Rates of convergence for the sup-norm risk in image models under sequential designs

✍ Scribed by Jae-Chun Kim; Alexander Korostelev

• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 118 KB
• Volume: 46
• Category: Article
• ISSN: 0167-7152
• DOI: 10.1016/s0167-7152(99)00128-5

✦ Synopsis

Let G be that portion of the unit square which lies below the graph of a smooth function. Assume that observations of the indicator function of G are available at any points X1; : : : ; Xn in the plane. If each consecutive point Xi can be chosen sequentially, on the basis of all the preceding data, then how accurately can the smooth function be estimated in sup-norm? Using the boundary fragment model, this question translates into a question about the large sample performance of the minimax risks under sup-norm loss. The asymptotic rates of these risks are found. The results are extended to additive noise models and multidimensional images.