In this paper we consider a class of nonparametric estimators of a distribution function F, with compact support, based on the theory of IFSs. The estimator of F is tought as the fixed point of a contractive operator T defined in terms of a vector of parameters p and a family of affine maps W which can be both depend of the sample (X_1, X_2, ...., X_n). Given W, the problem consists in finding a vector p such that the fixed point of T is ``sufficiently near'' to F. It turns out that this is a quadratic constrained optimization problem that we propose to solve by penalization techniques. If F has a density f, we can also provide an estimator of f based on Fourier techniques. IFS estimators for F are asymptotically equivalent to the empirical distribution function (e.d.f.) estimator. We will study relative efficiency of the IFS estimators with respect to the e.d.f. for small samples via Monte Carlo approach.
For well behaved distribution functions F and for a particular family of so-called wavelet maps the IFS estimators can be dramatically better than the e.d.f. (or the kernel estimator for density estimation) in presence of missing data, i.e. when it is only possibile to observe data on subsets of the whole support of F.
This research has also produced a free package for the R statistical environment which is ready to be used in applications.
In this paper, we consider a weighted local linear estimator based on the inverse selection probability for nonparametric regression with missing covariates at random. The asymptotic distribution of the maximal deviation between the estimator and the true regression function is derived and an asymptotically accurate simultaneous confidence band is constructed. The estimator for the regression function is shown to be oracally efficient in the sense that it is uniformly indistinguishable from that when the selection probabilities are known. Finite sample performance is examined via simulation st…
We consider an empirical likelihood inference for parameters defined by general estimating equations when some components of the random observations are subject to missingness. As the nature of the estimating equations is wide-ranging, we propose a nonparametric imputation of the missing values from a kernel estimator of the conditional distribution of the missing variable given the always observable variable. The empirical likelihood is used to construct a profile likelihood for the parameter of interest. We demonstrate that the proposed nonparametric imputation can remove the selection bias …
kNN estimation in semi-functional partial linear regression with missing responses at random
Universidade da Coruña（拉科鲁尼亚大学）；Centro de Investigación en Tecnologías de la Información y las Comunicaciones (CITIC)（信息与通信技术研究中心）；Universidad Carlos III de Madrid（马德里卡洛斯三世大学）
This article considers nonparametric regression models with multivariate covariates and with responses missing at random. We estimate the regression function with a local polynomial smoother. The residual-based empirical distribution function that only uses complete cases, i.e. residuals that can actually be constructed from the data, is shown to be efficient in the sense of Hájek and Le Cam. In the proofs we derive, more generally, the efficient influence function for estimating an arbitrary linear functional of the error distribution; this covers the distribution function as a special case. …
We consider nonparametric estimation for functional autoregressive processes with Markov switching. First, we study the case where complete data is available; i.e. when we observe the Markov switching regime. Then we estimate the regression function in each regime using a Nadaraya-Watson type estimator. Second, we introduce a nonparametric recursive algorithm in the case of hidden Markov switching regime. Our algorithm restores the missing data by means of a Monte-Carlo step and estimate the regression function via a Robbins-Monro step. Consistency of the estimators are proved in both cases. F…
Missingness in categorical data is a common problem in various real applications. Traditional approaches either utilize only the complete observations or impute the missing data by some ad hoc methods rather than the true conditional distribution of the missing data, thus losing or distorting the rich information in the partial observations. In this paper, we propose the Dirichlet Process Mixture of Collapsed Product-Multinomials (DPMCPM) to model the full data jointly and compute the model efficiently. By fitting an infinite mixture of product-multinomial distributions, DPMCPM is applicable f…
Quality of Life (QOL) outcomes are important in the management of chronic illnesses. In studies of efficacies of treatments or intervention modalities, QOL scales multi-dimensional constructs are routinely used as primary endpoints. The standard data analysis strategy computes composite (average) overall and domain scores, and conducts a mixed-model analysis for evaluating efficacy or monitoring medical conditions as if these scores were in continuous metric scale. However, assumptions of parametric models like continuity and homoscedastivity can be violated in many cases. Furthermore, it is e…
In this paper we propose an autoregressive wild bootstrap method to construct confidence bands around a smooth deterministic trend. The bootstrap method is easy to implement and does not require any adjustments in the presence of missing data, which makes it particularly suitable for climatological applications. We establish the asymptotic validity of the bootstrap method for both pointwise and simultaneous confidence bands under general conditions, allowing for general patterns of missing data, serial dependence and heteroskedasticity. The finite sample properties of the method are studied in…
Given $n$ samples from a population of individuals belonging to different types with unknown proportions, how do we estimate the probability of discovering a new type at the $(n+1)$-th draw? This is a classical problem in statistics, commonly referred to as the missing mass estimation problem. Recent results by Ohannessian and Dahleh \citet{Oha12} and Mossel and Ohannessian \citet{Mos15} showed: i) the impossibility of estimating (learning) the missing mass without imposing further structural assumptions on the type proportions; ii) the consistency of the Good-Turing estimator for the missing …
The modeling of dependence between maxima is an important subject in several applications in risk analysis. To this aim, the extreme value copula function, characterised via the madogram, can be used as a margin-free description of the dependence structure. From a practical point of view, the family of extreme value distributions is very rich and arise naturally as the limiting distribution of properly normalised component-wise maxima. In this paper, we investigate the nonparametric estimation of the madogram where data are completely missing at random. We provide the functional central limit …
We consider identification and inference about mean functionals of observed covariates and an outcome variable subject to nonignorable missingness. By leveraging a shadow variable, we establish a necessary and sufficient condition for identification of the mean functional even if the full data distribution is not identified. We further characterize a necessary condition for $\sqrt{n}$-estimability of the mean functional. This condition naturally strengthens the identifying condition, and it requires the existence of a function as a solution to a representer equation that connects the shadow va…
Statistical Laboratory, University of Cambridge（剑桥大学统计实验室）；Department of Data Sciences and Operations, University of Southern California（南加州大学数据科学与操作系）；Department of Statistics, University of Chicago（芝加哥大学统计学系）
Doubly Robust Machine Learning for Population Size Estimation with Missing Covariates: Application to Gaza Conflict Mortality
Center for Data Science New York University（纽约大学数据科学中心）；Department of Statistics & Data Science Carnegie Mellon University（卡内基梅隆大学统计与数据科学系）；Department of Medical Statistics London School of Hygiene and Tropical Medicine（伦敦卫生与热带医学院医学统计学系）
Identifying a biomarker or treatment-dose threshold that marks a specified level of risk is an important problem, especially in clinical trials. This risk, viewed as a function of thresholds and possibly adjusted for covariates, we call the threshold-response function. Extending the work of Donovan, Hudgens and Gilbert (2019), we propose a nonparametric efficient estimator for the covariate-adjusted threshold-response function, which utilizes machine learning and Targeted Minimum-Loss Estimation (TMLE). We additionally propose a more general estimator, based on sequential regression, that also…
Pattern-mixture models provide a transparent approach for handling missing data, where the full-data distribution is factorized in a way that explicitly shows the parts that can be estimated from observed data alone, and the parts that require identifying restrictions. We introduce a nonparametric estimator of the full-data distribution based on the pattern-mixture model factorization. Our approach uses the empirical observed-data distribution and augments it with a nonparametric estimator of the missing-data distributions under a given identifying restriction. Our results apply to a large cla…
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…
We propose a new reconstruction operator that aims to recover the missing parts of a function given the observed parts. This new operator belongs to a new, very large class of functional operators which includes the classical regression operators as a special case. We show the optimality of our reconstruction operator and demonstrate that the usually considered regression operators generally cannot be optimal reconstruction operators. Our estimation theory allows for autocorrelated functional data and considers the practically relevant situation in which each of the $n$ functions is observed a…
Patient-reported outcome (PRO) measures are increasingly collected as a means of measuring healthcare quality and value. The capability to predict such measures enables patient-provider shared decision making and the delivery of patient-centered care. However, due to their voluntary nature, PRO measures often suffer from a high missing rate, and the missingness may depend on many patient factors. Under such a complex missing mechanism, statistical inference of the parameters in prediction models for PRO measures is challenging, especially when flexible imputation models such as machine learnin…
The standard quantile regression model assumes a linear relationship at the quantile of interest and that all variables are observed. We relax these assumptions by considering a partial linear model while allowing for missing linear covariates. To handle the potential bias caused by missing data we propose a weighted objective function using inverse probability weighting. Our work examines estimators using parametric and nonparametric estimates of the missing probability. For variable selection of the linear terms in the presence of missing data we consider a penalized and weighted objective f…
We introduce a novel approach to estimation problems in settings with missing data. Our proposal -- the Correlation-Assisted Missing data (CAM) estimator -- works by exploiting the relationship between the observations with missing features and those without missing features in order to obtain improved prediction accuracy. In particular, our theoretical results elucidate general conditions under which the proposed CAM estimator has lower mean squared error than the widely used complete-case approach in a range of estimation problems. We showcase in detail how the CAM estimator can be applied t…
Practical problems with missing data are common, and statistical methods have been developed concerning the validity and/or efficiency of statistical procedures. On a central focus, there have been longstanding interests on the mechanism governing data missingness, and correctly deciding the appropriate mechanism is crucially relevant for conducting proper practical investigations. The conventional notions include the three common potential classes -- missing completely at random, missing at random, and missing not at random. In this paper, we present a new hypothesis testing approach for deci…
Dirichlet kernel density estimation on the simplex with missing data
Imam Mohammad Ibn Saud Islamic University（伊玛目穆罕默德·本·沙特伊斯兰大学）；Université de Tunis El-Manar（突尼斯埃尔曼纳尔大学）；Dalhousie University（达尔豪斯大学）；Université du Québec à Trois-Rivières（魁北克特罗瓦里维尔斯大学）
The estimation of coverage probabilities, and in particular of the missing mass, is a classical statistical problem with applications in numerous scientific fields. In this paper, we study this problem in relation to randomized data compression, or sketching. This is a novel but practically relevant perspective, and it refers to situations in which coverage probabilities must be estimated based on a compressed and imperfect summary, or sketch, of the true data, because neither the full data nor the empirical frequencies of distinct symbols can be observed directly. Our contribution is a Bayesi…
Given a set of incomplete observations, we study the nonparametric problem of testing whether data are Missing Completely At Random (MCAR). Our first contribution is to characterise precisely the set of alternatives that can be distinguished from the MCAR null hypothesis. This reveals interesting and novel links to the theory of Fréchet classes (in particular, compatible distributions) and linear programming, that allow us to propose MCAR tests that are consistent against all detectable alternatives. We define an incompatibility index as a natural measure of ease of detectability, establish it…
Feature allocation models generalize species sampling models by allowing every observation to belong to more than one species, now called features. Under the popular Bernoulli product model for feature allocation, given $n$ samples, we study the problem of estimating the missing mass $M_{n}$, namely the expected number hitherto unseen features that would be observed if one additional individual was sampled. This is motivated by numerous applied problems where the sampling procedure is expensive, in terms of time and/or financial resources allocated, and further samples can be only motivated by…
This paper proposes a simple and efficient estimation procedure for the model with non-ignorable missing data studied by Morikawa and Kim (2016). Their semiparametrically efficient estimator requires explicit nonparametric estimation and so suffers from the curse of dimensionality and requires a bandwidth selection. We propose an estimation method based on the Generalized Method of Moments (hereafter GMM). Our method is consistent and asymptotically normal regardless of the number of moments chosen. Furthermore, if the number of moments increases appropriately our estimator can achieve the sem…
We carry out ANOVA comparisons of multiple treatments for longitudinal studies with missing values. The treatment effects are modeled semiparametrically via a partially linear regression which is flexible in quantifying the time effects of treatments. The empirical likelihood is employed to formulate model-robust nonparametric ANOVA tests for treatment effects with respect to covariates, the nonparametric time-effect functions and interactions between covariates and time. The proposed tests can be readily modified for a variety of data and model combinations, that encompasses parametric, semip…
Missing data occur frequently in empirical studies in health and social sciences, often compromising our ability to make accurate inferences. An outcome is said to be missing not at random (MNAR) if, conditional on the observed variables, the missing data mechanism still depends on the unobserved outcome. In such settings, identification is generally not possible without imposing additional assumptions. Identification is sometimes possible, however, if an instrumental variable (IV) is observed for all subjects which satisfies the exclusion restriction that the IV affects the missingness proces…
We prove conditional asymptotic normality of a class of quadratic U-statistics that are dominated by their degenerate second order part and have kernels that change with the number of observations. These statistics arise in the construction of estimators in high-dimensional semi- and non-parametric models, and in the construction of nonparametric confidence sets. This is illustrated by estimation of the integral of a square of a density or regression function, and estimation of the mean response with missing data. We show that estimators are asymptotically normal even in the case that the rate…
Semiparametric Efficient Empirical Higher Order Influence Function Estimators
Institute of Natural Sciences, MOE-LSC, School of Mathematical Sciences, CMA-Shanghai and SJTU-Yale Joint Center for Biostatistics and Data Science, Shanghai Jiao Tong University（上海交通大学自然学院、教育部数学科学中心、CMA-上海和上海交通大学-耶鲁生物统计与数据科学联合中心）；Department of Biostatistics, Harvard University（哈佛大学生物统计系）；Department of Economics, Massachusetts Institute of Technology（麻省理工学院经济学系）；Department of Epidemiology and Biostatistics, Harvard University（哈佛大学流行病学与生物统计系）
Generative Modeling under Non-Monotone MAR Missingness via Approximate Wasserstein Gradient Flows
Department of Mathematics（数学系）；Research Institute for Statistics and Information Science（统计与信息科学研究所）；University of Geneva（日内瓦大学）；University of Hamburg（汉堡大学）
Handling Missingness and Censoring in Dirichlet Models
University of Pretoria（比勒陀利亚大学）；National Institute for Theoretical and Computational Sciences (NITheCS)（国家理论与计算科学研究所）；San José State University（圣何塞州立大学）；University of Catania（卡塔尼亚大学）
We consider statistical procedures for hypothesis testing of real valued functionals of matched pairs with missing values. In order to improve the accuracy of existing methods, we propose a novel multiplication combination procedure. Dividing the observed data into dependent (completely observed) pairs and independent (incompletely observed) components, it is based on combining separate results of adequate tests for the two sub datasets. Our methods can be applied for parametric as well as semi- and nonparametric models and make efficient use of all available data. In particular, the approache…
Imputation is a popular technique for handling missing data. We consider a nonparametric approach to imputation using the kernel ridge regression technique and propose consistent variance estimation. The proposed variance estimator is based on a linearization approach which employs the entropy method to estimate the density ratio. The root-n consistency of the imputation estimator is established when a Sobolev space is utilized in the kernel ridge regression imputation, which enables us to develop the proposed variance estimator. Synthetic data experiments are presented to confirm our theory.
In problems with large amounts of missing data one must model two distinct data generating processes: the outcome process which generates the response and the missing data mechanism which determines the data we observe. Under the ignorability condition of Rubin (1976), however, likelihood-based inference for the outcome process does not depend on the missing data mechanism so that only the former needs to be estimated; partially because of this simplification, ignorability is often used as a baseline assumption. We study the implications of Bayesian ignorability in the presence of high-dimensi…
In the near future, millions of load curves measuring the electricity consumption of French households in small time grids (probably half hours) will be available. All these collected load curves represent a huge amount of information which could be exploited using survey sampling techniques. In particular, the total consumption of a specific cus- tomer group (for example all the customers of an electricity supplier) could be estimated using unequal probability random sampling methods. Unfortunately, data collection may undergo technical problems resulting in missing values. In this paper we s…
Targeted maximum likelihood estimation (TMLE) is a general method for estimating parameters in semiparametric and nonparametric models. Each iteration of TMLE involves fitting a parametric submodel that targets the parameter of interest. We investigate the use of exponential families to define the parametric submodel. This implementation of TMLE gives a general approach for estimating any smooth parameter in the nonparametric model. A computational advantage of this approach is that each iteration of TMLE involves estimation of a parameter in an exponential family, which is a convex optimizati…
The statistical challenges in using big data for making valid statistical inference in the finite population have been well documented in literature. These challenges are due primarily to statistical bias arising from under-coverage in the big data source to represent the population of interest and measurement errors in the variables available in the data set. By stratifying the population into a big data stratum and a missing data stratum, we can estimate the missing data stratum by using a fully responding probability sample, and hence the population as a whole by using a data integration es…
Completely nonparametric transformation models with heteroscedastic errors are considered. Despite their flexibility, such models have rarely been used so far, since estimators of the model components have been missing and even identification of such models has not been clear until very recently. The results of Kloodt (2020) are used to construct the first two estimators of the transformation function in these models. While the first estimator converges to the true transformation function at a parametric rate, the second estimator can be obtained by an explicit formula and is less computationa…
Models phrased though moment conditions are central to much of modern inference. Here these moment conditions are embedded within a nonparametric Bayesian setup. Handling such a model is not probabilistically straightforward as the posterior has support on a manifold. We solve the relevant issues, building new probability and computational tools using Hausdorff measures to analyze them on real and simulated data. These new methods which involve simulating on a manifold can be applied widely, including providing Bayesian analysis of quasi-likelihoods, linear and nonlinear regression, missing da…
We introduce the concept of pattern graphs--directed acyclic graphs representing how response patterns are associated. A pattern graph represents an identifying restriction that is nonparametrically identified/saturated and is often a missing not at random restriction. We introduce a selection model and a pattern mixture model formulations using the pattern graphs and show that they are equivalent. A pattern graph leads to an inverse probability weighting estimator as well as an imputation-based estimator. We also study the semi-parametric efficiency theory and derive a multiply-robust estimat…
We consider the problem of tensor estimation from noisy observations with possibly missing entries. A nonparametric approach to tensor completion is developed based on a new model which we coin as sign representable tensors. The model represents the signal tensor of interest using a series of structured sign tensors. Unlike earlier methods, the sign series representation effectively addresses both low- and high-rank signals, while encompassing many existing tensor models -- including CP models, Tucker models, single index models, several hypergraphon models -- as special cases. We show that th…
In this paper, we use the empirical likelihood method to construct the confidence regions for the difference between the parameters of a two-phases nonlinear model with random design. We show that the empirical likelihood ratio has an asymptotic chi-squared distribution. The result is a nonparametric version of Wilk's theorem. Empirical likelihood method is also used to construct the confidence regions for the difference between the parameters of a two-phases nonlinear model with response variables missing at randoms (MAR). In order to construct the confidence regions of the parameter in quest…
During the past few decades, missing-data problems have been studied extensively, with a focus on the ignorable missing case, where the missing probability depends only on observable quantities. By contrast, research into non-ignorable missing data problems is quite limited. The main difficulty in solving such problems is that the missing probability and the regression likelihood function are tangled together in the likelihood presentation, and the model parameters may not be identifiable even under strong parametric model assumptions. In this paper we discuss a semiparametric model for non-ig…
Feature models are popular in machine learning and they have been recently used to solve many unsupervised learning problems. In these models every observation is endowed with a finite set of features, usually selected from an infinite collection $(F_{j})_{j\geq 1}$. Every observation can display feature $F_{j}$ with an unknown probability $p_{j}$. A statistical problem inherent to these models is how to estimate, given an initial sample, the conditional expected number of hitherto unseen features that will be displayed in a future observation. This problem is usually referred to as the missin…
