Abstract
In this paper, linear and nonlinear wavelet estimators are defined for a density in a Besov space based on a negatively dependent random sample, and their upper bounds on L^{p} (1leq p<infty ) risk are provided.
Highlights
Random variables X1, X2, . . . , Xn are said to be negatively dependent (ND), if for any x1, x2, . . . , xn ∈ R, nP(X1 ≤ x1, X2 ≤ x2, . . . , Xn ≤ xn) ≤ P(Xi ≤ xi), i=1 and nP(X1 > x1, X2 > x2, . . . , Xn > xn) ≤ P(Xi > xi). i=1The definition was introduced by Bozorgnia [3]
ND random variables are very useful in reliability theory and applications
For λ = {λk} ∈ lr(Z) and 1 ≤ r ≤ ∞, λk φjk Negatively dependent random variables possess the following property which will be used in this paper
Summary
Random variables X1, X2, . . . , Xn are said to be negatively dependent (ND), if for any x1, x2, . . . , xn ∈ R, n. ND random variables are very useful in reliability theory and applications. We consider density estimation for ND random variables in this paper. Donoho et al [6] defined wavelet estimators and showed their convergence rates on Lp-loss, when X1, X2, . They found that the convergence rate of the nonlinear estimator is better than that of the linear one. Doosti et al [8] proposed a linear wavelet estimator and evaluated its Lp (1 ≤ p < ∞) risks for negatively associated random variables. The above results were extended to the case of negatively dependent sequences [7]. Kou [11] defined linear and nonlinear wavelet estimators for mixing data and obtained their convergence rates
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