Moment convergence rates for the uniform empirical process and the uniform sample quantile process

Youyou Chen and Tailong Li

Communications in Statistics - Theory and Methods, 2017, vol. 46, issue 18, 9086-9091

Abstract: Let {U1, U2, …, Un} be a sequence of independent and identically U[0, 1]-distributed random variables. Define the uniform empirical process as αn(t) = n− 1/2∑ni = 1(I{Ui ⩽ t} − t), 0 ⩽ t ⩽ 1, where ∥αn∥=sup0≤t≤1|αn(t)|$\Vert \alpha _n\Vert =\sup _{0\le t\le 1}|\alpha _n(t)|$. This paper estimates the exact convergence rates of weighted infinite series of E{∥αn∥-ε2logn}+$\textsf {E}{\lbrace }\Vert \alpha _n\Vert -\varepsilon \sqrt{2\log n} {\rbrace }_{+}$ and E{∥αn∥-ε2loglogn}+$\textsf {E}{\lbrace }\Vert \alpha _n\Vert -\varepsilon \sqrt{2\log \log n} {\rbrace }_{+}$ as ϵ tends to a positive constant. The corresponding results for the uniform sample quantile process are also presented.

Date: 2017
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DOI: 10.1080/03610926.2016.1205614

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