Nonlinear wavelet-based estimators for spectral densities of non-Gaussian linear processes are considered. The convergence rates of mean integrated squared error (MISE) for those estimators over a large range of Besov function classes are derived, and it is shown that those rates are identical to minimax lower bounds in standard nonparametric regression model within a logarithmic term. Thus, those rates could be considered as nearly optimal. Therefore, the resulting wavelet-based estimators outperform traditional linear methods if the degree of smoothness of spectral densities varies considerably over the interval of interest, such as sharp spike, cusp, bump, etc., since linear estimators are not able to attain these rates. Unlike in classical nonparametric regression with Gaussian noise errors where thresholds are determined by normal distribution, we determine the thresholds based on a Bartlett type approximation of a quadratic form with dependent variables by its corresponding quadratic form with independent identically distributed (i.i.d.) random variables and Hanson-Wright inequality for quadratic forms in sub-gaussian random variables. The theory is illustrated with some numerical examples, and our simulation studies show that our proposed estimators are comparable to the current ones.

考虑了用于非高斯线性过程的谱密度的基于非线性小波的估计器。推导了这些估计器在大范围 Besov 函数类上的均方积分平方误差 (MISE) 的收敛速度，并表明这些收敛速度与标准非参数回归模型中的对数项内的极小极大下限相同。因此，可以认为这些速率几乎是最优的。因此，如果谱密度的平滑度在感兴趣的区间内发生显着变化，例如尖峰、尖点、凸点等，则所得的基于小波的估计器优于传统的线性方法，因为线性估计器无法达到这些速率. 与具有高斯噪声误差的经典非参数回归不同，其中阈值由正态分布确定，我们基于具有因变量的二次形式的 Bartlett 类型近似确定阈值，该二次形式通过其对应的具有独立同分布 (iid) 随机变量的二次形式和 Hanson - 亚高斯随机变量中二次形式的赖特不等式。该理论通过一些数值示例进行了说明，我们的模拟研究表明，我们提出的估计器与当前的估计器相当。) 亚高斯随机变量中二次形式的随机变量和 Hanson-Wright 不等式。该理论通过一些数值示例进行了说明，我们的模拟研究表明，我们提出的估计器与当前的估计器相当。) 亚高斯随机变量中二次形式的随机变量和 Hanson-Wright 不等式。该理论通过一些数值示例进行了说明，我们的模拟研究表明，我们提出的估计器与当前的估计器相当。