Asymptotic normality of a quadratic measure of orthogonal series type density estimate

Abstract

LetX₁,...,X_n be i.i.d. random variable with a common densityf. Let$f_n (x) = \sum\limits_{k = 0}^{q_n } {\hat a_k \phi _k } (x)$ be an estimate off(x) based on a complete orthonormal basis {φ_k:k≧0} ofL₂[a, b]. A Martingale central limit theorem is used to show that$(\sqrt 2 \sigma _n )^{ - 1} \left[ {n\int {(f_n (x) - f(x))^2 dx - \mu _n } } \right]\xrightarrow{\mathcal{L}}N(0,1)$, where$\mu _n = \sum\limits_{k = 0}^{q_n } {Var[\phi _k (X)]}$ and$\sigma _n^2 = \sum\limits_{k = 0}^{q_n } {\sum\limits_{k' = 0}^{q_n } {[Cov (\phi _k (X),\phi _{k'} (X))]^2 } }$.