Asymptotic normality of a smooth estimate of a random field distribution function under association

• Published in: Statistics & probability letters Vol. 24; no. 1; pp. 77 - 90
• Main Author: Roussas, George G.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01.07.1995; Elsevier
• Series: Statistics & Probability Letters
• Subjects: Asymptotic normality; Empirical distribution function; Exact sciences and technology; Mathematics; Nonparametric inference; Positive or negative association; Positive or negative quadrant dependence; Probability and statistics; Random field; Random field Positive or negative association Positive or negative quadrant dependence Stationarity Empirical distribution function Smooth estimate of a distribution function Asymptotic normality; Sciences and techniques of general use; Smooth estimate of a distribution function; Stationarity; Statistics; 62G05; Positive or negative association; Smooth estimate of a distribution function; Random field; 60G10; Positive or negative quadrant dependence; Asymptotic normality; 62E20; 60G60; Stationarity; Empirical distribution function; 60F05; Multidimensional space; Smoothing; Dependence; Statistical association; Stationary process; Integer lattice; Kernel method
• ISSN: 0167-7152; 1879-2103
• DOI: 10.1016/0167-7152(94)00151-W

Let Z d be the lattice of points in R d with integer coordinates, and let { X n }, n ϵ Z d , be a random field of real-valued translation invariant random variables with unknown distribution function F. For u and v in Z d with u < v let, B u v be the box in Z d defined by B u v = { n ϵ Z d ; u < n ⩽ v}, and for N = 1, 2, …, let k( N) = ( k 1 ( N), …, k d ( N)) with k i ( N) → ∞ as N → ∞, i = 1, …, d. On the basis of the random variables X n , n ϵ B 0 k( N) , let F ̌ ‖k(N)‖ be a smooth kernel-type estimate of F. Under suitable regularity conditions, including that of association, it is shown that F ̌ ‖k(N)‖ , properly normalized and centered, is asymptotically normal with specified parameters.