An orthogonal series estimate of time-varying regression
Let \((X_ 1,Y_ 1)\), \((X_ 2,Y_ 2),..\). be independent pairs of random variables with \(X_ i\in {\mathcal X}\subset {\mathbb{R}}^ p\), \(Y_ i\in {\mathbb{R}}^ 1\) and (1) a nonstationary regression function \(R_ n(x):=E[Y_ n| X_ n=x]=t_ n(x)\cdot R(x)\) where \(R,t_ n\) are unknown, but \(\sup_ x| t_ n(x)-c_ n| \to 0\) for some real sequence \(\{c_ n\}\); (2) \(Y_ n=t_ n(X_ n)R(X_ n)+Z_ n\) with centered i.i.d. variables \(Z_ n\); and (3) the marginal density \(f(\cdot)\) of \(X_ n\) being independent of n. Consider a complete orthogonal system \(\{g_ i(\cdot)\}\) of \(L_ 2({\mathcal X})\). The unknown Fourier coefficients \(b_ i=E[g_ i(X_ i)]\), \(a_{in}=E[Y_ ng_ i(X_ n)]\) of f(x) resp. \(h_ n(x):=R_ n(x)f(x)\) are estimated by \[ \hat b_{in}:=(g_ i(X_ 1)+...+g_ i(X_ n))/n\quad resp.\quad \hat a_{i,n+1}:=\hat a_{in}-\gamma_ n(a_{in}-Y_{n+1}g_ i(X_{n+1})) \] (Robbins-Monro; \(\gamma_ n>0)\). Thus, \(\hat f{}_ n(x):=\sum^{M(n)}_{0}\hat b_{in}g_ i(x)\), \(\hat h{}_ n(x):=\sum^{N(n)}_{0}\hat a_{in}g_ i(x)\) are orthogonal series estimates for f resp. \(h_ n.\) In the paper it is shown that (under suitable conditions for \(\gamma_ n,c_ n,N(n)\), etc.) \(\hat R_ n(x):=\hat h_ n(x)/\hat f_ n(x)\) is a consistent (nonparametric) estimate for \(R_ n: | \hat R_ n(x)- R_ n(x)| \to 0\) for \(f(x)>0\) and \(n\to \infty\) (in probability and a.s.). This result applies even to cases like \(t_ n(x)=(1+\rho(x)/n)n^ q\) with \(\sup | \rho(x)|<\infty\) and \(q>0\), where \(t_ n\) is unbounded for \(n\to \infty\) (use \(c_ n=n^ q\), \(\gamma_ n=\delta_ n^{-2/3}\), \(N(n)\sim n^{\alpha})\). As an example, Hermite and Legendre systems are considered.
scientific article; zbMATH DE number 3852258