Weak and universal consistency of moving weighted averages (Q1078951)

Consider the fixed design regression model \(y_{i,n}=g(t_{i,n})+\epsilon_{i,n}\), \(1\leq i\leq n\), where the random variables \(\epsilon_{i,n}\) form a triangular array and are independent for fixed n, and identically distributed with zero mean, \(t_{i,n}\in [0,1]\) are points where the measurements \(y_{i,n}\) are taken, and g is a smooth regression function to be estimated. For moving weighted averages \[ \hat g^{(\nu)}(t)=\sum^{n}_{i=1}w_{i,n}^{(\nu)}(t)y_{i,n}, \] results on weak consistency \(\hat g^{(\nu)}(t)\to^{P}g^{(\nu)}(t)\) for some \(\nu\geq 0\) are derived. Mofifying the definition of universal consistency given by \textit{C. J. Stone} [Ann. Stat. 5, 595-645 (1977; Zbl 0366.62051)], for the fixed design case, conditions for fixed design universal consistency are given. The results are then shown to apply to kernel estimators and local least squares estimators which are special cases of moving weighted averages.