Strong consistency of regression function estimator with martingale difference errors

Abstract

In this paper, we consider the regression model with fixed design: Yi=g(xi)+εi{Y}_{i}=g\left({x}_{i})+{\varepsilon }_{i}, 1≤i≤n1\le i\le n, where {xi}\left\{{x}_{i}\right\} are the nonrandom design points, and {εi}\left\{{\varepsilon }_{i}\right\} is a sequence of martingale, and gg is an unknown function. Nonparametric estimator gn(x){g}_{n}\left(x) of g(x)g\left(x) will be introduced and its strong convergence properties are established.

Keywords

• regression function
• martingale difference
• consistency
• 60f15
• 62g05