In recent years, the mixture of two-component normal distributions (MixN) has attracted considerable interest due to its flexibility in capturing a variety of density shapes. In this paper, we investigate the problem of discretizing a MixN by a fixed number of points under the minimum mean squared error (MSE-RPs). Motivated by the Fang-He algorithm, we provide an effective computational procedure with high precision for generating numerical approximations of MSE-RPs from a MixN. We have explored the properties of the nonlinear system used to generate MSE-RPs and demonstrated the convergence of the procedure. In numerical studies, the proposed computation procedure is compared with the k-means algorithm. From an application perspective, MSE-RPs have potential advantages in statistical inference.Our numerical studies show that MSE-RPs can significantly improve Kernel density estimation.

Deep generative models have demonstrated superior performance in lossless compression on identically distributed data. However, in real-world scenarios, data to be compressed are of various distributions and usually cannot be known in advance. Thus, commercially expected neural compression must have strong Out-of-Distribution (OoD) generalization capabilities. Compared with traditional compression methods, deep learning methods have intrinsic flaws for OoD generalization. In this work, we make the attempt to tackle this challenge via exploiting a zoo of Deep Autoregressive models (DAMix). We build a model zoo consisting of autoregressive models trained on data from diverse distributions. In the test phase, we select useful expert models by a simple model evaluation score and adaptively aggregate the predictions of selected models. By assuming the outputs from each expert model are biased in favor of their training distributions, a von Mises-Fisher based filter is proposed to recover the value of unbiased predictions that provides more accurate density estimations than a single model. We derive the posterior of unbiased predictions as well as concentration parameters in the filter, and a novel temporal Stein variational gradient descent for sequential data is proposed to adaptively update the posterior distributions. We evaluate DAMix on 22 image datasets, including in-distribution and OoD data, and demonstrate that making use of unbiased predictions has up to 45.6% improvement over the single model trained on ImageNet.

Similar to variable selection in the linear regression model, selecting significant components in the popular additive regression model is of great interest. However, such components are unknown smooth functions of independent variables, which are unobservable. As such, some approximation is needed. In this paper, we suggest a combination of penalized regression spline approximation and group variable selection, called the lasso-type spline method (LSM), to handle this component selection problem with a diverging number of strongly correlated variables in each group. It is shown that the proposed method can select significant components and estimate nonparametric additive function components simultaneously with an optimal convergence rate simultaneously. To make the LSM stable in computation and able to adapt its estimators to the level of smoothness of the component functions, weighted power spline bases and projected weighted power spline bases are proposed. Their performance is examined by simulation studies across two set-ups with independent predictors and correlated predictors, respectively, and appears superior to the performance of competing methods. The proposed method is extended to a partial linear regression model analysis with real data, and gives reliable results.