On moment inequalities of the supremum of empirical processes with applications to kernel estimation

✍ Scribed by Ibrahim A. Ahmad

• Publisher: Elsevier Science
• Year: 2002
• Tongue: English
• Weight: 97 KB
• Volume: 57
• Category: Article
• ISSN: 0167-7152
• DOI: 10.1016/s0167-7152(02)00029-9

✦ Synopsis

Let X 1 ; : : : ; X n be a random sample from a distribution function F. Let F n (x) = (1=n) n i=1 I (X i 6 x) denote the corresponding empirical distribution function. The empirical process is deÿned by

In this note, upper bounds are found for E(D n ) and for E(e tDn ); where D n =sup x D n (x): An extension to the two sample case is indicated. As one application, upper bounds are obtained for E(W n ), where,

) is the celebrated "kernel" density estimate of f(x); the density corresponding to F(x) and an optimal bandwidth is selected based on W n : Analogous results for the kernel estimate of F are also mentioned.