Abstract
The regression m(x)=E{Y|X=x} is estimated by the kernel regression estimate % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqpepeea0xe9qqVa0l% b9peea0lb9sq-JfrVkFHe9peea0dXdarVe0Fb9pgea0xa9pue9Fve9% Ffc8meGabaqaciGacaGaaeqabaWaaeaaeaaakeaaceWGTbGbambaaa% a!3888!\[\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over m} \](x) calculated from a sequence (X(1, Y 1), ..., (X n , Y n ) of independent identically distributed random vectors from R d×R. The second order asymptotic expansions for E% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqpepeea0xe9qqVa0l% b9peea0lb9sq-JfrVkFHe9peea0dXdarVe0Fb9pgea0xa9pue9Fve9% Ffc8meGabaqaciGacaGaaeqabaWaaeaaeaaakeaaceWGTbGbambaaa% a!3888!\[\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over m} \](x) and var % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqpepeea0xe9qqVa0l% b9peea0lb9sq-JfrVkFHe9peea0dXdarVe0Fb9pgea0xa9pue9Fve9% Ffc8meGabaqaciGacaGaaeqabaWaaeaaeaaakeaaceWGTbGbambaaa% a!3888!\[\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over m} \](x)} are derived. The expansions hold for almost all (μ) x∈R d, μ is the probability measure of X. No smoothing conditions on μ and m are imposed. As a result, the asymptotic distribution-free normality for a stochastic component of % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqpepeea0xe9qqVa0l% b9peea0lb9sq-JfrVkFHe9peea0dXdarVe0Fb9pgea0xa9pue9Fve9% Ffc8meGabaqaciGacaGaaeqabaWaaeaaeaaakeaaceWGTbGbambaaa% a!3888!\[\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over m} \](x)} is established. Also some bandwidth-selection rule is suggested and bias adjustment is proposed.
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This work was supported by NSERC Grant A8131.
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Pawlak, M. On the almost everywhere properties of the kernel regression estimate. Ann Inst Stat Math 43, 311–326 (1991). https://doi.org/10.1007/BF00118638
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DOI: https://doi.org/10.1007/BF00118638