We consider the situation of a univariate nonparametric regression where either the Gaussian error or the predictor follows a stationary strong mixing stochastic process and the other term follows an independent and identically distributed sequence. Also, we estimate the regression function by expanding it in a wavelet basis and applying a hard threshold to the coefficients. Since the observations of the predictor are unequally distant from each other, we work with wavelets warped by the density of the predictor variable. This choice enables us to retain some theoretical and computational properties of wavelets. We propose a unique estimator and show that some of its properties are the same for both model specifications. Specifically, in both cases the coefficients are unbiased and their variances decay at the same rate. Also the risk of the estimator, measured by the mean integrated square error is almost minimax and its maxiset remains unaltered. Simulations and an application illustrate the similarities and differences of the proposed estimator in both situations.

我们考虑单变量非参数回归的情况，其中高斯误差或预测变量遵循平稳的强混合随机过程，而另一项遵循独立且均匀分布的序列。另外，我们通过以小波为基础扩展回归函数并对系数应用硬阈值来估计回归函数。由于预测变量的观测值彼此之间的距离不相等，因此我们对受预测变量变量密度扭曲的小波进行处理。这种选择使我们能够保留小波的某些理论和计算属性。我们提出了一个唯一的估计量，并证明了这两个模型规格的某些属性都相同。具体地说，在两种情况下，系数都是无偏的，并且它们的方差以相同的速率衰减。同样，通过平均积分平方误差测得的估计器风险几乎是maxmax，并且其maxiset保持不变。仿真和应用说明了两种情况下拟议估计量的异同。