Limiting law results for a class of conditional mode estimates for functional stationary ergodic data

详细信息    查看全文

• 作者：S. Bouzebda ; M. Chaouch ; N. Laïb
• 关键词：conditional mode estimation ; uniform entropy integral ; ergodic processes ; functional data ; martingale difference array ; strong consistency ; VC ; class ; uniform CLT ; resampling
• 刊名：Mathematical Methods of Statistics
• 出版年：2016
• 出版时间：July 2016
• 年：2016
• 卷：25
• 期：3
• 页码：168-195
• 全文大小：821 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Statistics
  Statistical Theory and Methods
  Russian Library of Science
  
• 出版者：Allerton Press, Inc. distributed exclusively by Springer Science+Business Media LLC
• ISSN：1934-8045
• 卷排序：25

文摘

The main purpose of the present work is to establish the functional asymptotic normality of a class of kernel conditional mode estimates when functional stationary ergodic data are considered. More precisely, consider a random variable (X,Z) taking values in some semi-metric abstract space E × F. For a real function φ defined on F and for each x ∈ E, we consider the conditional mode, say ⊝_φ(x), of the real random variable φ(Z) given the event “X = x”. While estimating the conditional mode function by Θ̂_φ,n(x), using the kernel-type estimator, we establish the limiting law of the family of processes {Θ̂_φ(x) - Θ_φ(x)} (suitably normalized) over Vapnik–Chervonenkis class C of functions φ. Beyond ergodicity, no other assumption is imposed on the data. This paper extends the scope of some previous results established under mixing condition for a fixed function φ. From this result, the asymptotic normality of a class of predictors is derived and confidence bands are constructed. Finally, a general notion of bootstrapped conditional mode constructed by exchangeably weighting samples is presented. The usefulness of this result will be illustrated in the construction of confidence bands.