Asymptotic Normality of L₁-Error in Density Estimation

Abstract

Let f _n be a histogram estimate constructed from a sample of i.i.d. real-valued random variables with common continuously differentiable density f. In this paper we prove a central limit theorem for the L ₁ error . We determine a positive constant 0 < σ ² ≤ 1 − 2/π in order that, under the usual conditions of consistency, the law of be asymptotically Gaussian , N(0, 1).

Keywords:

• AMS 1991 subject classifications
• Primary 62H12
• Secondary 62G05

Keywords:

• Nonparametric density estimation
• central limit theorem
• histogram estimate