For a high-dimensional linear model with a finite number of covariates measured with error, we study statistical inference on the parameters associated with the error-prone covariates, and propose a new corrected decorrelated score test and the corresponding one-step estimator. We further establish asymptotic properties of the newly proposed test statistic and the one-step estimator. Under local alternatives, we show that the limiting distribution of our corrected decorrelated score test statistic is non-central normal. The finite-sample performance of the proposed inference procedure is examined through simulation studies. We further illustrate the proposed procedure via an empirical analysis of a real data example.
In this paper we introduce randomized $t$-type statistics that will be referred to as randomized pivots. We show that these randomized pivots yield central limit theorems with a significantly smaller magnitude of error as compared to that of their classical counterparts under the same conditions. This constitutes a desirable result when a relatively small number of data is available. When a data set is too big to be processed, we use our randomized pivots to make inference about the mean based on significantly smaller sub-samples. The approach taken is shown to relate naturally to estimating d…
We consider a heteroscedastic regression model in which some of the regression coefficients are zero but it is not known which ones. Penalized quantile regression is a useful approach for analyzing such data. By allowing different covariates to be relevant for modeling conditional quantile functions at different quantile levels, it provides a more complete picture of the conditional distribution of a response variable than mean regression. Existing work on penalized quantile regression has been mostly focused on point estimation. Although bootstrap procedures have recently been shown to be eff…
Post-selection inference is a statistical technique for determining salient variables after model or variable selection. Recently, selective inference, a kind of post-selection inference framework, has garnered the attention in the statistics and machine learning communities. By conditioning on a specific variable selection procedure, selective inference can properly control for so-called selective type I error, which is a type I error conditional on a variable selection procedure, without imposing excessive additional computational costs. While selective inference can provide a valid hypothes…
In randomised trials, continuous endpoints are often measured with some degree of error. This study explores the impact of ignoring measurement error, and proposes methods to improve statistical inference in the presence of measurement error. Three main types of measurement error in continuous endpoints are considered: classical, systematic and differential. For each measurement error type, a corrected effect estimator is proposed. The corrected estimators and several methods for confidence interval estimation are tested in a simulation study. These methods combine information about error-pron…
Matrix completion has become an extremely important technique as data scientists are routinely faced with large, incomplete datasets on which they wish to perform statistical inferences. We investigate how error introduced via matrix completion affects statistical inference. Furthermore, we prove recovery error bounds which depend upon the matrix recovery error for several common statistical inferences. We consider matrix recovery via nuclear norm minimization and a variant, $\ell_1$-regularized nuclear norm minimization for data with a structured sampling pattern. Finally, we run a series of …
High-dimensional vector autoregression with measurement error is frequently encountered in a large variety of scientific and business applications. In this article, we study statistical inference of the transition matrix under this model. While there has been a large body of literature studying sparse estimation of the transition matrix, there is a paucity of inference solutions, especially in the high-dimensional scenario. We develop inferential procedures for both the global and simultaneous testing of the transition matrix. We first develop a new sparse expectation-maximization algorithm to…
In various applications of regression analysis, in addition to errors in the dependent observations also errors in the predictor variables play a substantial role and need to be incorporated in the statistical modeling process. In this paper we consider a nonparametric measurement error model of Berkson type with fixed design regressors and centered random errors, which is in contrast to much existing work in which the predictors are taken as random observations with random noise. Based on an estimator that takes the error in the predictor into account and on a suitable Gaussian approximation,…
These notes shows how to do inference on the Demographic Parity (DP) metric. Although the metric is a complex statistic involving min and max computations, we propose a smooth approximation of those functions and derive its asymptotic distribution. The limit of these approximations and their gradients converge to those of the true max and min functions, wherever they exist. More importantly, when the true max and min functions are not differentiable, the approximations still are, and they provide valid asymptotic inference everywhere in the domain. We conclude with some directions on how to co…
This paper describes three methods for carrying out non-asymptotic inference on partially identified parameters that are solutions to a class of optimization problems. Applications in which the optimization problems arise include estimation under shape restrictions, estimation of models of discrete games, and estimation based on grouped data. The partially identified parameters are characterized by restrictions that involve the unknown population means of observed random variables in addition to structural parameters. Inference consists of finding confidence intervals for functions of the stru…
This work studies the properties of the maximum likelihood estimator (MLE) of a non-linear model with Gaussian errors and multidimensional parameter. The observations are collected in a two-stage experimental design and are dependent since the second stage design is determined by the observations at the first stage; the MLE maximizes the total likelihood. Differently from the most of the literature, the first stage sample size is small, and hence asymptotic approximation is used only in the second stage. It is proved that the MLE is consistent and that its asymptotic distribution is a specific…
To perform inference after model selection, we propose controlling the selective type I error; i.e., the error rate of a test given that it was performed. By doing so, we recover long-run frequency properties among selected hypotheses analogous to those that apply in the classical (non-adaptive) context. Our proposal is closely related to data splitting and has a similar intuitive justification, but is more powerful. Exploiting the classical theory of Lehmann and Scheffé (1955), we derive most powerful unbiased selective tests and confidence intervals for inference in exponential family models…
We consider the problem of testing for a difference in means between clusters of observations identified via k-means clustering. In this setting, classical hypothesis tests lead to an inflated Type I error rate. To overcome this problem, we take a selective inference approach. We propose a finite-sample p-value that controls the selective Type I error for a test of the difference in means between a pair of clusters obtained using k-means clustering, and show that it can be efficiently computed. We apply our proposal in simulation, and on hand-written digits data and single-cell RNA-sequencing …
Estimating linear regression using least squares and reporting robust standard errors is very common in financial economics, and indeed, much of the social sciences and elsewhere. For thick tailed predictors under heteroskedasticity this recipe for inference performs poorly, sometimes dramatically so. Here, we develop an alternative approach which delivers an unbiased, consistent and asymptotically normal estimator so long as the means of the outcome and predictors are finite. The new method has standard errors under heteroskedasticity which are easy to reliably estimate and tests which are cl…
In this paper we obtain an adjusted version of the likelihood ratio test for errors-in-variables multivariate linear regression models. The error terms are allowed to follow a multivariate distribution in the class of the elliptical distributions, which has the multivariate normal distribution as a special case. We derive a modified likelihood ratio statistic that follows a chi-squared distribution with a high degree of accuracy. Our results generalize those in Melo and Ferrari(Advances in Statistical Analysis, 2010, 94, 75-87) by allowing the parameter of interest to be vector-valued in the m…
Error-quantified Conformal Inference for Time Series
School of Statistics and Data Science, LPMC, KLMDASR and LEBPS, Nankai University（统计与数据科学学院，LPMC，KLMDASR和LEBPS，南开大学）；Tsinghua Shenzhen International Graduate School, Tsinghua University（清华大学深圳国际研究生院，清华大学）
Statistical inference on graphs is a burgeoning field in the applied and theoretical statistics communities, as well as throughout the wider world of science, engineering, business, etc. In many applications, we are faced with the reality of errorfully observed graphs. That is, the existence of an edge between two vertices is based on some imperfect assessment. In this paper, we consider a graph $G = (V,E)$. We wish to perform an inference task -- the inference task considered here is "vertex classification". However, we do not observe $G$; rather, for each potential edge $uv \in {{V}\choose{2…
We propose a statistical framework to investigate whether a given subpopulation lies between two other subpopulations in a multivariate feature space. This methodology is motivated by a biological question from a collaborator: Is a newly discovered cell type between two known types in several given features? We propose two in-betweenness indices (IBI) to quantify the in-betweenness exhibited by a random triangle formed by the summary statistics of the three subpopulations. Statistical inference methods are provided for triangle shape and IBI metrics. The application of our methods is demonstra…
We study semiparametric varying-coefficient partially linear models when some linear covariates are not observed, but ancillary variables are available. Semiparametric profile least-square based estimation procedures are developed for parametric and nonparametric components after we calibrate the error-prone covariates. Asymptotic properties of the proposed estimators are established. We also propose the profile least-square based ratio test and Wald test to identify significant parametric and nonparametric components. To improve accuracy of the proposed tests for small or moderate sample size…
Inference and prediction under the sparsity assumption have been a hot research topic in recent years. However, in practice, the sparsity assumption is difficult to test, and more importantly can usually be violated. In this paper, to study hypothesis test of any group of parameters under non-sparse high-dimensional linear models, we transform the null hypothesis to a testable moment condition and then use the self-normalization structure to construct moment test statistics under one-sample and two-sample cases, respectively. Compared to the one-sample case, the two-sample additionally require…
This chapter demystifies P-values, hypothesis tests and significance tests, and introduces the concepts of local evidence and global error rates. The local evidence is embodied in \textit{this} data and concerns the hypotheses of interest for \textit{this} experiment, whereas the global error rate is a property of the statistical analysis and sampling procedure. It is shown using simple examples that local evidence and global error rates can be, and should be, considered together when making inferences. Power analysis for experimental design for hypothesis testing are explained, along with the…
This paper studies higher-order inference properties of nonparametric local polynomial regression methods under random sampling. We prove Edgeworth expansions for $t$ statistics and coverage error expansions for interval estimators that (i) hold uniformly in the data generating process, (ii) allow for the uniform kernel, and (iii) cover estimation of derivatives of the regression function. The terms of the higher-order expansions, and their associated rates as a function of the sample size and bandwidth sequence, depend on the smoothness of the population regression function, the smoothness ex…
This note is an extended review of the book Error and Inference, edited by Deborah Mayo and Aris Spanos, about their frequentist and philosophical perspective on testing of hypothesis and on the criticisms of alternatives like the Bayesian approach.
I propose a novel approach to simultaneous inference that alleviates the need to specify a correlational structure among marginal errors. The vacuous orientation assumption retains what the normal i.i.d. assumption implies about the distribution of error configuration, but relaxes the implication that the error orientation is isotropic. When a large number of highly dependent hypotheses are tested simultaneously, the proposed model produces calibrated posterior inference by leveraging the logical relationship among them. This stands in contrast to the conservative performance of the Bonferroni…
A Gaussian measurement error assumption, i.e., an assumption that the data are observed up to Gaussian noise, can bias any parameter estimation in the presence of outliers. A heavy tailed error assumption based on Student's t distribution helps reduce the bias. However, it may be less efficient in estimating parameters if the heavy tailed assumption is uniformly applied to all of the data when most of them are normally observed. We propose a mixture error assumption that selectively converts Gaussian errors into Student's t errors according to latent outlier indicators, leveraging the best of …
This paper describes a method for performing inference on models chosen by cross-validation. When the test error being minimized in cross-validation is a residual sum of squares it can be written as a quadratic form. This allows us to apply the inference framework in Loftus et al. (2015) for models determined by quadratic constraints to the model that minimizes CV test error. Our only requirement on the model training pro- cedure is that its selection events are regions satisfying linear or quadratic constraints. This includes both Lasso and forward stepwise, which serve as our main examples t…
The paper presents a hierarchical Bayesian model for simultaneous inference of tournament graphs and informant error. From multiple informant reports or measurement instrument outputs, the model estimates the structure of a criterion (i.e., true) tournament graph with possibly tied outcomes. Tournament graphs with possibly tied outcomes are graphs in which there are three possible states for each unordered pair of graph nodes: node i wins and node j loses; node j wins and node i loses; neither node wins (i.e., tied outcome). The model also estimates the rates at which individual informants (or…
In this paper the Bayesian analysis is applied to assign a probability density to the value of a quantity having a definite sign. This analysis is logically consistent with the results, positive or negative, of repeated measurements. Results are used to estimate the atom density shift in a caesium fountain clock. The comparison with the classical statistical analysis is also reported and the advantages of the Bayesian approach for the realization of the time unit are discussed.
Statistical inference for discretely observed jump-diffusion processes is a complex problem which motivates new methodological challenges. Thus existing approaches invariably resort to time-discretisations which inevitably lead to approximations in inference. In this paper, we give the first general collection of methodologies for exact (in this context meaning discretisation-free) likelihood-based inference for discretely observed finite activity jump-diffusions. The only sources of error involved are Monte Carlo error and convergence of EM or MCMC algorithms. We shall introduce both frequent…
We develop methodology for causal inference in observational studies when using propensity score subclassification on data constructed with probabilistic record linkage techniques. We focus on scenarios where covariates and binary treatment assignments are in one file and outcomes are in another file, and the goal is to estimate an additive treatment effect by merging the files. We assume that the files can be linked using variables common to both files, e.g., names or birth dates, but that links are subject to errors, e.g., due to reporting errors in the linking variables. We develop methodol…
A great deal of inference in statistics is based on making the approximation that a statistic is normally distributed. The error in doing so is generally $O(n^{-1/2})$ and can be very considerable when the distribution is heavily biased or skew. This note shows how one may reduce this error to $O(n^{-(j+1)/2})$, where $j$ is a given integer. The case considered is when the statistic is the mean of the sample values from a continuous one-parameter distribution, after the sample has undergone an initial transformation.
Huazhong Agricultural University（华中农业大学）；Xi’an Jiaotong University（西安交通大学）；Southern University of Science and Technology（南方科技大学）
Recently, the scheme of model-X knockoffs was proposed as a promising solution to address controlled feature selection under high-dimensional finite-sample settings. However, the procedure of model-X knockoffs depends heavily on the coefficient-based feature importance and only concerns the control of false discovery rate (FDR). To further improve its adaptivity and flexibility, in this paper, we propose an error-based knockoff inference method by integrating the knockoff features, the error-based feature importance statistics, and the stepdown procedure together. The proposed inference proced…
Measurement Error-Robust Causal Inference via Constructed Instrumental Variables
Columbia Mailman School of Public Health（哥伦比亚大学麦金农公共卫生学院）；Department of Biostatistics, Columbia University（哥伦比亚大学生物统计系）；Harvard T.H. Chan School of Public Health（哈佛大学陈曾熙公共卫生学院）
We present a framework for inference for spatial processes that have actual values imperfectly represented by data. Environmental processes represented as spatial fields, either at fixed time points, or aggregated over fixed time periods, are studied. Data from both measurements and simulations performed by complex computer models are used to infer actual values of the spatial fields. Methods from geostatistics and statistical emulation are used to explicitly capture discrepancies between a spatial field's actual and simulated values. A geostatistical model captures spatial discrepancy: the di…
Measurement error in a covariate or the outcome of regression models is common, but is often ignored, even though measurement error can lead to substantial bias in the estimated covariate-outcome association. While several texts on measurement error correction methods are available, these methods remain seldomly applied. To improve the use of measurement error correction methodology, we developed mecor, an R package that implements measurement error correction methods for regression models with continuous outcomes. Measurement error correction requires information about the measurement error m…
University of California, Los Angeles（加利福尼亚大学洛杉矶分校）
This paper addresses the problem of measurement errors in causal inference and highlights several algebraic and graphical methods for eliminating systematic bias induced by such errors. In particulars, the paper discusses the control of partially observable confounders in parametric and non parametric models and the computational problem of obtaining bias-free effect estimates in such models.
University College London（伦敦大学学院）；University of Oxford（牛津大学）；The Alan Turing Institute（艾伦·图灵研究所）
We propose a kernel-based nonparametric estimator for the causal effect when the cause is corrupted by error. We do so by generalizing estimation in the instrumental variable setting. Despite significant work on regression with measurement error, additionally handling unobserved confounding in the continuous setting is non-trivial: we have seen little prior work. As a by-product of our investigation, we clarify a connection between mean embeddings and characteristic functions, and how learning one simultaneously allows one to learn the other. This opens the way for kernel method research to le…
