The L₂risk of an isotonic estimate

Abstract

Suppose that observations (t₁, y₁), …, (t_n, y_n) are collected from isotonic regression y = f(t) + ϵ, where f is a bounded non-increasing function. Denote by [ftilde]_n be the isotonic estimate of f. This paper derives a nonasymptotic bound on the L₂ risk of [ftilde]_n for given n and f. This bound is then used to study both the limit of the L₂ risk of [ftilde]_n as n goes to infinity and the maximum of the L₂ risk of [ftilde]_n over a class of bounded non-increasing functions.

Keywords:

• asymptotic risk
• nonasymptotic risk
• isotonic estimate
• L₂ risk