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Zbl 0596.62040
Müller, H.-G.
Weak and universal consistency of moving weighted averages. (English)
[J] Period. Math. Hung. 18, 241-250 (1987). ISSN 0031-5303; ISSN 1588-2829/e

Consider the fixed design regression model $y\sb{i,n}=g(t\sb{i,n})+\epsilon\sb{i,n}$, $1\le i\le n$, where the random variables $\epsilon\sb{i,n}$ form a triangular array and are independent for fixed n, and identically distributed with zero mean, $t\sb{i,n}\in [0,1]$ are points where the measurements $y\sb{i,n}$ are taken, and g is a smooth regression function to be estimated. For moving weighted averages $$ \hat g\sp{(\nu)}(t)=\sum\sp{n}\sb{i=1}w\sb{i,n}\sp{(\nu)}(t)y\sb{i,n}, $$ results on weak consistency $\hat g\sp{(\nu)}(t)\to\sp{P}g\sp{(\nu)}(t)$ for some $\nu\ge 0$ are derived. \par Mofifying the definition of universal consistency given by {\it 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.

MSC 2000:: *62G05 Nonparametric estimation
62G20 Nonparametric asymptotic efficiency
62J02 General nonlinear regression

Keywords: moving weighted averages; weak consistency; universal consistency; fixed design case; kernel estimators; local least squares estimators; regression function

Citations: Zbl 0366.62051

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