Bootstrap method for misspecified ergodic Lévy driven stochastic differential equation models

A Bootstrap Lasso + Partial Ridge Method to Construct Confidence Intervals for Parameters in High-dimensional Sparse Linear Models

Constructing confidence intervals for the coefficients of high-dimensional sparse linear models remains a challenge, mainly because of the complicated limiting distributions of the widely used estimators, such as the lasso. Several methods have been developed for constructing such intervals. Bootstrap lasso+ols is notable for its technical simplicity, good interpretability, and performance that is comparable with that of other more complicated methods. However, bootstrap lasso+ols depends on the beta-min assumption, a theoretic criterion that is often violated in practice. Thus, we introduce a…

Variance estimation in pseudo-expected estimating equations for missing data

Missing data is a common challenge in biomedical research. This fact, along with growing dataset volumes of the modern era, make the issue of computationally-efficient analysis with missing data of crucial practical importance. A general computationally-efficient estimation framework for dealing with missing data is the pseudo-expected estimating equations (PEEE) approach. The method is applicable with any parametric model for which estimation involves the solution of a set of estimating equations, such as likelihood score equations. A key limitation of the PEEE approach is that there is curre…

Schwartz type model selection for ergodic stochastic differential equation models

We study the construction of the theoretical foundation of model comparison for ergodic stochastic differential equation (SDE) models and an extension of the applicable scope of the conventional Bayesian information criterion. Different from previous studies, we suppose that the candidate models are possibly misspecified models, and we consider both Wiener and a pure-jump Lévy noise driven SDE. Based on the asymptotic behavior of the marginal quasi-log likelihood, the Schwarz type statistics and stepwise model selection procedure are proposed. We also prove the model selection consistency of t…

Generalized regression operator estimation for continuous time functional data processes with missing at random response

In this paper, we are interested in nonparametric kernel estimation of a generalized regression function, including conditional cumulative distribution and conditional quantile functions, based on an incomplete sample $(X_t, Y_t, ζ_t)_{t\in \mathbb{ R}^+}$ copies of a continuous-time stationary ergodic process $(X, Y, ζ)$. The predictor $X$ is valued in some infinite-dimensional space, whereas the real-valued process $Y$ is observed when $ζ= 1$ and missing whenever $ζ= 0$. Pointwise and uniform consistency (with rates) of these estimators as well as a central limit theorem are established. Con…

M-estimation in GARCH Models in the Absence of Higher-Order Moments

We consider a class of M-estimators of the parameters of a GARCH (p,q) model. These estimators involve score functions and, for adequate choices of the score functions, are asymptotically normal under milder moment assumptions than the usual quasi maximum likelihood, which makes them more reliable in the presence of heavy tails. We also consider weighted bootstrap approximations of the distributions of these M-estimators and establish their validity. Through extensive simulations, we demonstrate the robustness of these M-estimators under heavy tails and conduct a comparative study of the perfo…

通过交叉验证实现高维模型平均

High-Dimensional Model Averaging via Cross-Validation

本文提出交叉验证驱动的高维模型平均方法，结合一般损失与正则化，证明错误设定下的最优性、正确模型下的近 oracle 速率，并给出去偏推断与 FGMA 快速算法。

CLT 与 Edgeworth 展开：m-out-of-n 自助法对学生化中位数估计量的应用

CLT and Edgeworth Expansion for m-out-of-n Bootstrap Estimators of The Studentized Median

本文为 m-out-of-n 自助法估计样本分位数建立了无参数保证，证明中心极限定理并给出 Edgeworth 展开与 Berry-Esseen 界，且应用于 MCMC 和 MDP 等现代任务。

Assessing the global and local uncertainty in scientific evidence in the presence of model misspecification

Scientists need to compare the support for models based on observed phenomena. The main goal of the evidential paradigm is to quantify the strength of evidence in the data for a reference model relative to an alternative model. This is done via an evidence function, such as $ΔSIC$, an estimator of the sample size scaled difference of divergences between the generating mechanism and the competing models. To use evidence, either for decision making or as a guide to the accumulation of knowledge, an understanding of the uncertainty in the evidence is needed. This uncertainty is well characterized…

Long-term stability of sequential Monte Carlo methods under verifiable conditions

This paper discusses particle filtering in general hidden Markov models (HMMs) and presents novel theoretical results on the long-term stability of bootstrap-type particle filters. More specifically, we establish that the asymptotic variance of the Monte Carlo estimates produced by the bootstrap filter is uniformly bounded in time. On the contrary to most previous results of this type, which in general presuppose that the state space of the hidden state process is compact (an assumption that is rarely satisfied in practice), our very mild assumptions are satisfied for a large class of HMMs wit…

Parametric estimation of stochastic differential equations via online gradient descent

We propose an online parametric estimation method of stochastic differential equations with discrete observations and misspecified modelling based on online gradient descent. Our study provides uniform upper bounds for the risks of the estimators over a family of stochastic differential equations. The derivation of the bounds involves three underlying theoretical results: the analysis of the stochastic mirror descent algorithm based on dependent and biased subgradients, the simultaneous exponential ergodicity of classes of diffusion processes, and the proposal of loss functions whose approxima…