Abstract
Abstract In this paper, we consider the regression model with fixed design: Y i = g ( x i ) + ε i {Y}_{i}=g\left({x}_{i})+{\varepsilon }_{i} , 1 ≤ i ≤ n 1\le i\le n , where { x i } \left\{{x}_{i}\right\} are the nonrandom design points, and { ε i } \left\{{\varepsilon }_{i}\right\} is a sequence of martingale, and g g is an unknown function. Nonparametric estimator g n ( x ) {g}_{n}\left(x) of g ( x ) g\left(x) will be introduced and its strong convergence properties are established.
Highlights
The estimation of a regression function g(x) = E(y∣x) is an important statistical problem
When the errors are normal random variables, we can test the appropriateness of the hypothesized model
Priestley and Chao [1] considered the problem of estimating an unknown regression function g(x) given observations at a fixed set of points
Summary
The estimation of a regression function g(x) = E(y∣x) is an important statistical problem. G(x) has a specified functional form and parameter estimates are obtained according to certain desirable criteria. When the errors are normal random variables, we can test the appropriateness of the hypothesized model. One may wish to have an estimation technique applicable for an arbitrary g(x). Priestley and Chao [1] considered the problem of estimating an unknown regression function g(x) given observations at a fixed set of points. Their estimate is nonparametric in the sense that g(x) is restricted only by certain smoothing requirements
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