由于对参数模型错误设定的担忧，在评估暴露对结局的因果效应时，人们希望利用机器学习来调整混杂。遗憾的是，依赖机器学习预测的暴露效应估计量通常会受到所谓的插件偏差影响，这会使朴素的 p 值和置信区间失效。通过 targeted minimum loss estimation（靶向最小损失估计）以及较新的 double machine learning（双重机器学习）等方案，研究已取得进展；这些方法依赖于同时学习结局和暴露的条件均值。只要两类预测都（足够快地）收敛到真值，就可以获得有效的推断。本文聚焦部分线性回归模型，证明机器学习技术的一种特定实现所产生的暴露效应估计量，即使其中一个第一阶段预测未收敛到真值，也能具有较小偏差。由此得到的检验和置信区间具有双重稳健性。我们还表明，当仅一致估计一个冗余参数时，所提出的估计量可能不具备正则性；尽管如此，模拟研究观察到，在中等到较大样本中，本文方案能够降低偏差并改善置信区间覆盖率。
英文摘要：
Due to concerns about parametric model misspecification, there is interest in using machine learning to adjust for confounding when evaluating the causal effect of an exposure on an outcome. Unfortunately, exposure effect estimators that rely on machine learning predictions are generally subject to so-called plug-in bias, which can render naive $p$-values and confidence intervals invalid. Progress has been made via proposals like targeted minimum loss estimation and more recently double machine learning, which rely on learning the conditional mean of both the outcome and exposure. Valid inference can then be obtained so long as both predictions converge (sufficiently fast) to the truth. Focusing on partially linear regression models, we show that a specific implementation of the machine learning techniques can yield exposure effect estimators that have small bias even when one of the first-stage predictions does not converge to the truth. The resulting tests and confidence intervals are doubly robust. We also show that the proposed estimators may fail to be regular when only one nuisance parameter is consistently estimated; nevertheless, we observe in simulation studies that our proposal can lead to reduced bias and improved confidence interval coverage in moderate-to-large samples.
Consider semiparametric estimation where a doubly robust estimating function for a low-dimensional parameter is available, depending on two working models. With high-dimensional data, we develop regularized calibrated estimation as a general method for estimating the parameters in the two working models, such that valid Wald confidence intervals can be obtained for the parameter of interest under suitable sparsity conditions if either of the two working models is correctly specified. We propose a computationally tractable two-step algorithm and provide rigorous theoretical analysis which justi…
In fitting a mixture of linear regression models, normal assumption is traditionally used to model the error and then regression parameters are estimated by the maximum likelihood estimators (MLE). This procedure is not valid if the normal assumption is violated. To relax the normal assumption on the error distribution hence reduce the modeling bias, we propose semiparametric mixture of linear regression models with unspecified error distributions. We establish a more general identifiability result under weaker conditions than existing results, construct a class of new estimators, and establis…
Semiparametric Copula Quantile Regression for Complete or Censored Data
当面对多元协变量时，一般的半参数回归技术可以提出灵活的模型，这些模型不受维数灾难的影响。在这项工作中，半参数Copula为基础的估计条件分位数完全或右删失数据进行了研究。在精神上，该方法扩展了Noh et al.（2013）和Noh et al.（2015）的近期工作，因为主要思想在于根据多变量copula和边缘分布适当定义分位数回归。事先估计后者和简单的插件导致一个容易实现的估计表示，无论上下文中有或没有审查，作为一个加权分位数的观察到的响应变量。此外，与文献中的初始建议相反，研究了多变量Copula密度的半参数估计方案，其动机是纯参数方法的可能缺点，并由回归上下文驱动。由此得到的分位数回归估计具有在分位数水平上自动单调的有价值的性质，并且在经典的正则条件下得到了完全数据和删失数据的渐近正态性。最后，数值例子以及实际数据的应用，以说明所提出的方法的有效性和有限样本性能。
One fundamental statistical question for research areas such as precision medicine and health disparity is about discovering effect modification of treatment or exposure by observed covariates. We propose a semiparametric framework for identifying such effect modification. Instead of using the traditional outcome models, we directly posit semiparametric models on contrasts, or expected differences of the outcome under different treatment choices or exposures. Through semiparametric estimation theory, all valid estimating equations, including the efficient scores, are derived. Besides doubly ro…
This research is motivated by discovering and underpinning genetic causes for the progression of a bilateral eye disease, Age-related Macular Degeneration (AMD), of which the primary outcomes, progression times to late-AMD, are bivariate and interval-censored due to intermittent assessment times. We propose a novel class of copula-based semiparametric transformation models for bivariate data under general interval censoring, which includes the case 1 interval censoring (current status data) and case 2 interval censoring. Specifically, the joint likelihood is modeled through a two-parameter Arc…
We propose a new class of semiparametric regression models of mean residual life for censored outcome data. The models, which enable us to estimate the expected remaining survival time and generalize commonly used mean residual life models, also conduct covariate dimension reduction. Using the geometric approaches in semiparametrics literature and the martingale properties with survival data, we propose a flexible inference procedure that relaxes the parametric assumptions on the dependence of mean residual life on covariates and how long a patient has lived. We show that the estimators for th…
Analyzing polytomous response from a complex survey scheme, like stratified or cluster sampling is very crucial in several socio-economics applications. We present a class of minimum quasi weighted density power divergence estimators for the polytomous logistic regression model with such a complex survey. This family of semiparametric estimators is a robust generalization of the maximum quasi weighted likelihood estimator exploiting the advantages of the popular density power divergence measure. Accordingly robust estimators for the design effects are also derived. Robust testing of general li…
Structural failure time models are causal models for estimating the effect of time-varying treatments on a survival outcome. G-estimation and artificial censoring have been proposed to estimate the model parameters in the presence of time-dependent confounding and administrative censoring. However, most of existing methods require manually preprocessing data into regularly spaced data, which may invalidate the subsequent causal analysis. Moreover, the computation and inference are challenging due to the non-smoothness of artificial censoring. We propose a class of continuous-time structural fa…
"For how many days during the past 30 days was your mental health not good?" The responses to this question measure self-reported mental health and can be linked to important covariates in the National Health and Nutrition Examination Survey (NHANES). However, these count variables present major distributional challenges: the data are overdispersed, zero-inflated, bounded by 30, and heaped in five- and seven-day increments. To meet these challenges, we design a semiparametric estimation and inference framework for count data regression. The data-generating process is defined by simultaneously …
Interval-censored multi-state data arise in many studies of chronic diseases, where the health status of a subject can be characterized by a finite number of disease states and the transition between any two states is only known to occur over a broad time interval. We formulate the effects of potentially time-dependent covariates on multi-state processes through semiparametric proportional intensity models with random effects. We adopt nonparametric maximum likelihood estimation (NPMLE) under general interval censoring and develop a stable expectation-maximization (EM) algorithm. We show that …
Estimating the effective dimension reduction (EDR) space, related to the semiparametric regression model introduced by Li \cite{sir}, is based on the estimation of the covariance matrix $Λ$ of the conditional expectation of the vector of predictors given the response. An estimator $\widehatΛ_n$ of $Λ$ based on kernel method was introduced by Zhu and Fang \cite{Asymptotics} who then derived, under some conditions, the asymptotic distribution of $\sqrt{n}\left(\widehatΛ_n-Λ\right)$, as $n\rightarrow +\infty$. In this paper, we obtain, under specified conditions, the almost sure convergence of $\…
We consider semiparametric transformation models, where after pre-estimation of a parametric transformation of the response the data are modeled by means of nonparametric regression. We suggest subsequent procedures for testing lack-of-fit of the regression function and for significance of covariables, which - in contrast to procedures from the literature - are asymptotically not influenced by the pre-estimation of the transformation. The test statistics are asymptotically pivotal and have the same asymptotic distribution as in regression models without transformation. We show validity of a mu…
Institute of Statistics and Data Science, Nanjing Audit University（南京审计大学统计与数据科学学院）
China has made great achievements in electric power industry during the long-term deepening of reform and opening up. However, the complex regional economic, social and natural conditions, electricity resources are not evenly distributed, which accounts for the electricity deficiency in some regions of China. It is desirable to develop a robust electricity forecasting model. Motivated by which, we propose a Panel Semiparametric Quantile Regression Neural Network (PSQRNN) by utilizing the artificial neural network and semiparametric quantile regression. The PSQRNN can explore a potential linear…
Parameter shrinkage applied optimally can always reduce error and projection variances from those of maximum likelihood estimation. Many variables that actuaries use are on numerical scales, like age or year, which require parameters at each point. Rather than shrinking these towards zero, nearby parameters are better shrunk towards each other. Semiparametric regression is a statistical discipline for building curves across parameter classes using shrinkage methodology. It is similar to but more parsimonious than cubic splines. We introduce it in the context of Bayesian shrinkage and apply it …
Triangular systems with nonadditively separable unobserved heterogeneity provide a theoretically appealing framework for the modelling of complex structural relationships. However, they are not commonly used in practice due to the need for exogenous variables with large support for identification, the curse of dimensionality in estimation, and the lack of inferential tools. This paper introduces two classes of semiparametric nonseparable triangular models that address these limitations. They are based on distribution and quantile regression modelling of the reduced form conditional distributio…
In this review we cover the basics of efficient nonparametric parameter estimation (also called functional estimation), with a focus on parameters that arise in causal inference problems. We review both efficiency bounds (i.e., what is the best possible performance for estimating a given parameter?) and the analysis of particular estimators (i.e., what is this estimator's error, and does it attain the efficiency bound?) under weak assumptions. We emphasize minimax-style efficiency bounds, worked examples, and practical shortcuts for easing derivations. We gloss over most technical details, in …
We consider identification and estimation with an outcome missing not at random (MNAR). We study an identification strategy based on a so-called shadow variable. A shadow variable is assumed to be correlated with the outcome, but independent of the missingness process conditional on the outcome and fully observed covariates. We describe a general condition for nonparametric identification of the full data law under MNAR using a valid shadow variable. Our condition is satisfied by many commonly-used models; moreover, it is imposed on the complete cases, and therefore has testable implications w…
It is well known that the minimax rates of convergence of nonparametric density and regression function estimation of a random variable measured with error is much slower than the rate in the error free case. Surprisingly, we show that if one is willing to impose a relatively mild assumption in requiring that the error-prone variable has a compact support, then the results can be greatly improved. We describe new and constructive methods to take full advantage of the compact support assumption via spline-assisted semiparametric methods. We further prove that the new estimator achieves the usua…
