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Asymptotic normality of series estimators for nonparametric and semiparametric regression models. (English) Zbl 0727.62047
The purpose of this paper is to establish some asymptotic distribution theory for series estimators in various nonparametric and semiparametric regression models. Fourier flexible form series estimators, trigonometric series estimators and polynomial series estimators are covered by the results. Multiple regressors are allowed, the values of the regressors are not restricted and the errors may be independent, nonidentically distributed.
The results provide conditions under which a series estimator is asymptotically normally distributed after being centered at either its expectation or the estimand and being normalized by premultiplication by the square root of its covariance matrix. The estimands considered here include the regression function itself, derivatives of the regression function and integrated values of the regression function. The errors in the model may be homoskedastic or heteroskedastic.
The author also investigates series estimators for additive interactive regression models, the partially linear regression model and semiparametric index regression models. Consistency and asymptotic normality results for the different estimands, estimators and models considered are all obtained from a single set of results for series estimators.

62G07 Density estimation
62G20 Asymptotic properties of nonparametric inference
62E20 Asymptotic distribution theory in statistics
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