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How best to estimate the accuracy of a predictive rule has been a longstanding question in statistics. Approaches to this task range from simple methods like Mallow’s Cp to algorithmic techniques l...

In this article I review the modeling of capture, recapture, and removal statistics for the purpose of estimating demographic parameters of fish and wildlife populations. Topics considered include capture-recapture models, band or tag return models, removal and catch per unit effort models, selective removal or change-in-ratio models, radio-tagging survival models, and nest survival models. The purpose is to present important concepts in a general manner for the benefit of a wide audience of statisticians. I will not attempt to be comprehensive, and I indulge in speculation about future directions. I indicate the importance of different statistical tools to this subject, such as Bayesian inference, “boot strapping,” robustness studies, goodness-of-fit tests. I also emphasize connections to other application areas of statistics. Capture-recapture methods, for example, are being considered for estimation of a variety of elusive human populations, such as the homeless and people missed in the census...

Advances in computation and the fast and cheap computational facilities now available to statisticians have had a significant impact upon statistical research, and especially the development of nonparametric data analysis procedures. In particular, theoretical and applied research on nonparametric density estimation has had a noticeable influence on related topics, such as nonparametric regression, nonparametric discrimination, and nonparametric pattern recognition. This article reviews recent developments in nonparametric density estimation and includes topics that have been omitted from review articles and books on the subject. The early density estimation methods, such as the histogram, kernel estimators, and orthogonal series estimators are still very popular, and recent research on them is described. Different types of restricted maximum likelihood density estimators, including order-restricted estimators, maximum penalized likelihood estimators, and sieve estimators, are discussed, where re...

The determination of plotting positions on probability graph paper so that the associated weighted least squares estimators of the scale parameter and the percentiles of a continuous distribution have certain properties is discussed. Necessary and sufficient conditions are given for an invariant optimal plot for percentile estimation. Also discussed is the derivation of ordered plotting positions.

This paper is concerned with constructing a confidence interval for a target policy's value offline based on a pre-collected observational data in infinite horizon settings. Most of the existing works assume no unmeasured variables exist that confound the observed actions. This assumption, however, is likely to be violated in real applications such as healthcare and technological industries. In this paper, we show that with some auxiliary variables that mediate the effect of actions on the system dynamics, the target policy's value is identifiable in a confounded Markov decision process. Based on this result, we develop an efficient off-policy value estimator that is robust to potential model misspecification and provide rigorous uncertainty quantification. Our method is justified by theoretical results, simulated and real datasets obtained from ridesharing companies. A Python implementation of the proposed procedure is available at https://github.com/Mamba413/cope.

Classical two-sample permutation tests for equality of distributions have exact size in finite samples, but they fail to control size for testing equality of parameters that summarize each distribution. This paper proposes permutation tests for equality of parameters that are estimated at root-$n$ or slower rates. Our general framework applies to both parametric and nonparametric models, with two samples or one sample split into two subsamples. Our tests have correct size asymptotically while preserving exact size in finite samples when distributions are equal. They have no loss in local asymptotic power compared to tests that use asymptotic critical values. We propose confidence sets with correct coverage in large samples that also have exact coverage in finite samples if distributions are equal up to a transformation. We apply our theory to four commonly-used hypothesis tests of nonparametric functions evaluated at a point. Lastly, simulations show good finite sample properties, and two empirical examples illustrate our tests in practice., One PDF file contains main paper (35 pages) plus supplement (64 pages)

In this paper, we propose a new projection test for linear hypotheses on regression coefficient matrices in linear models with high dimensional responses. We systematically study the theoretical properties of the proposed test. We first derive the optimal projection matrix for any given projection dimension to achieve the best power and provide an upper bound for the optimal dimension of projection matrix. We further provide insights into how to construct the optimal projection matrix. One- and two-sample mean problems can be formulated as special cases of linear hypotheses studied in this paper. We both theoretically and empirically demonstrate that the proposed test can outperform the existing ones for one- and two-sample mean problems. We conduct Monte Carlo simulation to examine the finite sample performance and illustrate the proposed test by a real data example.

We would like to congratulate the authors on two beautifully motivated and technically flawless papers and thank the editors for the opportunity to discuss both papers. We would like to make two po...

The Model-free Prediction Principle of Politis (2015) has been successfully applied to general regression problems, as well as problems involving stationary time series. However, with long time series, e.g. annual temperature measurements spanning over 100 years or daily financial returns spanning several years, it may be unrealistic to assume stationarity throughout the span of the dataset. In the paper at hand, we show how Model-free Prediction can be applied to handle time series that are only locally stationary, i.e., they can be assumed to be as stationary only over short time-windows. Surprisingly there is little literature on point prediction for general locally stationary time series even in model-based setups and there is no literature on the construction of prediction intervals of locally stationary time series. We attempt to fill this gap here as well. Both one-step-ahead point predictors and prediction intervals are constructed, and the performance of model-free is compared to model-based prediction using models that incorporate a trend and/or heteroscedasticity. Both aspects of the paper, model-free and model-based, are novel in the context of time-series that are locally (but not globally) stationary. We also demonstrate the application of our Model-based and Model-free prediction methods to speleothem climate data which exhibits local stationarity and show that our best model-free point prediction results outperform that obtained with the RAMPFIT algorithm previously used for analysis of this data.

Lyme disease is an infectious disease that is caused by a bacterium called Borrelia burgdorferi sensu stricto. In the United States, Lyme disease is one of the most common infectious diseases. The major endemic areas of the disease are New England, Mid-Atlantic, East-North Central, South Atlantic, and West North-Central. Virginia is on the front-line of the disease's diffusion from the northeast to the south. One of the research objectives for the infectious disease community is to identify environmental and economic variables that are associated with the emergence of Lyme disease. In this paper, we use a spatial Poisson regression model to link the spatial disease counts and environmental and economic variables, and develop a spatial variable selection procedure to effectively identify important factors by using an adaptive elastic net penalty. The proposed methods can automatically select important covariates, while adjusting for possible spatial correlations of disease counts. The performance of the proposed method is studied and compared with existing methods via a comprehensive simulation study. We apply the developed variable selection methods to the Virginia Lyme disease data and identify important variables that are new to the literature. Supplementary materials for this paper are available online., Comment: 34 pages

This JASA rejoinder concerns the problem of individualized decision making under point, sign, and partial identification. The paper unifies various classical decision making strategies through a lower bound perspective proposed in Cui and Tchetgen Tchetgen (2020b) in the context of optimal treatment regimes under uncertainty due to unmeasured confounding. Building on this unified framework, the paper also provides a novel minimax solution (i.e., a rule that minimizes the maximum regret for so-called opportunists) for individualized decision making/policy assignment.

The goal of this paper is to develop a novel statistical model for studying cross-neuronal spike train interactions during decision making. For an individual to successfully complete the task of decision-making, a number of temporally-organized events must occur: stimuli must be detected, potential outcomes must be evaluated, behaviors must be executed or inhibited, and outcomes (such as reward or no-reward) must be experienced. Due to the complexity of this process, it is likely the case that decision-making is encoded by the temporally-precise interactions between large populations of neurons. Most existing statistical models, however, are inadequate for analyzing such a phenomenon because they provide only an aggregated measure of interactions over time. To address this considerable limitation, we propose a dynamic Bayesian model which captures the time-varying nature of neuronal activity (such as the time-varying strength of the interactions between neurons). The proposed method yielded results that reveal new insight into the dynamic nature of population coding in the prefrontal cortex during decision making. In our analysis, we note that while some neurons in the prefrontal cortex do not synchronize their firing activity until the presence of a reward, a different set of neurons synchronize their activity shortly after stimulus onset. These differentially synchronizing sub-populations of neurons suggests a continuum of population representation of the reward-seeking task. Secondly, our analyses also suggest that the degree of synchronization differs between the rewarded and non-rewarded conditions. Moreover, the proposed model is scalable to handle data on many simultaneously-recorded neurons and is applicable to analyzing other types of multivariate time series data with latent structure. Supplementary materials (including computer codes) for our paper are available online.

Statistical analysis of multimodal imaging data is a challenging task, since the data involves high-dimensionality, strong spatial correlations and complex data structures. In this paper, we propose rigorous statistical testing procedures for making inferences on the complex dependence of multimodal imaging data. Motivated by the analysis of multi-task fMRI data in the Human Connectome Project (HCP) study, we particularly address three hypothesis testing problems: (a) testing independence among imaging modalities over brain regions, (b) testing independence between brain regions within imaging modalities, and (c) testing independence between brain regions across different modalities. Considering a general form for all the three tests, we develop a global testing procedure and a multiple testing procedure controlling the false discovery rate. We study theoretical properties of the proposed tests and develop a computationally efficient distributed algorithm. The proposed methods and theory are general and relevant for many statistical problems of testing independence structure among the components of high-dimensional random vectors with arbitrary dependence structures. We also illustrate our proposed methods via extensive simulations and analysis of five task fMRI contrast maps in the HCP study.

One of the major concerns of targeting interventions on individuals in social welfare programs is discrimination: individualized treatments may induce disparities across sensitive attributes such as age, gender, or race. This paper addresses the question of the design of fair and efficient treatment allocation rules. We adopt the non-maleficence perspective of first do no harm: we select the fairest allocation within the Pareto frontier. We cast the optimization into a mixed-integer linear program formulation, which can be solved using off-the-shelf algorithms. We derive regret bounds on the unfairness of the estimated policy function and small sample guarantees on the Pareto frontier under general notions of fairness. Finally, we illustrate our method using an application from education economics.

The theory of optimal design of experiments has been traditionally developed on an Euclidean space. In this paper, new theoretical results and an algorithm for finding the optimal design of an experiment located on a Riemannian manifold are provided. It is shown that analogously to the results in Euclidean spaces, D-optimal and G-optimal designs are equivalent on manifolds, and we provide a lower bound for the maximum prediction variance of the response evaluated over the manifold. In addition, a converging algorithm that finds the optimal experimental design on manifold data is proposed. Numerical experiments demonstrate the importance of considering the manifold structure in a designed experiment when present, and the superiority of the proposed algorithm.

This article studies the problem of modeling relationship between two spherical (or directional) random variables in a regression setup. Here the predictor and the response variables are constrained to be on a unit sphere and, due to this nonlinear condition, the standard Euclidean regression models do not apply. Several past papers have studied this problem, termed spherical regression, by modeling the response variable with a von Mises-Fisher (VMF) density with the mean given by a rotation of the predictor variable. The few papers that go beyond rigid rotations are limited to one- or two-dimensional spheres. This article extends the mean transformations to a larger group—the projective linear group of transformations—on unit spheres of arbitrary dimensions, while keeping the VMF density to model the noise. It develops a Newton–Raphson algorithm on the special linear group for estimating the MLE of regression parameter and establishes its asymptotic properties when the sample-size becomes large. Through ...

This paper investigates the theoretical and empirical performance of Fisher-Pitman-type permutation tests for assessing the equality of unknown Poisson mixture distributions. Building on nonparametric maximum likelihood estimators (NPMLEs) of the mixing distribution, these tests are theoretically shown to be able to adapt to complicated unspecified structures of count data and also consistent against their corresponding ANOVA-type alternatives; the latter is a result in parallel to classic claims made by Robinson (Robinson, 1973). The studied methods are then applied to a single-cell RNA-seq data obtained from different cell types from brain samples of autism subjects and healthy controls; empirically, they unveil genes that are differentially expressed between autism and control subjects yet are missed using common tests. For justifying their use, rate optimality of NPMLEs is also established in settings similar to nonparametric Gaussian (Wu and Yang, 2020a) and binomial mixtures (Tian et al., 2017; Vinayak et al., 2019)., 52 pages

We propose a new estimator of high-dimensional spectral density matrices, called ALgebraic Spectral Estimator (ALSE), under the assumption of an underlying low rank plus sparse structure, as typically assumed in dynamic factor models. The ALSE is computed by minimizing a quadratic loss under a nuclear norm plus l1 norm constraint to control the latent rank and the residual sparsity pattern. The loss function requires as input the classical smoothed periodogram estimator and two threshold parameters, the choice of which is thoroughly discussed. We prove consistency of ALSE as both the dimension p and the sample size T diverge to infinity, as well as the recovery of latent rank and residual sparsity pattern with probability one. We then propose the UNshrunk ALgebraic Spectral Estimator (UNALSE), which is designed to minimize the Frobenius loss with respect to the pre-estimator while retaining the optimality of the ALSE. When applying UNALSE to a standard US quarterly macroeconomic dataset, we find evidence of two main sources of comovements: a real factor driving the economy at business cycle frequencies, and a nominal factor driving the higher frequency dynamics. The paper is also complemented by an extensive simulation exercise.

Classical tests for a difference in means control the type I error rate when the groups are defined a priori. However, when the groups are instead defined via clustering, then applying a classical test yields an extremely inflated type I error rate. Notably, this problem persists even if two separate and independent data sets are used to define the groups and to test for a difference in their means. To address this problem, in this paper, we propose a selective inference approach to test for a difference in means between two clusters. Our procedure controls the selective type I error rate by accounting for the fact that the choice of null hypothesis was made based on the data. We describe how to efficiently compute exact p-values for clusters obtained using agglomerative hierarchical clustering with many commonly-used linkages. We apply our method to simulated data and to single-cell RNA-sequencing data., Comment: Final accepted version

The James-Stein estimator is an estimator of the multivariate normal mean and dominates the maximum likelihood estimator (MLE) under squared error loss. The original work inspired great interest in developing shrinkage estimators for a variety of problems. Nonetheless, research on shrinkage estimation for manifold-valued data is scarce. In this paper, we propose shrinkage estimators for the parameters of the Log-Normal distribution defined on the manifold of $N \times N$ symmetric positive-definite matrices. For this manifold, we choose the Log-Euclidean metric as its Riemannian metric since it is easy to compute and is widely used in applications. By using the Log-Euclidean distance in the loss function, we derive a shrinkage estimator in an analytic form and show that it is asymptotically optimal within a large class of estimators including the MLE, which is the sample Fr\'echet mean of the data. We demonstrate the performance of the proposed shrinkage estimator via several simulated data experiments. Furthermore, we apply the shrinkage estimator to perform statistical inference in diffusion magnetic resonance imaging problems., Comment: 54 pages, 5 figures

We consider reinforcement learning (RL) methods in offline domains without additional online data collection, such as mobile health applications. Most of existing policy optimization algorithms in the computer science literature are developed in online settings where data are easy to collect or simulate. Their generalizations to mobile health applications with a pre-collected offline dataset remain unknown. The aim of this paper is to develop a novel advantage learning framework in order to efficiently use pre-collected data for policy optimization. The proposed method takes an optimal Q-estimator computed by any existing state-of-the-art RL algorithms as input, and outputs a new policy whose value is guaranteed to converge at a faster rate than the policy derived based on the initial Q-estimator. Extensive numerical experiments are conducted to back up our theoretical findings. A Python implementation of our proposed method is available at https://github.com/leyuanheart/SEAL.

In this paper we consider estimation of sparse covariance matrices and propose a thresholding procedure which is adaptive to the variability of individual entries. The estimators are fully data driven and enjoy excellent performance both theoretically and numerically. It is shown that the estimators adaptively achieve the optimal rate of convergence over a large class of sparse covariance matrices under the spectral norm. In contrast, the commonly used universal thresholding estimators are shown to be sub-optimal over the same parameter spaces. Support recovery is also discussed. The adaptive thresholding estimators are easy to implement. Numerical performance of the estimators is studied using both simulated and real data. Simulation results show that the adaptive thresholding estimators uniformly outperform the universal thresholding estimators. The method is also illustrated in an analysis on a dataset from a small round blue-cell tumors microarray experiment. A supplement to this paper which contains additional technical proofs is available online., Comment: To appear in Journal of the American Statistical Association

Shrinkage estimates of small domain parameters typically utilize a combination of a noisy "direct" estimate that only uses data from a specific small domain and a more stable regression estimate. When the regression model is misspecified, estimation performance for the noisier domains can suffer due to substantial shrinkage towards a poorly estimated regression surface. In this paper, we introduce a new class of robust, empirically-driven regression weights that target estimation of the small domain means under potential misspecification of the global regression model. Our regression weights are a convex combination of the model-based weights associated with the best linear unbiased predictor (BLUP) and those associated with the observed best predictor (OBP). The compromise parameter in this convex combination is found by minimizing a novel, unbiased estimate of the mean-squared prediction error for the small domain means, and we label the associated small domain estimates the "compromise best predictor", or CBP. Using a data-adaptive mixture for the regression weights enables the CBP to possess the robustness of the OBP while retaining the main advantages of the EBLUP whenever the regression model is correct. We demonstrate the use of the CBP in an application estimating gait speed in older adults.

This paper presents a novel data-driven bandwidth selector compatible with the small bandwidth asymptotics developed in Cattaneo, Crump, and Jansson (2009) for density-weighted average derivatives. The new bandwidth selector is of the plug-in variety, and is obtained based on a mean squared error expansion of the estimator of interest. An extensive Monte Carlo experiment shows a remarkable improvement in performance when the bandwidth-dependent robust inference procedures proposed by Cattaneo, Crump, and Jansson (2009) are coupled with this new data-driven bandwidth selector. The resulting robust data-driven confidence intervals compare favorably to the alternative procedures available in the literature. The online supplemental material to this paper contains further results from the simulation study.

In a series of recent articles, Bradley Efron has pointed out that in large-scale multiple testing problems, the observed distribution of the test statistics often does not match the theoretical null distribution (Efron, 2004, 2007a,b, 2008). The correction, which he termed “empirical null”, has been a subject of controversy in the statistical community. In Efron (2007a), he made the case that, even when the theoretical model is correct, the observed distribution of the test statistics can look different from the theoretical null distribution simply because of correlation between the test statistics. Efron’s present article represents an important step forward in the understanding of this problem. As opposed to previous papers, where the the effect of correlation was treated within the context of multiple testing problems and false discovery rates, Efron’s present article breaks through the confusion by separating these two concepts and focusing on the core issue, which is the behavior of a large collection of correlated normal variables. Only then, as applications, he presents the implications for false discovery rate analysis when the correlated normal variables are z-scores in a large-scale multiple testing problem. I think this separation is crucial and helps get us nearer a new theory of inference for high-dimensional data. In what follows, I present my own interpretation of Efron’s results on how correlation affects the empirical distribution of normal variables. As a shortcut, I work in the countinous domain directly rather than with histogram bins and avoid the inclusion of unnecessary constraints such as normalization, which applies very specifically to microarray data. For simplicity, I assume all the variables are standard normal rather than belonging to a mixture model, but may have an arbitrary correlation structure. Using these results, I hope to answer the question raised in Efron (2007a) of whether large-scale correlation can substantially widen the observed histogram, as in Figure 1 in that paper and Efron’s current one. The theoretical arguments below indicate that this is unlikely. On the contrary, large-scale correlation has to be mostly positive in order to satisfy positive definite contraints, and as a result, correlation will most likely narrow and shift the observed histogram rather than widen it. An important implication is that a wide histogram may be an indication of the presence of true signal, rather than an artifact of correlation. Having a model for the observed distribution under correlation, I will suggest a new way of fitting an empirical null, which may be more appropriate in cases where the observed widening is larger than could be explained by correlation alone. I will conclude commenting briefly on the possibility of performing this kind of analysis with χ2 variates rather than normal. I will try to keep the notation close to that of Efron, but some discrepancies in notation will be inevitable.

In this paper, we propose a computationally efficient approach -- space(Sparse PArtial Correlation Estimation)-- for selecting non-zero partial correlations under the high-dimension-low-sample-size setting. This method assumes the overall sparsity of the partial correlation matrix and employs sparse regression techniques for model fitting. We illustrate the performance of space by extensive simulation studies. It is shown that space performs well in both non-zero partial correlation selection and the identification of hub variables, and also outperforms two existing methods. We then apply space to a microarray breast cancer data set and identify a set of hub genes which may provide important insights on genetic regulatory networks. Finally, we prove that, under a set of suitable assumptions, the proposed procedure is asymptotically consistent in terms of model selection and parameter estimation., Comment: A paper based on this report has been accepted for publication on Journal of the American Statistical Association(http://www.amstat.org/publications/JASA/)

Starting with the seminal paper of Haario, Saksman and Tamminen (Haario et al. (2001)), a substantial amount of work has been done to validate adaptive Markov chain Monte Carlo algorithms. In this paper we focus on two practical aspects of adaptive Metropolis samplers. First, we draw attention to the deficient performance of standard adaptation when the target distribution is multi-modal. We propose a parallel chain adaptation strategy that incorporates multiple Markov chains which are run in parallel. Second, we note that

Real-time traffic flow data across entire networks can be used in a traffic management system to monitor current traffic flows so that traffic can be directed and managed efficiently. Reliable short-term forecasting models of traffic flows are crucial for the success of any traffic management system.\ud \ud The model proposed in this paper for forecasting traffic flows is a multivariate Bayesian dynamic model called the multiregression dynamic model (MDM). This model is an example of a dynamic Bayesian network and is designed to preserve the conditional independences and causal drive exhibited by the traffic flow series.\ud \ud Sudden changes can occur in traffic flow series in response to such events as traffic accidents or roadworks. A traffic management system is particularly useful at such times of change. To ensure that the associated forecasting model continues to produce reliable forecasts, despite the change, the MDM uses the technique of external intervention. This paper will demonstrate how intervention works in the MDM and how it can improve forecast performance at times of change.\ud \ud External intervention has also been used in the context of Bayesian networks to identify causal relationships between variables, and in dynamic Bayesian networks to identify lagged causal relationships between time series. This paper goes beyond the identification of lagged causal relationships previously addressed using intervention in dynamic Bayesian networks, to show how intervention in the MDM can be used to identify contemporaneous causal relationships between time series.

In the analysis of microarray data, and in some other contemporary statistical problems, it is not uncommon to apply hypothesis tests in a highly simultaneous way. The number, $\nu$ say, of tests used can be much larger than the sample sizes, $n$, to which the tests are applied, yet we wish to calibrate the tests so that the overall level of the simultaneous test is accurate. Often the sampling distribution is quite different for each test, so there may not be an opportunity for combining data across samples. In this setting, how large can $\nu$ be, as a function of $n$, before level accuracy becomes poor? In the present paper we answer this question in cases where the statistic under test is of Student's $t$ type. We show that if either Normal or Student's $t$ distribution is used for calibration then the level of the simultaneous test is accurate provided $\log\nu$ increases at a strictly slower rate than $n^{1/3}$ as $n$ diverges. On the other hand, if bootstrap methods are used for calibration then we may choose $\log\nu$ almost as large as $n\half$ and still achieve asymptotic level accuracy. The implications of these results are explored both theoretically and numerically., Comment: 25 pages paper

This paper proposes a hierarchical approximate-factor approach to analyzing high-dimensional, large-scale heterogeneous time series data using distributed computing. The new method employs a multiple-fold dimension reduction procedure using Principal Component Analysis (PCA) and shows great promises for modeling large-scale data that cannot be stored nor analyzed by a single machine. Each computer at the basic level performs a PCA to extract common factors among the time series assigned to it and transfers those factors to one and only one node of the second level. Each 2nd-level computer collects the common factors from its subordinates and performs another PCA to select the 2nd-level common factors. This process is repeated until the central server is reached, which collects common factors from its direct subordinates and performs a final PCA to select the global common factors. The noise terms of the 2nd-level approximate factor model are the unique common factors of the 1st-level clusters. We focus on the case of 2 levels in our theoretical derivations, but the idea can easily be generalized to any finite number of hierarchies. We discuss some clustering methods when the group memberships are unknown and introduce a new diffusion index approach to forecasting. We further extend the analysis to unit-root nonstationary time series. Asymptotic properties of the proposed method are derived for the diverging dimension of the data in each computing unit and the sample size $T$. We use both simulated data and real examples to assess the performance of the proposed method in finite samples, and compare our method with the commonly used ones in the literature concerning the forecastability of extracted factors., Comment: 48 pages, 10 figures

We aim to provably complete a sparse and highly-missing tensor in the presence of covariate information along tensor modes. Our motivation comes from online advertising where users click-through-rates (CTR) on ads over various devices form a CTR tensor that has about 96% missing entries and has many zeros on non-missing entries, which makes the standalone tensor completion method unsatisfactory. Beside the CTR tensor, additional ad features or user characteristics are often available. In this paper, we propose Covariate-assisted Sparse Tensor Completion (COSTCO) to incorporate covariate information for the recovery of the sparse tensor. The key idea is to jointly extract latent components from both the tensor and the covariate matrix to learn a synthetic representation. Theoretically, we derive the error bound for the recovered tensor components and explicitly quantify the improvements on both the reveal probability condition and the tensor recovery accuracy due to covariates. Finally, we apply COSTCO to an advertisement dataset consisting of a CTR tensor and ad covariate matrix, leading to 23% accuracy improvement over the baseline. An important by-product is that ad latent components from COSTCO reveal interesting ad clusters, which are useful for better ad targeting., To Appear in Journal of the American Statistical Association

Self-supervised metric learning has been a successful approach for learning a distance from an unlabeled dataset. The resulting distance is broadly useful for improving various distance-based downstream tasks, even when no information from downstream tasks is utilized in the metric learning stage. To gain insights into this approach, we develop a statistical framework to theoretically study how self-supervised metric learning can benefit downstream tasks in the context of multi-view data. Under this framework, we show that the target distance of metric learning satisfies several desired properties for the downstream tasks. On the other hand, our investigation suggests the target distance can be further improved by moderating each direction’s weights. In addition, our analysis precisely characterizes the improvement by self-supervised metric learning on four commonly used downstream tasks: sample identification, two-sample testing, k-means clustering, and k-nearest neighbor classification. When the distance is estimated from an unlabeled dataset, we establish the upper bound on distance estimation’s accuracy and the number of samples sufficient for downstream task improvement. Finally, numerical experiments are presented to support the theoretical results in the paper.

Extending rank-based inference to a multivariate setting such as multiple-output regression or MANOVA with unspecified d-dimensional error density has remained an open problem for more than half a century. None of the many solutions proposed so far is enjoying the combination of distribution-freeness and efficiency that makes rank-based inference a successful tool in the univariate setting. A concept of center- outward multivariate ranks and signs based on measure transportation ideas has been introduced recently. Center-outward ranks and signs are not only distribution-free but achieve in dimension d > 1 the (essential) maximal ancillarity property of traditional univariate ranks. In the present case, we show that fully distribution-free testing procedures based on center-outward ranks can achieve parametric efficiency. We establish the Hajek representation and asymptotic normality results required in the construction of such tests in multiple-output regression and MANOVA models. Simulations and an empirical study demonstrate the excellent performance of the proposed procedures., info:eu-repo/semantics/published