Asymptotic properties of nonparametric curve estimates

Abstract

We consider a class of nonparametric estimators for the regression function re(t) in the model: Yl = m(tl) + el, 1 ~ i ~ n, t I ( [0, 1], which are linear in the observations Yl. Several limit theorems concerning local and global stochastic and a.s. convergence and limit distributions are given.

On confidence bands for multivariate nonparametric regression

“…To this end we extend an approach introduced by Stadtmüller (1986) or Eubank and Speckman (1993) for one-dimensional models with deterministic (close to) uniform design.…”

Moderate deviations principle for the kernel estimator of nonrandom regression functions

“…(where D → denotes the convergence in distribution) and that, for ρ ∈]0, 1/2 [, Stadtmüller (1986) proved that 2 log(1/h n ) sup…”

“…where Z * is a random variable whose distribution function is z → exp(−2 exp(−z)), and where (η n ) is a nonrandom sequence (explicitly given in Stadtmüller (1986)) satisfying lim n→∞ η n = 0. It follows that, for the sequences (v n ) such that nh n v −2 n → ∞ and nh n [v 2 n log(1/h n )] −1 → 0, the sequence (v n [µ n (t) − E(µ n (t))]) converges in probability to zero, whereas the sequence (v n sup t∈[ρn,1−ρn] |µ n (t) − E(µ n (t))|) does not, so that a A.…”

“…To conclude this section, we now compare the confidence bands obtained by application of the MDP with those constructed with the help of result (5) of Stadtmüller (1986).…”

“…To construct confidence bands by applying the result (5) of Stadtmüller (1986) and by slightly undersmoothing, (h * n ) must be chosen such that nh * 5 n log(1/h * n ) → 0 (and, let us say, such that nh * 3…”