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An introduction to the paper "Independent functions: Probability and analysis in Poland between the Wars," by Professor Philip Holgate, is presented.

Score-driven models have recently been introduced as a general framework to specify time-varying parameters of conditional densities. The score enjoys stochastic properties that make these models easy to implement and convenient to apply in several contexts, ranging from biostatistics to finance. Score-driven parameter updates have been shown to be optimal in terms of locally reducing a local version of the Kullback–Leibler divergence between the true conditional density and the postulated density of the model. A key limitation of such an optimality property is that it holds only locally both in the parameter space and sample space, yielding to a definition of local Kullback–Leibler divergence that is in fact not a divergence measure. The current paper shows that score-driven updates satisfy stronger optimality properties that are based on a global definition of Kullback–Leibler divergence. In particular, it is shown that score-driven updates reduce the distance between the expected updated parameter and the pseudo-true parameter. Furthermore, depending on the conditional density and the scaling of the score, the optimality result can hold globally over the parameter space, which can be viewed as a generalization of the monotonicity property of the stochastic gradient descent scheme. Several examples illustrate how the results derived in the paper apply to specific models under different easy-to-check assumptions, and provide a formal method to select the link function and the scaling of the score. [ABSTRACT FROM AUTHOR]

Quality control charts aim at raising an alarm as soon as sequentially obtained observations of an underlying random process no longer seem to be within stochastic fluctuations prescribed by an in-control scenario. Such random processes can often be modelled using the concept of stationarity, or even independence as in most classical works. An important out-of-control scenario is the changepoint alternative, for which the distribution of the process changes at an unknown point in time. In his seminal 1954 Biometrika paper, E. S. Page introduced the famous cumulative sum control charts for changepoint monitoring. Innovatively, decision rules based on cumulative sum procedures took the full history of the process into account, whereas previous procedures were based only on a fixed and typically small number of the most recent observations. The extreme case of using only the most recent observation, often referred to as the Shewhart chart, is more akin to serial outlier than changepoint detection. Page's cumulative sum approach, introduced seven decades ago, is ubiquitous in modern changepoint analysis, and his original paper has led to a multitude of follow-up papers in different research communities. This review is focused on a particular subfield of this research, namely nonparametric sequential, or online, changepoint tests that are constructed to maintain a desired Type-1 error as opposed to the more traditional approach seeking to minimize the average run length of the procedures. Such tests have originated at the intersection of econometrics and statistics. We trace the development of these tests and highlight their properties, mostly using a simple location model for clarity of exposition, but we also review more complex situations such as regression and time series models. [ABSTRACT FROM AUTHOR]

Exposure mappings facilitate investigations of complex causal effects when units interact in experiments. Current methods require experimenters to use the same exposure mappings to define the effect of interest and to impose assumptions on the interference structure. However, the two roles rarely coincide in practice, and experimenters are forced to make the often questionable assumption that their exposures are correctly specified. This paper argues that the two roles exposure mappings currently serve can, and typically should, be separated, so that exposures are used to define effects without necessarily assuming that they are capturing the complete causal structure in the experiment. The paper shows that this approach is practically viable by providing conditions under which exposure effects can be precisely estimated when the exposures are misspecified. Some important questions remain open. [ABSTRACT FROM AUTHOR]

The paper shows that matching without replacement on propensity scores produces estimators that generally are inconsistent for the average treatment effect of the treated. To achieve consistency, practitioners must either assume that no units exist with propensity scores greater than 1/2 or assume that there is no confounding among such units. The result is not driven by the use of propensity scores, and similar artifacts arise when matching on other scores as long as it is without replacement. [ABSTRACT FROM AUTHOR]

In many real-world networks, it is often observed that subgraphs or higher-order structures of certain configurations, e.g. triangles and by-fans, are overly abundant compared to standard randomly generated networks (Milo et al. 2002). However, statistical models accounting for this phenomenon are limited, especially when community structure is of interest. This limitation is coupled with a lack of community detection methods that leverage subgraphs or higher-order structures. In this paper, we propose a new community detection method that effectively uses higher-order structures in a network. Furthermore, for the community detection accuracy, under an edge-dependent network model that consists of both community and triangle structures, we develop a finite-sample error bound characterized by the expected triangle degree, which leads to the consistency of the proposed method. To the best of our knowledge, this is the first statistical error bound and consistency result for community detection of a single network considering a network model with dependent edges. We also show, in both simulation studies and a real-world data example, that our method unveils network communities that are otherwise invisible to methods that ignore higher-order structures. [ABSTRACT FROM AUTHOR]

While researchers commonly use the bootstrap to quantify the uncertainty of an estimator, it has been noticed that the standard bootstrap, in general, does not work for Chatterjee's rank correlation. In this paper, we provide proof of this issue under an additional independence assumption, and complement our theory with simulation evidence for general settings. Chatterjee's rank correlation thus falls into a category of statistics that are asymptotically normal, but bootstrap inconsistent. Valid inferential methods in this case are Chatterjee's original proposal for testing independence and the analytic asymptotic variance estimator of Lin & Han (2022) for more general purposes. [ Received on 5 April 2023. Editorial decision on 10 January 2024] [ABSTRACT FROM AUTHOR]

Estimation of the time-average variance constant is important for statistical analyses involving dependent data. This problem is difficult as it relies on a bandwidth parameter. Specifically, the optimal choices of the bandwidths of all existing estimators depend on the estimand itself and another unknown parameter that is very difficult to estimate. Thus, optimal variance estimation is unachievable. In this paper, we introduce a concept of converging flat-top kernels for constructing variance estimators whose optimal bandwidths are free of unknown parameters asymptotically and hence can be computed easily. We prove that the new estimator has an asymptotically constant risk and is locally asymptotically minimax. [ABSTRACT FROM AUTHOR]

Statistical hypotheses are translations of scientific hypotheses into statements about one or more distributions, often concerning their centre. Tests that assess statistical hypotheses of centre implicitly assume a specific centre, e.g. the mean or median. Yet, scientific hypotheses do not always specify a particular centre. This ambiguity leaves the possibility for a gap between scientific theory and statistical practice that can lead to rejection of a true null. In the face of replicability crises in many scientific disciplines, significant results of this kind are concerning. Rather than testing a single centre, this paper proposes testing a family of plausible centres, such as that induced by the Huber loss function. Each centre in the family generates a testing problem, and the resulting family of hypotheses constitutes a familial hypothesis. A Bayesian nonparametric procedure is devised to test familial hypotheses, enabled by a novel pathwise optimization routine to fit the Huber family. The favourable properties of the new test are demonstrated theoretically and experimentally. Two examples from psychology serve as real-world case studies. [ABSTRACT FROM AUTHOR]

We propose a novel method for testing serial independence of object-valued time series in metric spaces, which are more general than Euclidean or Hilbert spaces. The proposed method is fully nonparametric, free of tuning parameters and can capture all nonlinear pairwise dependence. The key concept used in this paper is the distance covariance in metric spaces, which is extended to the autodistance covariance for object-valued time series. Furthermore, we propose a generalized spectral density function to account for pairwise dependence at all lags and construct a Cramér–von Mises-type test statistic. New theoretical arguments are developed to establish the asymptotic behaviour of the test statistic. A wild bootstrap is also introduced to obtain the critical values of the nonpivotal limiting null distribution. Extensive numerical simulations and two real data applications on cumulative intraday returns and human mortality data are conducted to illustrate the effectiveness and versatility of our proposed test. [ABSTRACT FROM AUTHOR]

Direct use of the likelihood function typically produces severely biased estimates when the dimension of the parameter vector is large relative to the effective sample size. With linearly separable data generated from a logistic regression model, the loglikelihood function asymptotes and the maximum likelihood estimator does not exist. We show that an exact analysis for each regression coefficient produces half-infinite confidence sets for some parameters when the data are separable. Such conclusions are not vacuous, but an honest portrayal of the limitations of the data. Finite confidence sets are only achievable when additional, perhaps implicit, assumptions are made. Under a notional double-asymptotic regime in which the dimension of the logistic coefficient vector increases with the sample size, the present paper considers the implications of enforcing a natural constraint on the vector of logistic transformed probabilities. We derive a relationship between the logistic coefficients and a notional parameter obtained as a probability limit of an ordinary least-squares estimator. The latter exists even when the data are separable. Consistency is ascertained under weak conditions on the design matrix. [ABSTRACT FROM AUTHOR]

In this paper, we develop a systematic theory for high-dimensional analysis of variance in multivariate linear regression, where the dimension and the number of coefficients can both grow with the sample size. We propose a new U -type statistic to test linear hypotheses and establish a high-dimensional Gaussian approximation result under fairly mild moment assumptions. Our general framework and theory can be used to deal with the classical one-way multivariate analysis of variance, and the nonparametric one-way multivariate analysis of variance in high dimensions. To implement the test procedure, we introduce a sample-splitting-based estimator of the second moment of the error covariance and discuss its properties. A simulation study shows that our proposed test outperforms some existing tests in various settings. [ABSTRACT FROM AUTHOR]

In observational causal inference, exact covariate matching plays two statistical roles: (i) it effectively controls for bias due to measured confounding; (ii) it justifies assumption-free inference based on randomization tests. In this paper we show that inexact covariate matching does not always play these same roles. We find that inexact matching often leaves behind statistically meaningful bias, and that this bias renders standard randomization tests asymptotically invalid. We therefore recommend additional model-based covariate adjustment after inexact matching. In the framework of local misspecification, we prove that matching makes subsequent parametric analyses less sensitive to model selection or misspecification. We argue that gaining such robustness is the primary statistical role of inexact matching. [ABSTRACT FROM AUTHOR]

Consider a survival time |$T$| that is subject to random right censoring, and suppose that |$T$| is stochastically dependent on the censoring time |$C$|⁠. We are interested in the marginal distribution of |$T$|⁠. This situation is often encountered in practice. Consider, for example, the case where |$T$| is a patient's time to death from a certain disease. Then the censoring time |$C$| could be the time until the patient leaves the study or the time until death from another cause. If the reason for leaving the study is related to the health condition of the patient, or if the patient dies from a disease that has similar risk factors to the disease of interest, then |$T$| and |$C$| are likely to be dependent. In this paper we propose a new model that takes such dependence into account. The model is based on a parametric copula for the relationship between |$T$| and |$C$|⁠ , and on parametric marginal distributions for |$T$| and |$C$|⁠. Unlike most other authors, we do not assume that the parameter defining the copula is known. We give sufficient conditions on these parametric copulas and marginals under which the bivariate distribution of |$(T,C)$| is identified. These sufficient conditions are then checked for a wide range of common copulas and marginals. We also study the estimation of the model, and carry out extensive simulations and analysis on a pancreatic cancer dataset to illustrate the proposed model and estimation procedure. [ABSTRACT FROM AUTHOR]

The micro-randomized trial is a sequential randomized experimental design to empirically evaluate the effectiveness of mobile health intervention components that may be delivered at hundreds or thousands of decision points. Micro-randomized trials have motivated a new class of causal estimands, termed causal excursion effects , for which semiparametric inference can be conducted via a weighted, centred least-squares criterion (Boruvka et al. 2018). Causal excursion effects allow health scientists to answer important scientific questions about how intervention effectiveness may change over time or may be moderated by individual characteristics, time-varying context or past responses. Existing definitions and associated methods assume between-subject independence and noninterference. Deviations from these assumptions often occur. In this paper, causal excursion effects are revisited under potential cluster-level treatment effect heterogeneity and interference, where the treatment effect of interest may depend on cluster-level moderators. Utility of the proposed methods is shown by analysing data from a multi-institution cohort of first-year medical residents in the United States. [ABSTRACT FROM AUTHOR]

The Kronecker product covariance structure provides an efficient way to model the inter-correlations of matrix-variate data. In this paper, we propose test statistics for the Kronecker product covariance matrix based on linear spectral statistics of renormalized sample covariance matrices. A central limit theorem is proved for the linear spectral statistics, with explicit formulas for the mean and covariance functions, thereby filling a gap in the literature. We then show theoretically that the proposed test statistics have well-controlled size and high power. We further propose a bootstrap resampling algorithm to approximate the limiting distributions of the associated linear spectral statistics. Consistency of the bootstrap procedure is guaranteed under mild conditions. The proposed test procedure is also applicable to the Kronecker product covariance model with additional random noise. In our simulations, the empirical sizes of the proposed test procedure and its bootstrapped version are close to the corresponding theoretical values, while the power converges to |$1$| quickly as the dimension and sample size increase. [ABSTRACT FROM AUTHOR]

A basic principle in the design of observational studies is to approximate the randomized experiment that would have been conducted under ideal circumstances. At present, linear regression models are commonly used to analyse observational data and estimate causal effects. How do linear regression adjustments in observational studies emulate key features of randomized experiments, such as covariate balance, self-weighted sampling and study representativeness? In this paper, we provide answers to this and related questions by analysing the implied individual-level data weights of various linear regression methods. We derive new closed-form expressions of these implied weights, and examine their properties in both finite and large samples. Among others, in finite samples we characterize the implied target population of linear regression, and in large samples demonstrate the multiply robust properties of regression estimators from the perspective of their implied weights. We show that the implied weights of general regression methods can be equivalently obtained by solving a convex optimization problem. This equivalence allows us to bridge ideas from the regression modelling and causal inference literatures. As a result, we propose novel regression diagnostics for causal inference that are part of the design stage of an observational study. We implement the weights and diagnostics in the new lmw package for R. [ABSTRACT FROM AUTHOR]

A key challenge in causal inference from observational studies is the identification and estimation of causal effects in the presence of unmeasured confounding. In this paper, we introduce a novel approach for causal inference that leverages information in multiple outcomes to deal with unmeasured confounding. An important assumption in our approach is conditional independence among multiple outcomes. In contrast to existing proposals in the literature, the roles of multiple outcomes in the conditional independence assumption are symmetric; hence, the name parallel outcomes. We show nonparametric identifiability with at least three parallel outcomes and provide parametric estimation tools under a set of linear structural equation models. Our proposal is evaluated through a set of synthetic and real data analyses. [ABSTRACT FROM AUTHOR]

This paper introduces an assumption-lean method that constructs valid and efficient lower predictive bounds for survival times with censored data. We build on recent work by Candès et al. (2023) , whose approach first subsets the data to discard any data points with early censoring times and then uses a reweighting technique, namely, weighted conformal inference (Tibshirani et al. 2019), to correct for the distribution shift introduced by this subsetting procedure. For our new method, instead of constraining to a fixed threshold for the censoring time when subsetting the data, we allow for a covariate-dependent and data-adaptive subsetting step, which is better able to capture the heterogeneity of the censoring mechanism. As a result, our method can lead to lower predictive bounds that are less conservative and give more accurate information. We show that in the Type-I right-censoring setting, if either the censoring mechanism or the conditional quantile of the survival time is well estimated, our proposed procedure achieves nearly exact marginal coverage, where in the latter case we additionally have approximate conditional coverage. We evaluate the validity and efficiency of our proposed algorithm in numerical experiments, illustrating its advantage when compared with other competing methods. Finally, our method is applied to a real dataset to generate lower predictive bounds for users' active times on a mobile app. [ABSTRACT FROM AUTHOR]

We propose an empirical Bayes formulation of the structure learning problem, where the prior specification assumes that all node variables have the same error variance, an assumption known to ensure the identifiability of the underlying causal directed acyclic graph. To facilitate efficient posterior computation, we approximate the posterior probability of each ordering by that of a best directed acyclic graph model, which naturally leads to an order-based Markov chain Monte Carlo algorithm. Strong selection consistency for our model in high-dimensional settings is proved under a condition that allows heterogeneous error variances, and the mixing behaviour of our sampler is theoretically investigated. Furthermore, we propose a new iterative top-down algorithm, which quickly yields an approximate solution to the structure learning problem and can be used to initialize the Markov chain Monte Carlo sampler. We demonstrate that our method outperforms other state-of-the-art algorithms under various simulation settings, and conclude the paper with a single-cell real-data study illustrating practical advantages of the proposed method. [ABSTRACT FROM AUTHOR]

This document is a correction notice for an article titled "Nonparametric efficient causal mediation with intermediate confounders." The correction addresses a typographical error in the proof of Theorem 5 of the Supplementary Material, which resulted in incorrect calculations and results in the paper. The authors provide the corrected versions of the affected theorems and lemma, along with an explanation of the error and its impact. The text includes a mathematical equation and a reference to the original article for further information. [Extracted from the article]

Likelihood-based inference under nonconvex constraints on model parameters has become increasingly common in biomedical research. In this paper, we establish large-sample properties of the maximum likelihood estimator when the true parameter value lies at the boundary of a nonconvex parameter space. We further derive the asymptotic distribution of the likelihood ratio test statistic under nonconvex constraints on model parameters. A general Monte Carlo procedure for generating the limiting distribution is provided. The theoretical results are demonstrated by five examples in Anderson's stereotype logistic regression model, genetic association studies, gene-environment interaction tests, cost-constrained linear regression and fairness-constrained linear regression. [ABSTRACT FROM AUTHOR]

Varimax factor rotations, while popular among practitioners in psychology and statistics since being introduced by Kaiser (1958) , have historically been viewed with skepticism and suspicion by some theoreticians and mathematical statisticians. Now, work by Rohe & Zeng (2023) provides new, fundamental insight: varimax rotations provably perform statistical estimation in certain classes of latent variable models when paired with spectral-based matrix truncations for dimensionality reduction. We build on this new-found understanding of varimax rotations by developing further connections to network analysis and spectral methods rooted in entrywise matrix perturbation analysis. Concretely, this paper establishes the asymptotic multivariate normality of vectors in varimax-transformed Euclidean point clouds that represent low-dimensional node embeddings in certain latent space random graph models. We address related concepts including network sparsity, data denoising and the role of matrix rank in latent variable parameterizations. Collectively, these findings, at the confluence of classical and contemporary multivariate analysis, reinforce methodology and inference procedures grounded in matrix factorization-based techniques. Numerical examples illustrate our findings and supplement our discussion. [ABSTRACT FROM AUTHOR]

In this paper, we develop an eigenvector-assisted estimation framework for a collection of signal-plus-noise matrix models arising in high-dimensional statistics and many applications. The framework is built upon a novel asymptotically unbiased estimating equation using the leading eigenvectors of the data matrix. However, the estimator obtained by directly solving the estimating equation could be numerically unstable in practice and lacks robustness against model misspecification. We propose to use the quasi-posterior distribution by exponentiating a criterion function whose maximizer coincides with the estimating equation estimator. The proposed framework can incorporate heteroskedastic variance information, but does not require the complete specification of the sampling distribution and is also robust to the potential misspecification of the distribution of the noise matrix. Computationally, the quasi-posterior distribution can be obtained via a Markov chain Monte Carlo sampler, which exhibits superior numerical stability over some of the existing optimization-based estimators and is straightforward for uncertainty quantification. Under mild regularity conditions, we establish the large sample properties of the quasi-posterior distributions. In particular, the quasi-posterior credible sets have the correct frequentist nominal coverage probability provided that the criterion function is carefully selected. The validity and usefulness of the proposed framework are demonstrated through the analysis of synthetic datasets and the real-world ENZYMES network datasets. [ABSTRACT FROM AUTHOR]

While model selection is a well-studied topic in parametric and nonparametric regression or density estimation, selection of possibly high-dimensional nuisance parameters in semiparametric problems is far less developed. In this paper, we propose a selective machine learning framework for making inferences about a finite-dimensional functional defined on a semiparametric model, when the latter admits a doubly robust estimating function and several candidate machine learning algorithms are available for estimating the nuisance parameters. We introduce a new selection criterion aimed at bias reduction in estimating the functional of interest based on a novel definition of pseudo risk inspired by the double robustness property. Intuitively, the proposed criterion selects a pair of learners with the smallest pseudo risk, so that the estimated functional is least sensitive to perturbations of a nuisance parameter. We establish an oracle property for a multi-fold cross-validation version of the new selection criterion that states that our empirical criterion performs nearly as well as an oracle with a priori knowledge of the pseudo risk for each pair of candidate learners. Finally, we apply the approach to model selection of a semiparametric estimator of average treatment effect given an ensemble of candidate machine learners to account for confounding in an observational study that we illustrate in simulations and a data application. [ABSTRACT FROM AUTHOR]

One of the most interesting problems in the recent renaissance of the studies in kernel regression might be whether kernel interpolation can generalize well, since it may help us understand the 'benign overfitting phenomenon' reported in the literature on deep networks. In this paper, under mild conditions, we show that, for any ε > 0 ⁠ , the generalization error of kernel interpolation is lower bounded by Ω (n − ε) ⁠. In other words, the kernel interpolation generalizes poorly for a large class of kernels. As a direct corollary, we can show that overfitted wide neural networks defined on the sphere generalize poorly. [ABSTRACT FROM AUTHOR]

It is frequently observed in practice that the Wald statistic gives a poor assessment of the statistical significance of a variance component. This paper provides detailed analytic insight into the phenomenon by way of two simple models, which point to an atypical geometry as the source of the aberration. The latter can in principle be checked numerically to cover situations of arbitrary complexity, such as those arising from elaborate forms of blocking in an experimental context, or models for longitudinal or clustered data. The salient point, echoing Dickey (2020) , is that a suitable likelihood-ratio test should always be used for the assessment of variance components. [ABSTRACT FROM AUTHOR]

Fitting parametric models by optimizing frequency-domain objective functions is an attractive approach of parameter estimation in time series analysis. Whittle estimators are a prominent example in this context. Under weak conditions and the assumption that the true spectral density of the underlying process does not necessarily belong to the parametric class of spectral densities fitted, the distribution of Whittle estimators typically depends on difficult to estimate characteristics of the underlying process. This makes the implementation of asymptotic results for the construction of confidence intervals or for assessing the variability of estimators difficult in practice. In this paper we propose a frequency-domain bootstrap method to estimate the distribution of Whittle estimators that is asymptotically valid under assumptions that not only allow for possible model misspecification, but also for weak dependence conditions that are satisfied by a wide range of stationary stochastic processes. Adaptations of the bootstrap procedure developed to incorporate different modifications of Whittle estimators proposed in the literature, such as, for instance, tapered, debiased or boundary extended Whittle estimators, are also considered. Simulations demonstrate the capabilities of the bootstrap method proposed and its good finite sample performance. A real-life data analysis on sunspots is also presented. [ABSTRACT FROM AUTHOR]

For the case with a single causal variable, Dawid et al. (2014) defined the probability of causation, and Pearl (2000) defined the probability of necessity to assess the causes of effects. For a case with multiple causes that could affect each other, this paper defines the posterior total and direct causal effects based on the evidence observed for post-treatment variables, which could be viewed as measurements of causes of effects. Posterior causal effects involve the probabilities of counterfactual variables. Thus, as with the probability of causation, the probability of necessity and direct causal effects, the identifiability of posterior total and direct causal effects requires more assumptions than the identifiability of traditional causal effects conditional on pre-treatment variables. We present assumptions required for the identifiability of posterior causal effects and provide identification equations. Further, when the causal relationships between multiple causes and an endpoint can be depicted by causal networks, we can simplify both the required assumptions and the identification equations of the posterior total and direct causal effects. Finally, using numerical examples, we compare the posterior total and direct causal effects with other measures for evaluating the causes of effects and the population attributable risks. [ABSTRACT FROM AUTHOR]

We consider the problem of estimating and inferring treatment effects in randomized experiments. In practice, stratified randomization, or more generally, covariate-adaptive randomization, is routinely used in the design stage to balance treatment allocations with respect to a few variables that are most relevant to the outcomes. Then, regression is performed in the analysis stage to adjust the remaining imbalances to yield more efficient treatment effect estimators. Building upon and unifying recent results obtained for ordinary-least-squares adjusted estimators under covariate-adaptive randomization, this paper presents a general theory of regression adjustment that allows for model mis-specification and the presence of a large number of baseline covariates. We exemplify the theory on two lasso-adjusted treatment effect estimators, both of which are optimal in their respective classes. In addition, nonparametric consistent variance estimators are proposed to facilitate valid inferences, which work irrespective of the specific randomization methods used. The robustness and improved efficiency of the proposed estimators are demonstrated through numerical studies. [ABSTRACT FROM AUTHOR]