Asymptotic normality of a quadratic measure of orthogonal series type density estimate期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

按检索

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

Authors:	Jugal Ghorai

Institution:	(1) University of Wisconsin-Milwaukee, Milwaukee, USA

Abstract:	LetX ₁,...,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} ofL ₂a, 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 $\sigma _n^2 = \sum\limits_{k = 0}^{q_n } {\sum\limits_{k' = 0}^{q_n } {Cov (\phi _k (X),\phi _{k'} (X))]^2 } }$ .

Keywords:	Primary 62E20 Secondary 62G05
本文献已被 SpringerLink 等数据库收录！