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Estimating regression functions and their derivatives by the kernel method. (English) Zbl 0548.62028
The fixed sign model of the nonparametric regression problem is considered. Namely, it is required to estimate the unknown regression function \(\mu\) and its derivatives in the following model: \(X(t_ i)=\mu (t_ i)+\epsilon_ i\), \(i=1,...,n\), where \(\epsilon_ i\) are i.i.d. r.v.’s with \(E(\epsilon_ i)=0\), \(Var(\epsilon_ i)=\sigma^ 2<\infty\), \(0\leq t_ 1\leq...\leq t_ n\leq 1\) (the ”design”). The points \(t_ 1,...,t_ n\) depend on n and are determined by the experimenter.
The kernel estimates of the regression function \(\mu\) and its derivatives proposed in the paper are defined as follows: \[ {\hat\mu }_{n,\nu}(t)=b^{-(\nu +1)}\sum^{n}_{j=1}\int^{s_ j}_{s_{j-1}}W_{\nu}(t-u/b)du\cdot X(t_ j),\quad\nu \geq 0, \] where \(t_ j\leq s_ j\leq t_{j+1}\), \(j=1,...,n-1\), and the kernel \(W_{\nu}\) is defined as the \(\nu\) -th derivation of the \(\nu\) -times differentiable density function W.
Under some smoothness conditions on W and \(\mu\) weak and strong consistency asymptotic normality, rates of convergence for the mean square error and integrated mean square error of the considered estimates are obtained. Two examples of applications in biomedicine are given.
Reviewer: R.Mnatsakanov

62G05 Nonparametric estimation
62J02 General nonlinear regression
62P10 Applications of statistics to biology and medical sciences; meta analysis