On iterated logarithm laws for linear arrays and nonparametric regression estimators
Let \[ \hat g(x)=\sum^{n}_{i=1}Y_ iL\{(x-x_ i)/h_ n\}/\sum^{n}_{i=1}L\{(x-x_ i)/h_ n\} \] be the kernel estimator for the smooth function g in the nonparametric regression model \(Y_ i=g(x_ i)+e_ i\), \(i\leq n\), where each \(x_ i\) is a fixed d- dimensional vector, \(Y_ i\) is a random variable, \(e_ 1,e_ 2,...,e_ n\) are i.i.d. with E \(e_ 1=0\), E \(e^ 2_ 1=1\), L is a kernel function and \(h_ n\) is the bandwidth. Under modest hypotheses, a law of the iterated logarithm for \(\hat g(x)-E\hat g(x)\) is proved, thus shedding light on the rate of convergence of \(\hat g(x)-E\hat g(x)\). This extends work in the case \(d=1\) due to \textit{W. Härdle} [Ann. Stat. 12, 624-635 (1984; Zbl 0591.62030)] and \textit{U. Stadtmüller} [Ann. Probab. 12, 35-44 (1984; Zbl 0537.60023)].
scientific article; zbMATH DE number 4213126