Smoothing the difference-based estimates of variance using variational mode decomposition

We propose a variational mode decomposition approach to estimate the variance function in a nonparametric heteroscedastic fixed design regression model. A data-driven estimator is constructed by applying variational mode decomposition technique to the difference-based initial estimates. The numerical results show that the proposed estimator performs better than the existing variance estimation procedures in the mean square sense.     

Estimation of Variance Function in Heteroscedastic Regression Models by Generalized Coiflets

A wavelet approach is presented to estimate the variance function in heteroscedastic nonparametric regression model. The initial variance estimates are obtained as squared weighted sums of neighboring observations. The initial estimator of a smooth variance function is improved by means of wavelet smoothers under the situation that the samples at the dyadic points are not available. Since the traditional wavelet system for the variance function estimation is not appropriate in this situation, we demonstrate that the choice of the wavelet system is significant to have better performance. This is accomplished by choosing a suitable wavelet system known as the generalized coiflets. We conduct extensive simulations to evaluate finite sample performance of our method. We also illustrate our method using a real dataset.     

Variance Estimation in Heteroscedastic Models by Undecimated Haar Transform

We propose a method in order to maximize the accuracy in the estimation of piecewise constant and piecewise smooth variance functions in a nonparametric heteroscedastic fixed design regression model. The difference-based initial estimates are obtained from the given observations. Then an estimator is constructed by using iterative regularization method with the analysis-prior undecimated three-level Haar transform as regularizer term. We notice that this method shows better results in the mean square sense over an existing adaptive estimation procedure considering all the standard test functions used in addition to the functions that we target. Some simulations and comparisons with other methods are conducted to assess the performance of the proposed method.