This study proposes a hierarchical Bayes regression model designed to distinguish and estimate the multiple heterogeneous regression coefficients for a single explanatory variable, such as the heterogeneity in product offerings and time. We examine the effectiveness of the proposed model by both numerical experiments and analysis using actual POS data. In the numerical experiment, the proposed estimation algorithm's effectiveness is shown in that the estimated model based on the simulated data can reproduce the original model with high accuracy. Besides, this paper reports POS data analysis to show how the proposed framework works in actual data well by distinguishing products and time heterogeneities in the market responses. The proposed framework achieves the factor decomposition of a single market response explicitly and can be effectively utilized in modeling various social science phenomena and decision making based on the models.

In this article, we introduce a study which presents a minimax error rate analysis of nonparametric regression aimed at elucidating the superiority of deep neural networks over standard methods. In the problem of nonparametric regression, it is well known that many standard methods achieve minimax optimal rates of generalization error for smooth functions, and it is not easy to reveal the theoretical advantage of deep neural networks. The work presented in this paper fills this theoretical gap by considering estimation for a class of non-smooth functions with singularities on hypersurfaces. The results obtained are as follows: (i) Analyzes the generalization error of the function estimator by deep neural networks and prove that its convergence rate is optimal (excluding logarithmic order effects). (ii) Identifies situations in which the deep neural network outperforms standard methods such as the kernel method, Gaussian process method, etc., then constructs a phase diagram with respect to shape parameters for smoothness and singularity. The superiority of this deep neural network comes from the fact that its multilayer structure can properly handle the shape of singularities.

We consider the statistical inference for time series from the perspective of the prediction problem. We start with the harmonizable stable processes to deal with the prediction problem in a unified manner. The prediction and interpolation error can be expressed as a functional of the spectral density function of the process. A contrast function, based on the prediction and interpolation error between a parametric spectral density and the true spectral density, can be introduced when we fit the parametric one to the true one. The parameters in the time series model are estimated by the minimum contrast estimation, since many probability densities of stable distribution do not have the closed form, which hampers the use of the maximum likelihood. The consistency and the asymptotic distribution of the minimum contrast estimators are shown. We also discuss the empirical likelihood method for the pivotal quantity of the time series model and provide the recent research achievement in this area.