Testing network effects in weighted directed networks is a foundational problem in econometrics, sociology, and psychology. Yet, the prevalent edge dependency poses a significant methodological challenge. Most existing methods are model-based and come with stringent assumptions, limiting their applicability. In response, we introduce a novel, fully nonparametric framework that requires only minimal regularity assumptions. While inspired by recent developments in $U$-statistic literature (arXiv:1712.00771, arXiv:2004.06615), our approach notably broadens their scopes. Specifically, we identified and carefully addressed the challenge of indeterminate degeneracy in the test statistics $-$ a problem that aforementioned tools do not handle. We established Berry-Esseen type bound for the accuracy of type-I error rate control. Using original analysis, we also proved the minimax optimality of our test's power. Simulations underscore the superiority of our method in computation speed, accuracy, and numerical robustness compared to competing methods. We also applied our method to the U.S. faculty hiring network data and discovered intriguing findings.
Nonparametric kernel density and local polynomial regression estimators are very popular in Statistics, Economics, and many other disciplines. They are routinely employed in applied work, either as part of the main empirical analysis or as a preliminary ingredient entering some other estimation or inference procedure. This article describes the main methodological and numerical features of the software package nprobust, which offers an array of estimation and inference procedures for nonparametric kernel-based density and local polynomial regression methods, implemented in both the R and Stata…
The class of composite likelihood functions provides a flexible and powerful toolkit to carry out approximate inference for complex statistical models when the full likelihood is either impossible to specify or unfeasible to compute. However, the strenght of the composite likelihood approach is dimmed when considering hypothesis testing about a multidimensional parameter because the finite sample behavior of likelihood ratio, Wald, and score-type test statistics is tied to the Godambe information matrix. Consequently inaccurate estimates of the Godambe information translate in inaccurate p-val…
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…
This paper studies multivariate nonparametric change point localization and inference problems. The data consists of a multivariate time series with potentially short range dependence. The distribution of this data is assumed to be piecewise constant with densities in a Hölder class. The change points, or times at which the distribution changes, are unknown. We derive the limiting distributions of the change point estimators when the minimal jump size vanishes or remains constant, a first in the literature on change point settings. We are introducing two new features: a consistent estimator th…
Given discrete time observations over a growing time interval, we consider a nonparametric Bayesian approach to estimation of the Lévy density of a Lévy process belonging to a flexible class of infinite activity subordinators. Posterior inference is performed via MCMC, and we circumvent the problem of the intractable likelihood via the data augmentation device, that in our case relies on bridge process sampling via Gamma process bridges. Our approach also requires the use of a new infinite-dimensional form of a reversible jump MCMC algorithm. We show that our method leads to good practical res…
Given a sample of size $n$ from a population of individuals belonging to different species with unknown proportions, a popular problem of practical interest consists in making inference on the probability $D_{n}(l)$ that the $(n+1)$-th draw coincides with a species with frequency $l$ in the sample, for any $l=0,1,\ldots,n$. This paper contributes to the methodology of Bayesian nonparametric inference for $D_{n}(l)$. Specifically, under the general framework of Gibbs-type priors we show how to derive credible intervals for a Bayesian nonparametric estimation of $D_{n}(l)$, and we investigate th…
We investigate hypothesis testing in nonparametric additive models estimated using simplified smooth backfitting (Huang and Yu, Journal of Computational and Graphical Statistics, \textbf{28(2)}, 386--400, 2019). Simplified smooth backfitting achieves oracle properties under regularity conditions and provides closed-form expressions of the estimators that are useful for deriving asymptotic properties. We develop a generalized likelihood ratio (GLR) and a loss function (LF) based testing framework for inference. Under the null hypothesis, both the GLR and LF tests have asymptotically rescaled ch…
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…
Spurred on by recent successes in causal inference competitions, Bayesian nonparametric (and high-dimensional) methods have recently seen increased attention in the causal inference literature. In this paper, we present a comprehensive overview of Bayesian nonparametric applications to causal inference. Our aims are to (i) introduce the fundamental Bayesian nonparametric toolkit; (ii) discuss how to determine which tool is most appropriate for a given problem; and (iii) show how to avoid common pitfalls in applying Bayesian nonparametric methods in high-dimensional settings. Unlike standard fi…
The method of instrumental variables provides a fundamental and practical tool for causal inference in many empirical studies where unmeasured confounding between the treatments and the outcome is present. Modern data such as the genetical genomics data from these studies are often high-dimensional. The high-dimensional linear instrumental-variables regression has been considered in the literature due to its simplicity albeit a true nonlinear relationship may exist. We propose a more data-driven approach by considering the nonparametric additive models between the instruments and the treatment…
Understanding the pathways whereby an intervention has an effect on an outcome is a common scientific goal. A rich body of literature provides various decompositions of the total intervention effect into pathway specific effects. Interventional direct and indirect effects provide one such decomposition. Existing estimators of these effects are based on parametric models with confidence interval estimation facilitated via the nonparametric bootstrap. We provide theory that allows for more flexible, possibly machine learning-based, estimation techniques to be considered. In particular, we establ…
The Highly-Adaptive-LASSO Targeted Minimum Loss Estimator (HAL-TMLE) is an efficient plug-in estimator of a pathwise differentiable parameter in a statistical model that at minimal (and possibly only) assumes that the sectional variation norm of the true nuisance functional parameters (i.e., the relevant part of data distribution) are finite. It relies on an initial estimator (HAL-MLE) of the nuisance functional parameters by minimizing the empirical risk over the parameter space under the constraint that the sectional variation norm of the candidate functions are bounded by a constant, where …
Using and extending fractional order statistic theory, we characterize the $O(n^{-1})$ coverage probability error of the previously proposed confidence intervals for population quantiles using $L$-statistics as endpoints in Hutson (1999). We derive an analytic expression for the $n^{-1}$ term, which may be used to calibrate the nominal coverage level to get $O\bigl(n^{-3/2}[\log(n)]^3\bigr)$ coverage error. Asymptotic power is shown to be optimal. Using kernel smoothing, we propose a related method for nonparametric inference on conditional quantiles. This new method compares favorably with as…
We consider inference procedures, conditional on an observed ancillary statistic, for regression coefficients under a linear regression setup where the unknown error distribution is specified nonparametrically. We establish conditional asymptotic normality of the regression coefficient estimators under regularity conditions, and formally justify the approach of plugging in kernel-type density estimators in conditional inference procedures. Simulation results show that the approach yields accurate conditional coverage probabilities when used for constructing confidence intervals. The plug-in ap…
Missing exposure information is a very common feature of many observational studies. Here we study identifiability and efficient estimation of causal effects on vector outcomes, in such cases where treatment is unconfounded but partially missing. We consider a missing at random setting where missingness in treatment can depend not only on complex covariates, but also on post-treatment outcomes. We give a new identifying expression for average treatment effects in this setting, along with the efficient influence function for this parameter in a nonparametric model, which yields a nonparametric …
Since Stein's 1956 seminal paper, shrinkage has played a fundamental role in both parametric and nonparametric inference. This article discusses minimaxity and adaptive minimaxity in nonparametric function estimation. Three interrelated problems, function estimation under global integrated squared error, estimation under pointwise squared error, and nonparametric confidence intervals, are considered. Shrinkage is pivotal in the development of both the minimax theory and the adaptation theory. While the three problems are closely connected and the minimax theories bear some similarities, the ad…
We propose two classes of nonparametric point estimators of $θ=P(X<Y)$ in the case where $(X,Y)$ are paired, possibly dependent, absolutely continuous random variables. The proposed estimators are based on nonparametric estimators of the joint density of $(X,Y)$ and the distribution function of $Z=Y - X$. We explore the use of several density and distribution function estimators and characterise the convergence of the resulting estimators of $θ$. We consider the use of bootstrap methods to obtain confidence intervals. The performance of these estimators is illustrated using simulated and real …
Instrumental variable methods allow for inference about the treatment effect by controlling for unmeasured confounding in randomized experiments with noncompliance. However, many studies do not consider the observed compliance behavior in the testing procedure, which can lead to a loss of power. In this paper, we propose a novel nonparametric likelihood approach, referred to as the binomial likelihood (BL) method, that incorporates information on compliance behavior while overcoming several limitations of previous techniques and utilizing the advantages of likelihood methods. Our proposed meth…
This study examines a nonparametric inference on a stationary Lévy-driven Ornstein-Uhlenbeck (OU) process $X = (X_{t})_{t \geq 0}$ with a compound Poisson subordinator. We propose a new spectral estimator for the Lévy measure of the Lévy-driven OU process $X$ under macroscopic observations. We also derive, for the estimator, multivariate central limit theorems over a finite number of design points, and high-dimensional central limit theorems in the case wherein the number of design points increases with an increase in the sample size. Built on these asymptotic results, we develop methods to co…
Due to their great flexibility, nonparametric Bayes methods have proven to be a valuable tool for discovering complicated patterns in data. The term "nonparametric Bayes" suggests that these methods inherit model-free operating characteristics of classical nonparametric methods, as well as coherent uncertainty assessments provided by Bayesian procedures. However, as the authors say in the conclusion to their article, nonparametric Bayesian methods may be more aptly described as "massively parametric." Furthermore, I argue that many of the default nonparametric Bayes procedures are only Bayesia…
This study extends the Bayesian nonparametric instrumental variable regression model to determine the structural effects of covariates on the conditional quantile of the response variable. The error distribution is nonparametrically modelled using a Dirichlet mixture of bivariate normal distributions. The mean functions include the smooth effects of the covariates represented using the spline functions in an additive manner. The conditional variance of the second-stage error is also modelled using the spline functions such that it varies smoothly with the covariates. Accordingly, the proposed …
This paper studies the validity of nonparametric tests used in the regression discontinuity design. The null hypothesis of interest is that the average treatment effect at the threshold in the so-called sharp design equals a pre-specified value. We first show that, under assumptions used in the majority of the literature, for \emph{any} test the power against any alternative is bounded above by its size. This result implies that, under these assumptions, any test with nontrivial power will exhibit size distortions. We next provide a sufficient strengthening of the standard assumptions under wh…
In this paper, the panel count data analysis for recurrent events is considered. Such analysis is useful for studying tumor or infection recurrences in both clinical trial and observational studies. A bivariate Gaussian Cox process model is proposed to jointly model the observation process and the recurrent event process. Bayesian nonparametric inference is proposed for simultaneously estimating regression parameters, bivariate frailty effects and baseline intensity functions. Inference is done through Markov chain Monte Carlo, with fully developed computational techniques. Predictive inferenc…
We develop a novel method of constructing confidence bands for nonparametric regression functions under shape constraints. This method can be implemented via a linear programming, and it is thus computationally appealing. We illustrate a usage of our proposed method with an application to the regression kink design (RKD). Econometric analyses based on the RKD often suffer from wide confidence intervals due to slow convergence rates of nonparametric derivative estimators. We demonstrate that economic models and structures motivate shape restrictions, which in turn contribute to shrinking the co…
Causal inference from observational data is an ambitious but highly relevant task, with diverse applications ranging from natural to social sciences. Within the scope of nonparametric time series, causal inference defined through interventions (cf. Pearl (2000)) is largely unexplored, although time order simplifies the problem substantially. We consider a marginal integration scheme for inferring causal effects from observational time series data, MINT-T (marginal integration in time series), which is an adaptation for time series of a method proposed by Ernest and Bühlmann (Electron. J. Stati…
To improve nonparametric estimates of lifetime distributions, we propose using the increasing odds rate (IOR) model as an alternative to other popular, but more restrictive, ``adverse ageing'' models, such as the increasing hazard rate one. This extends the scope of applicability of some methods for statistical inference under order restrictions, since the IOR model is compatible with heavy-tailed and bathtub distributions. We study a strongly uniformly consistent estimator of the cumulative distribution function of interest under the IOR constraint. Numerical evidence shows that this estimato…
Nonparametric Bayesian models are used routinely as flexible and powerful models of complex data. Many times, a statistician may have additional informative beliefs about data distribution of interest, e.g., its mean or subset components, that is not part of, or even compatible with, the nonparametric prior. An important challenge is then to incorporate this partial prior belief into nonparametric Bayesian models. In this paper, we are motivated by settings where practitioners have additional distributional information about a subset of the coordinates of the observations being modeled. Our ap…
We develop a scalable multi-step Monte Carlo algorithm for inference under a large class of nonparametric Bayesian models for clustering and classification. Each step is "embarrassingly parallel" and can be implemented using the same Markov chain Monte Carlo sampler. The simplicity and generality of our approach makes inference for a wide range of Bayesian nonparametric mixture models applicable to large datasets. Specifically, we apply the approach to inference under a product partition model with regression on covariates. We show results for inference with two motivating data sets: a large s…
Fusion learning refers to synthesizing inferences from multiple sources or studies to provide more effective inference and prediction than from any individual source or study alone. Most existing methods for synthesizing inferences rely on parametric model assumptions, such as normality, which often do not hold in practice. In this paper, we propose a general nonparametric fusion learning framework for synthesizing inferences of the target parameter from multiple sources. The main tool underlying the proposed framework is the notion of depth confidence distribution (depth-CD), which is also de…
Hypothesis testing and other statistical inference procedures are most efficient when a reliable low-dimensional parametric family can be specified. We propose a method that learns such a family when one exists but its form is not known a priori, by examining samples from related populations and fitting a low-dimensional exponential family that approximates all the samples as well as possible. We propose a computationally efficient spectral method that allows us to carry out hypothesis tests that are valid whether or not the fit is good, and recover asymptotically optimal power if it is. Our m…
In broad applications, it is routinely of interest to assess whether there is evidence in the data to refute the assumption of conditional independence of $Y$ and $X$ conditionally on $Z$. Such tests are well developed in parametric models but are not straightforward in the nonparametric case. We propose a general Bayesian approach, which relies on an encompassing nonparametric Bayes model for the joint distribution of $Y$, $X$ and $Z$. The framework allows $Y$, $X$ and $Z$ to be random variables on arbitrary spaces, and can accommodate different dimensional vectors having a mixture of discret…
New methods and theory have recently been developed to nonparametrically estimate cumulative incidence functions for competing risks survival data subject to current status censoring. In particular, the limiting distribution of the nonparametric maximum likelihood estimator and a simplified "naive estimator" have been established under certain smoothness conditions. In this paper, we establish the large-sample behavior of these estimators in two additional models, namely when the observation time distribution has discrete support and when the observation times are grouped. These asymptotic res…
During the last decades there has been increasing interest in modeling the volatility of financial data. Several parametric models have been proposed to this aim, starting from ARCH, GARCH and their variants, but often it is hard to evaluate which one is the most suitable for the analyzed financial data. In this paper we focus on nonparametric analysis of the volatility function for mixing processes. Our approach encompasses many parametric frameworks and supplies several tools which can be used to give evidence against or in favor of a specific parametric model: nonparametric function estimat…
Here we develop a method for performing nonparametric Bayesian inference on quantiles. Relying on geometric measure theory and employing a Hausdorff base measure, we are able to specify meaningful priors for the quantile while treating the distribution of the data otherwise nonparametrically. We further extend the method to a hierarchical model for quantiles of subpopulations, linking subgroups together solely through their quantiles. Our approach is computationally straightforward, allowing for censored and noisy data. We demonstrate the proposed methodology on simulated data and an applied p…
Kernel Bayesian inference is a principled approach to nonparametric inference in probabilistic graphical models, where probabilistic relationships between variables are learned from data in a nonparametric manner. Various algorithms of kernel Bayesian inference have been developed by combining kernelized basic probabilistic operations such as the kernel sum rule and kernel Bayes' rule. However, the current framework is fully nonparametric, and it does not allow a user to flexibly combine nonparametric and model-based inferences. This is inefficient when there are good probabilistic models (or …
It has historically been a challenge to perform Bayesian inference in a design-based survey context. The present paper develops a Bayesian model for sampling inference in the presence of inverse-probability weights. We use a hierarchical approach in which we model the distribution of the weights of the nonsampled units in the population and simultaneously include them as predictors in a nonparametric Gaussian process regression. We use simulation studies to evaluate the performance of our procedure and compare it to the classical design-based estimator. We apply our method to the Fragile Famil…
Bayesian nonparametric methods are a popular choice for analysing survival data due to their ability to flexibly model the distribution of survival times. These methods typically employ a nonparametric prior on the survival function that is conjugate with respect to right-censored data. Eliciting these priors, particularly in the presence of covariates, can be challenging and inference typically relies on computationally intensive Markov chain Monte Carlo schemes. In this paper, we build on recent work that recasts Bayesian inference as assigning a predictive distribution on the unseen values …
In this paper we review recently developed methods for nonparametric Bayesian inference for one-dimensional diffusion models. We discuss different possible prior distributions, computational issues, and asymptotic results.
