Estimating nonquadratic functionals of a density using haar wavelets

• 成果类型: Article
• 署名作者: Kerkyacharian, G; Picard, D
• 署名单位: Universite Paris Cite
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• 发表日期: 1996
• 页码: 485-507
• 关键词: besov-spaces; derivatives; kernel

摘要

Consider the problem of estimating integral Phi(f), where Phi is a smooth function and f is a density with given order of regularity s. Special attention is paid to the case Phi(t) = t(3). It has been shown that for low values of s the n (-1/2) late of convergence is not achievable uniformly over the class of objects of regularity s. In fact, a lower bound for this rate is n(-4s/(1+4s)) for 0 < s less than or equal to 1/4. AS for the upper bound, using a Taylor expansion, it can be seen that it is enough to provide an estimate for the case Phi(x) = x(3). That is the aim of this paper. Our method makes intensive use of special algebraic and wavelet properties of the Haar basis.