We introduce a smooth variant of the SCAD thresholding rule for wavelet denoising by replacing its piecewise linear transition with a raised cosine. The resulting shrinkage function is odd, continuous on R, and continuously differentiable away from the main threshold, yet retains the hallmark SCAD properties of sparsity for small coefficients and near unbiasedness for large ones. This smoothness places the rule within the continuous thresholding class for which Stein's unbiased risk estimate is valid. As a result, unbiased risk computation, stable data-driven threshold selection, and the asymptotic theory of Kudryavtsev and Shestakov apply. A corresponding nonconvex prior is obtained whose posterior mode coincides with the estimator, yielding a transparent Bayesian interpretation. We give an explicit SURE risk expression, discuss the oracle scale of the optimal threshold, and describe both global and level-dependent adaptive versions. The smooth SCAD rule therefore offers a tractable refinement of SCAD, combining low bias, exact sparsity, and analytical convenience in a single wavelet shrinkage procedure.

翻译：我们通过将分段线性过渡替换为升余弦函数，提出了一种用于小波去噪的平滑SCAD阈值规则变体。所得收缩函数为奇函数，在实数域上连续，且在主要阈值之外连续可微，同时保留了SCAD的核心特性：对小系数具有稀疏性，对大系数具有近似无偏性。这种平滑性使该规则属于连续阈值类别，从而适用于Stein无偏风险估计。因此，无偏风险计算、稳定的数据驱动阈值选择以及Kudryavtsev和Shestakov的渐近理论均适用。我们推导出相应的非凸先验分布，其后验众数与估计量一致，从而提供了清晰的贝叶斯解释。我们给出了显式的SURE风险表达式，讨论了最优阈值的理想尺度，并描述了全局和分层自适应的实现版本。因此，平滑SCAD规则为SCAD提供了一种易于处理的改进方案，在单一小波收缩过程中融合了低偏差、精确稀疏性和解析便利性。