Nonlinear wavelet estimation of a bivariate density function under NSD assumption

In this paper, we construct two kinds of wavelet estimators of a lag-k' bivari-ate density function for a strictly stationary sequence of negatively superadditive dependent (NSD) random variables. The first one is a linear estimator based on simple projections and the second one is an adaptive nonlinear estimator inspired by a block thresholding rule. For a stationary NSD sequence {X_n, n ≥ 1}, we estimate the joint density h(x, y) of (X_i, X_i+k') using pairs of observations separated by lag k'. We provide an asymptotic expression for the mean integrated squared error (MISE) of both estimators over Besov balls Ω^s_pq (M) with s > 2/p − 1 and p, q ≥ 1: Under the assumptions that the underlying wavelets have bounded variation, the sequence is strictly stationary, and the bivariate density function is bounded, both pro-posed estimators achieve the optimal convergence rate of (n − k')^{−s∗/(1+s∗)}, where n is the sample size, k' is a fixed lag parameter, and s^∗ is the effective smoothness parameter defined as s^∗ = s + 1 − 2/p when p ∈ [1, 2] and s^∗ = s when p ≥ 2. The block thresholding estimator achieves this optimal rate adaptively without prior knowledge of the smoothness parameters. A simulation study demonstrates the superior performance of the proposed non-parametric wavelet estimators compared to kernel-based density estimators for stationary NSD observations, particularly showing the advantage of the nonlinear block thresholding approach over both linear wavelet and kernel methods.