, Hendrik P. Lopuhaä
2012 Annals of Statistics  
Let $f$ be a nonincreasing function defined on $[0,1]$. Under standard regularity conditions, we derive the asymptotic distribution of the supremum norm of the difference between $f$ and its Grenander-type estimator on sub-intervals of $[0,1]$. The rate of convergence is found to be of order $(n/\log n)^{-1/3}$ and the limiting distribution to be Gumbel.
doi:10.1214/12-aos1015 fatcat:7iptazzcsnhvlkfdnhzlnbahdu