Asymptotic Distribution of $L_2$ Norms of the Deviations of Density Function Estimates

Abstract

Let $(\phi_\nu)^\infty_{\nu=0}$ be a complete orthonormal system of functions on a interval $\lbrack a, b\rbrack$ and let $w$ be a function defined on $\mathbb{R}$ with support $\lbrack a, b\rbrack$ and strictly positive on $(a, b)$. Let $(x_j)^\infty_{j=1}$ be i.i.d.rv's absolutely continuous with density function $f$ with respect to Lebesgue measure $\mu$ on $\mathbb{R}$. Let $f_n$ be the estimate of $f$ defined for $x\in (a, b)$ by $f_n(x) = \Sigma^m_{\nu=0}a_\nu(m)\hat{d}_\nu\phi_\nu(x)/w(x)$, where $a_\nu(m), (m = 0, 1, \cdots, \nu = 0, 1, \cdots, m)$ is a sequence in $\mathbb{R}$ and $\hat{d}_\nu = n^{-1}\Sigma^n_{j=1}\phi_\nu(X_j)w(X_j)$. In this work the asymptotic distributions of the functionals $T_n = n(m + 1)^{-1} \int^b_a(f_n - Ef_n)^2w^2 dx$ and $T^\ast_n = n(m + 1)^{-1} \int^b_a(f_n - f)^2w^2 dx$ are found. These results are used to construct tests of goodness-of-fit analogous to those proposed by Bickel and Rosenblatt. The basic idea in obtaining the results consists in finding the asymptotic distribution of $T_n(T^\ast_n)$ with $f_n$ replaced by a conveniently chosen Gaussian process and showing that the two functionals converge to the same law. For this the normalized and centered sample distribution function is approximated by an appropriate Brownian motion process by using a Skorohod-like imbedding due to Brillinger and Breiman.