Oct 2024, Volume 12 Issue 1

    

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• Weak Solutions of McKean–Vlasov SDEs with Supercritical Drifts

  Xicheng Zhang

  2023, 12(1): 1-14. https://doi.org/10.1007/s40304-021-00277-0

  Consider the following McKean–Vlasov SDE:

  \begin{array}{c}\hfill \text{d}{X}_t=\sqrt{2}\text{d}{W}_t+{\int }_{{\mathbb{R}}^d}K(t,X_t-y){\mu }_{X_t}(\text{d}y)\text{d}t,X_0=x,\end{array}

  \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \textrm{d} X_t=\sqrt{2}\textrm{d} W_t+\int _{{\mathbb {R}}^d}K(t,X_t-y)\mu _{X_t}(\textrm{d} y)\textrm{d} t,\ \ X_0=x, \end{aligned}$$\end{document}

  where

  {\mu }_{X_t}

  \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _{X_t}$$\end{document}

  stands for the distribution of

  X_t

  \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_t$$\end{document}

  and

  K(t,x):{\mathbb{R}}_{+}\times {\mathbb{R}}^d\to {\mathbb{R}}^d

  \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K(t,x): {{\mathbb {R}}}_+\times {{\mathbb {R}}}^d\rightarrow {{\mathbb {R}}}^d$$\end{document}

  is a time-dependent divergence free vector field. Under the assumption

  K\in L_t^q({\tilde{L}}_x^p)

  \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K\in L^q_t({\widetilde{L}}_x^p)$$\end{document}

  with

  \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{d}{p}+\frac{2}{q}<2$$\end{document}

  , where

  {\tilde{L}}_x^p

  \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\widetilde{L}}^p_x$$\end{document}

  stands for the localized

  L^p

  \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^p$$\end{document}

  -space, we show the existence of weak solutions to the above SDE. As an application, we provide a new proof for the existence of weak solutions to 2D Navier–Stokes equations with measure as initial vorticity.

• Identification and Estimation of Generalized Additive Partial Linear Models with Nonignorable Missing Response

  Jierui Du, Yuan Li, Xia Cui

  2023, 12(1): 113-156. https://doi.org/10.1007/s40304-022-00284-9

  The generalized additive partial linear models (GAPLM) have been widely used for flexible modeling of various types of response. In practice, missing data usually occurs in studies of economics, medicine, and public health. We address the problem of identifying and estimating GAPLM when the response variable is nonignorably missing. Three types of monotone missing data mechanism are assumed, including logistic model, probit model and complementary log-log model. In this situation, likelihood based on observed data may not be identifiable. In this article, we show that the parameters of interest are identifiable under very mild conditions, and then construct the estimators of the unknown parameters and unknown functions based on a likelihood-based approach by expanding the unknown functions as a linear combination of polynomial spline functions. We establish asymptotic normality for the estimators of the parametric components. Simulation studies demonstrate that the proposed inference procedure performs well in many settings. We apply the proposed method to the household income dataset from the Chinese Household Income Project Survey 2013.

• Incorporating Relative Error Criterion to Conformal Prediction for Positive Data

  Yuxiang Luo, Yang Wei, Zhouping Li, Bing-Yi Jing

  2023, 12(1): 157-186. https://doi.org/10.1007/s40304-023-00360-8

  Positive data are very common in many scientific fields and applications; for these data, it is known that estimation and inference based on relative error criterion are superior to that of absolute error criterion. In prediction problems, conformal prediction provides a useful framework to construct flexible prediction intervals based on hypothesis testing, which has been actively studied in the past decade. In view of the advantages of the relative error criterion for regression problems with positive responses, in this paper, we combine the relative error criterion (REC) with conformal prediction to develop a novel REC-based predictive inference method to construct prediction intervals for the positive response. The proposed method satisfies the finite sample global coverage guarantee and to some extent achieves the local validity. We conduct extensive simulation studies and two real data analysis to demonstrate the competitiveness of the new proposed method.