This paper proposes a uniqueness Shapley measure to compare the extent to which different variables are able to identify a subject. Revealing the value of a variable on subject t shrinks the set of possible subjects that t could be. The extent of the shrinkage depends on which other variables have also been revealed. We use Shapley value to combine all of the reductions in log cardinality due to revealing a variable after some subset of the other variables has been revealed. This uniqueness Shapley measure can be aggregated over subjects where it becomes a weighted sum of conditional entropies. Aggregation over subsets of subjects can address questions like how identifying is age for people of a given zip code. Such aggregates have a corresponding expression in terms of cross entropies. We use uniqueness Shapley to investigate the differential effects of revealing variables from the North Carolina voter registration rolls and in identifying anomalous solar flares. An enormous speedup (approaching 2000 fold in one example) is obtained by using the all dimension trees of Moore and Lee (1998) to store the cardinalities we need.

The logistic and probit link functions are the most common choices for regression models with a binary response. However, these choices are not robust to the presence of outliers/unexpected observations. The robit link function, which is equal to the inverse CDF of the Student’s t-distribution, provides a robust alternative to the probit and logistic link functions. A multivariate normal prior for the regression coefficients is the standard choice for Bayesian inference in robit regression models. The resulting posterior density is intractable and a Data Augmentation (DA) Markov chain is used to generate approximate samples from the desired posterior distribution. Establishing geometric ergodicity for this DA Markov chain is important as it provides theoretical guarantees for asymptotic validity of MCMC standard errors for desired posterior expectations/quantiles. Previous work [16] established geometric ergodicity of this robit DA Markov chain assuming (i) the sample size n dominates the number of predictors p, and (ii) an additional constraint which requires the sample size to be bounded above by a fixed constant which depends on the design matrix X. In particular, modern high-dimensional settings where $n<p$ are not considered. In this work, we show that the robit DA Markov chain is trace-class (i.e., the eigenvalues of the corresponding Markov operator are summable) for arbitrary choices of the sample size n, the number of predictors p, the design matrix X, and the prior mean and variance parameters. The trace-class property implies geometric ergodicity. Moreover, this property allows us to conclude that the sandwich robit chain (obtained by inserting an inexpensive extra step in between the two steps of the DA chain) is strictly better than the robit DA chain in an appropriate sense, and enables the use of recent methods to estimate the spectral gap of trace class DA Markov chains.

A method for the detection of changes in the expectation in univariate sequences is provided. Moving sum processes (MOSUM) are studied. These rely on the selection of a tuning bandwidth. Here, a framework to overcome bandwidth selection is presented – the bandwidth adjusts gradually. For that, MOSUM are made dependent on both time and the bandwidth: the domain becomes a triangle. On the triangle, paths are constructed which systematically lead to change points. An algorithm is provided that estimates change points by subsequent consideration of paths. Strong consistency for the number and location of change points is shown. Simulation studies corroborate estimation precision and reveal competitiveness with state of the art change point detection methods. A companion $\mathtt{\text{R}}$-package mscp is made available on CRAN.

In many statistical problems the hypotheses are naturally divided into groups, and the investigators are interested to perform group-level inference, possibly along with inference on individual hypotheses. We consider the goal of discovering groups containing u or more signals with group-level false discovery rate (FDR) control. This goal can be addressed by multiple testing of partial conjunction hypotheses with a parameter u, which reduce to global null hypotheses for $u=1$. We consider the case where the partial conjunction p-values are combinations of within-group p-values, and obtain sufficient conditions on (1) the dependencies among the p-values within and across the groups, (2) the combining method for obtaining partial conjunction p-values, and (3) the multiple testing procedure, for obtaining FDR control on partial conjunction discoveries. We consider separately the dependencies encountered in the meta-analysis setting, where multiple features are tested in several independent studies, and the p-values within each study may be dependent. Based on the results for this setting, we generalize the procedure of Benjamini, Heller, and Yekutieli (2009) for assessing replicability of signals across studies, and extend their theoretical results regarding FDR control with respect to replicability claims.

In this work we consider the problem of estimating function-on-scalar regression models when the functions are observed over multi-dimensional or manifold domains and with potentially multivariate output. We establish the minimax rates of convergence and present an estimator based on reproducing kernel Hilbert spaces that achieves the minimax rate. To better interpret the derived rates, we extend well-known links between RKHS and Sobolev spaces to the case where the domain is a compact Riemannian manifold. This is accomplished using an interesting connection to Weyl’s Law from partial differential equations. We conclude with a numerical study and an application to 3D facial imaging.

In this paper, we consider the inverse problem of estimating the product $fg$ of two densities, given a d-dimensional n-sample of i.i.d. observations drawn from each distribution. We propose a general method of estimation encompassing both projection estimators with model selection device and kernel estimators with bandwidth selection strategies. The procedures do not consist in making the product of each density estimator, but in plugging an overfitted estimator of one of the two densities, in an estimator based on the second sample. Our findings are a first step toward a better understanding of the good performances of overfitting in regression Nadaraya-Watson estimator.

Tie-breaker experimental designs are hybrids of Randomized Controlled Trials (RCTs) and Regression Discontinuity Designs (RDDs) in which subjects with moderate scores are placed in an RCT while subjects with extreme scores are deterministically assigned to the treatment or control group. In settings where it is unfair or uneconomical to deny the treatment to the more deserving recipients, the tie-breaker design (TBD) trades off the practical advantages of the RDD with the statistical advantages of the RCT. The practical costs of the randomization in TBDs can be hard to quantify in generality, while the statistical benefits conferred by randomization in TBDs have only been studied under linear and quadratic models. In this paper, we discuss and quantify the statistical benefits of TBDs without using parametric modelling assumptions. If the goal is estimation of the average treatment effect or the treatment effect at more than one score value, the statistical benefits of using a TBD over an RDD are apparent. If the goal is nonparametric estimation of the mean treatment effect at merely one score value, we prove that about 2.8 times more subjects are needed for an RDD in order to achieve the same asymptotic mean squared error. We further demonstrate using both theoretical results and simulations from the Angrist and Lavy (1999) classroom size dataset, that larger experimental radii choices for the TBD lead to greater statistical efficiency.

Let $x_1,\cdots ,x_n$ be independent observations of size p, each of them belonging to one of c distinct classes. We assume that observations within the class a are characterized by their distribution $N(0,\frac{1}{p}{C}_a)$ where here $C_1,\cdots ,C_c$ are some non-negative definite $p\times p$ matrices. This paper studies the asymptotic behavior of the symmetric matrix ${\tilde{\mathrm{\Phi }}}_{kl}=\sqrt{p}\left({(x_k^T{x}_l)}^2{\mathit{\delta }}_{k\ne l}\right)$ when p and n grow to infinity with $\frac{n}{p}\to c_0$. Particularly, we prove that, if the class covariance matrices are sufficiently close in a certain sense, the matrix $\tilde{\mathrm{\Phi }}$ behaves like a low-rank perturbation of a Wigner matrix, presenting possibly some isolated eigenvalues outside the bulk of the semi-circular law. We carry out a careful analysis of some of the isolated eigenvalues of $\tilde{\mathrm{\Phi }}$ and their associated eigenvectors and illustrate how these results can help understand spectral clustering methods that use $\tilde{\mathrm{\Phi }}$ as a kernel matrix.

In recent years, generative adversarial networks (GANs) have demonstrated impressive experimental results while there are only a few works that foster statistical learning theory for GANs. In this work, we propose an infinite dimensional theoretical framework for generative adversarial learning. We assume that the probability density functions of the underlying measure are uniformly bounded, k-times α-Hölder differentiable ($C^{k,\mathit{\alpha }}$) and uniformly bounded away from zero. Under these assumptions, we show that the Rosenblatt transformation induces an optimal generator, which is realizable in the hypothesis space of $C^{k,\mathit{\alpha }}$-generators. With a consistent definition of the hypothesis space of discriminators, we further show that the Jensen-Shannon divergence between the distribution induced by the generator from the adversarial learning procedure and the data generating distribution converges to zero. Under certain regularity assumptions on the density of the data generating process, we also provide rates of convergence based on chaining and concentration.

In this paper, we aim to test the overall significance of regression coefficients in high-dimensional single-index models. We first reformulate the hypothesis testing problem under elliptical distributions for predictors. Applying distribution-based transformation, we introduce a high-dimensional score-type test statistic. Notably, no moment condition for the error term is required. Our introduced procedures are thus robust with respect to outliers in response. Moreover our procedure is free of variance estimation of the error term. We establish the test statistic’s asymptotic normality under null hypothesis. Power analysis is also investigated. To further improve computational efficiency and enhance empirical powers, we also introduce a two-stage test procedure under ultrahigh-dimensional settings based on random data splitting. To eliminate the additional randomness induced by data splitting, we further develop a powerful ensemble algorithm based on multiple data splitting. We show that the ensemble algorithm can control the type I error rate at a given significance level. Extension to partial significance testing problem is also investigated. Lastly, numerical studies and real data analysis are conducted to compare with existing approaches and to illustrate the robustness and validity of our proposed test procedures.

A bipartite experiment consists of one set of units being assigned treatments and another set of units for which we measure outcomes. The two sets of units are connected by a bipartite graph, governing how the treated units can affect the outcome units. In this paper, we consider estimation of the average total treatment effect in the bipartite experimental framework under a linear exposure-response model. We introduce the Exposure Reweighted Linear (ERL) estimator, and show that the estimator is unbiased, consistent and asymptotically normal, provided that the bipartite graph is sufficiently sparse. To facilitate inference, we introduce an unbiased and consistent estimator of the variance of the ERL point estimator. Finally, we introduce a cluster-based design, Exposure-Design, that uses heuristics to increase the precision of the ERL estimator by realizing a desirable exposure distribution.

Envelope methodology is succinctly pitched as a class of procedures for increasing efficiency in multivariate analyses without altering traditional objectives [5, first sentence of page 1]. This description comes with the additional caveat that efficiency gains obtained by envelope methodology are mitigated by model selection volatility to an unknown degree. Recent strides to account for model selection volatility have been made on two fronts: 1) development of a weighted envelope estimator to account for this variability directly in the context of the multivariate linear regression model; 2) development of model selection criteria that facilitate consistent dimension selection for more general settings. We unify these two directions and provide weighted envelope estimators that directly account for the variability associated with model selection and are appropriate for general multivariate estimation settings. Our weighted estimation technique provides practitioners with robust and useful variance reduction in finite samples. Theoretical and empirical justification is given for our estimators and validity of a nonparametric bootstrap procedure for estimating their asymptotic variance are established. Simulation studies and a real data analysis support our claims and demonstrate the advantage of our weighted envelope estimator when model selection variability is present.

Predictive classification considered in this paper concerns the problem of identifying subgroups based on a continuous biomarker through estimation of an unknown cutpoint and assessing whether these subgroups differ in treatment effect relative to some clinical outcome. The problem is considered under a generalized linear model framework for clinical outcomes and formulated as testing the significance of the interaction between the treatment and the subgroup indicator. When the main effect of the subgroup indicator does not exist, the cutpoint is non-identifiable under the null. Existing procedures are not adaptive to the identifiability issue, and do not work well when the main effect is small. In this work, we propose profile score-type and Wald-type test statistics, and further m-out-of-n bootstrap techniques to obtain their critical values. The proposed procedures do not rely on the knowledge about the model identifiability, and we establish their asymptotic size validity and study the power under local alternatives in both cases. Further, we show that the standard bootstrap is inconsistent for the non-identifiable case. Simulation results corroborate our theory, and the proposed method is applied to a dataset from a clinical trial on advanced colorectal cancer.

We consider the deconvolution problem for densities supported on a $(d-1)$-dimensional sphere with unknown center and unknown radius, in the situation where the distribution of the noise is unknown and without any other observations. We propose estimators of the radius, of the center, and of the density of the signal on the sphere that are proved consistent without further information. The estimator of the radius is proved to have almost parametric convergence rate for any dimension d. When $d=2$, the estimator of the density is proved to achieve the same rate of convergence over Sobolev regularity classes of densities as when the noise distribution is known.

We propose a new autocorrelation measure for functional time series that we term “spherical autocorrelation”. It is based on measuring the average angle between lagged pairs of series after having been projected onto the unit sphere. This new measure enjoys several complimentary advantages compared to existing autocorrelation measures for functional data, since it both 1) describes a notion of “sign” or “direction” of serial dependence in the series, and 2) is more robust to outliers. The asymptotic properties of estimators of the spherical autocorrelation are established, and are used to construct confidence intervals and portmanteau white noise tests. These confidence intervals and tests are shown to be effective in simulation experiments, and demonstrated in applications to model selection for daily electricity price curves, and measuring the volatility in densely observed asset price data.

Local general depth ($LGD$) functions are used for describing the local geometric features and mode(s) in multivariate distributions. In this paper, we undertake a rigorous systematic study of $LGD$ and establish several analytical and statistical properties. First, we show that, when the underlying probability distribution is absolutely continuous with density $f(\cdot )$, the scaled version of $LGD$ (referred to as τ-approximation) converges, uniformly and in $L^d({\mathbb{R}}^p)$ to $f(\cdot )$ when τ converges to zero. Second, we establish that, as the sample size diverges to infinity the centered and scaled sample $LGD$ converge in distribution to a centered Gaussian process uniformly in the space of bounded functions on $H_G$, a class of functions yielding $LGD$. Third, using the sample version of the τ-approximation ($S\mathit{\tau }A$) and the gradient system analysis, we develop a new clustering algorithm. The validity of this algorithm requires several results concerning the uniform finite difference approximation of the gradient system associated with $S\mathit{\tau }A$. For this reason, we establish Bernstein-type inequality for deviations between the centered and scaled sample $LGD$, which is also of independent interest. Finally, invoking the above results, we establish consistency of the clustering algorithm. Applications of the proposed methods to mode estimation and upper level set estimation are also provided. Finite sample performance of the methodology are evaluated using numerical experiments and data analysis.

We consider ill-posed inverse problems where the forward operator T is unknown, and instead we have access to training data consisting of functions $f_i$ and their noisy images $T{f}_i$. This is a practically relevant and challenging problem which current methods are able to solve only under strong assumptions on the training set. Here we propose a new method that requires minimal assumptions on the data, and prove reconstruction rates that depend on the number of training points and the noise level. We show that, in the regime of “many” training data, the method is minimax optimal. The proposed method employs a type of convolutional neural networks (U-nets) and empirical risk minimization in order to “fit” the unknown operator. In a nutshell, our approach is based on two ideas: the first is to relate U-nets to multiscale decompositions such as wavelets, thereby linking them to the existing theory, and the second is to use the hierarchical structure of U-nets and the low number of parameters of convolutional neural nets to prove entropy bounds that are practically useful. A significant difference with the existing works on neural networks in nonparametric statistics is that we use them to approximate operators and not functions, which we argue is mathematically more natural and technically more convenient.

We propose a new method under the Bayesian framework to perform valid inference for low dimensional parameters in high dimensional linear models under sparsity constraints. Our approach is to use surrogate Bayesian posteriors based on partial regression models to remove the effect of high dimensional nuisance variables. We name the final distribution we used to conduct inference “conditional Bayesian posterior” as it is a surrogate posterior constructed conditional on quasi posterior distributions of other parameters and does not admit a fully Bayesian interpretation. Unlike existing Bayesian regularization methods, our method can be used to quantify the estimation uncertainty for arbitrarily small signals and therefore does not require variable selection consistency to guarantee its validity. Theoretically, we show that the resulting Bayesian credible intervals achieve desired coverage probabilities in the frequentist sense. Methodologically, our proposed Bayesian framework can easily incorporate popular Bayesian regularization procedures such as those based on spike and slab priors and horseshoe priors to facilitate high accuracy estimation and inference. Numerically, our proposed method rectifies the uncertainty underestimation of Bayesian shrinkage approaches and has a comparable empirical performance with state-of-the-art frequentist methods based on extensive simulation studies and a real data analysis.

We consider the nonparametric regression problem with multiple predictors and an additive error, where the regression function is assumed to be coordinatewise nondecreasing. We propose a Bayesian approach to make inferences on the multivariate monotone regression function, obtain the posterior contraction rate, and construct a universally consistent Bayesian testing procedure for multivariate monotonicity. To facilitate posterior analysis, we temporarily set aside the shape restrictions, and endow a prior on blockwise constant regression functions with independently normally distributed heights. The unknown variance of the error term is either estimated by the marginal maximum likelihood estimate, or is equipped with an inverse-gamma prior. Then the unrestricted block-heights are a posteriori also independently normally distributed given the error variance, by conjugacy. To comply with the shape restrictions, we project samples from the unrestricted posterior onto the class of multivariate monotone functions, inducing the “projection-posterior distribution”, to be used for making an inference. Under an ${\mathbb{L}}_1$-metric, we show that the projection-posterior based on n independent samples contracts around the true monotone regression function at the optimal rate $n^{-1∕(2+d)}$. Then we construct a Bayesian test for multivariate monotonicity based on the posterior probability of a shrinking neighborhood of the class of multivariate monotone functions. We show that the test is universally consistent, that is, the level of the Bayesian test goes to zero, and the power at any fixed alternative goes to one. Moreover, we show that for a smooth alternative function, power goes to one as long as its distance to the class of multivariate monotone functions is at least of the order of the estimation error for a smooth function. To the best of our knowledge, no other test for multivariate monotonicity is available in the Bayesian or the frequentist literature.

Epidemiological evidence suggests that simultaneous exposures to multiple environmental risk factors (Es) can increase disease risk larger than the additive effect of individual exposure acting alone. The interaction between a gene and multiple Es on a disease risk is termed as synergistic gene-environment interactions (synG $\times$ E). Single-index varying-coefficients models (SIVCM) have been a promising tool to model synergistic G $\times$ E effect and to understand how multiple Es jointly influence genetic risks on a disease outcome. In this work, we proposed a unified variable selection approach for SIVCM to estimate different effects of gene variables: varying, non-zero constant and zero effects which respectively correspond to nonlinear synG $\times$ E, no synG $\times$ E and no genetic effect. For multiple environmental exposure variables, we also estimated and selected important environmental variables that contribute to the synergistic interaction effect. We theoretically evaluated the oracle property of the proposed variable selection approach. Extensive simulation studies were conducted to evaluate the finite sample performance of the method, considering both continuous and discrete gene variables. Application to a real dataset further demonstrated the utility of the method. Our method has broad applications in areas where the purpose is to identify synergistic interaction effect.

Change-points are a routine feature of ‘big data’ observed in the form of high-dimensional data streams. In many such data streams, the component series possess group structures and it is natural to assume that changes only occur in a small number of all groups. We propose a new change point procedure, called groupInspect, that exploits the group sparsity structure to estimate a projection direction so as to aggregate information across the component series to successfully estimate the change-point in the mean structure of the series. We prove that the estimated projection direction is minimax optimal, up to logarithmic factors, when all group sizes are of comparable order. Moreover, our theory provide strong guarantees on the rate of convergence of the change-point location estimator. Numerical studies demonstrates the competitive performance of groupInspect in a wide range of settings and a real data example confirms the practical usefulness of our procedure.

Divide-and-conquer Bayesian methods consist of three steps: dividing the data into smaller computationally manageable subsets, running a sampling algorithm in parallel on all the subsets, and combining parameter draws from all the subsets. These methods use the combined parameter draws for efficient posterior inference in massive data settings. Existing divide-and-conquer methods have a major limitation in that their first two steps assume that the observations are independent. We address this problem by developing a divide-and-conquer method for Bayesian inference in parametric hidden Markov models, where the state space is known and finite. Our main contributions are two-fold. First, we partition the data into smaller blocks of consecutive observations and modify the likelihood on every time block. For any time block, the posterior distribution computed using the modified likelihood is such that its variance has the same asymptotic order as that of the true posterior. Second, suppose the number of subsets is chosen appropriately depending on the mixing properties of the hidden Markov chain. In that case, we show that the subset posterior distributions defined using the modified likelihood are asymptotically normal as the subset sample size tends to infinity. This result facilitates using any existing combination algorithm in the third step. We show that the combined posterior distribution obtained using one such algorithm is close to the true posterior distribution in 1-Wasserstein distance under widely used regularity assumptions. Our numerical results show that the proposed method provides an accurate approximation of the true posterior distribution than its competitors in simulation studies and a real data analysis.

The proof of Lemma 3 (pp 1841) in [1] is incorrect and the error part is actually not needed. We can directly prove (3.6) because $A_n(\mathit{\beta })$ is the left-hand side of (3.6), i.e., we do not need the terms $B_n$ and $C_n(\mathit{\beta })$.

Estimating the proportion of signals hidden in a large number of noise variables is a pervasive objective in scientific research. In this paper, we consider realistic, yet theoretically challenging scenarios with arbitrary covariance dependencies between variables. We quantify the overall level of covariance dependence using mean absolute correlation (MAC), and investigate the performance of a family of estimators across the full range of MAC values. We explore the joint effect of MAC dependence, signal sparsity, and signal intensity on estimator performance, and find that no single estimator in the family performs optimally across all MAC dependence levels. Based on this theoretical insight, we propose a new estimator that is better suited to arbitrary covariance dependencies. Our method compares favorably to several existing methods in a variety of finite-sample settings, including those with strong or weak covariance dependencies and real dependence structures from genetic association studies.

We introduce a forward sufficient dimension reduction method for categorical or ordinal responses by extending the outer product of gradients and minimum average variance estimator to categorical and ordinal-categorical generalized linear models. Previous works in this direction extend forward regression to binary responses, and are applied in a pairwise manner for multi-category data, which is less efficient than our approach. Like other forward regression-based sufficient dimension reduction methods, our approach avoids the relatively stringent distributional requirements necessary for inverse regression alternatives. We show the consistency of our proposed estimator and derive its convergence rate. We develop an algorithm for our methods based on repeated applications of available algorithms for forward regression. We also propose a clustering-based tuning procedure to estimate the bandwidth. The effectiveness of our estimator and related algorithms is demonstrated via simulations and applications.

Estimation of the mixing distribution under a general mixture model is a very difficult problem, especially when the mixing distribution is assumed to have a density. Predictive recursion (PR) is a fast, recursive algorithm for nonparametric estimation of a mixing distribution/density in general mixture models. However, the existing PR consistency results make rather strong assumptions, some of which fail for practically relevant mixture models. In this paper, we first develop new consistency results for PR under weaker conditions and then we apply this theory in the important case of mixtures of scaled uniform kernels.

Two-phase outcome dependent sampling (ODS) is widely used in many fields, especially when certain covariates are expensive and/or difficult to measure. For two-phase ODS, the conditional maximum likelihood (CML) method is very attractive because it can handle zero Phase 2 selection probabilities and avoids modeling the covariate distribution. However, most existing CML-based methods use only the Phase 2 sample and thus may be less efficient than other methods. We propose a general empirical likelihood method that uses CML augmented with additional information in the whole Phase 1 sample to improve estimation efficiency. The proposed method maintains the ability to handle zero selection probabilities and avoids modeling the covariate distribution, but can lead to substantial efficiency gains over CML in the inexpensive covariates, or in the influential covariate when a surrogate is available, because of an effective use of the Phase 1 data. Simulations and a real data illustration using NHANES data are presented.

With the increasing availability of non-Euclidean data objects, statisticians are faced with the task of developing appropriate statistical methods for their analysis. For regression models in which the predictors lie in ${\mathbb{R}}^p$ and the response variables are situated in a metric space, conditional Fréchet means can be used to define the Fréchet regression function. Global and local Fréchet methods have recently been developed for modeling and estimating this regression function as extensions of multiple and local linear regression, respectively. This paper expands on these methodologies by proposing the Fréchet single index model, in which the Fréchet regression function is assumed to depend only on a scalar projection of the multivariate predictor. Estimation is performed by combining local Fréchet along with M-estimation to estimate both the coefficient vector and the underlying regression function, and these estimators are shown to be consistent. The method is illustrated by simulations for response objects on the surface of the unit sphere and through an analysis of human mortality data in which lifetable data are represented by distributions of age-of-death, viewed as elements of the Wasserstein space of distributions.

Extreme U-statistics arise when the kernel of a U-statistic has a high degree but depends only on its arguments through a small number of top order statistics. As the kernel degree of the U-statistic grows to infinity with the sample size, estimators built out of such statistics form an intermediate family in between those constructed in the block maxima and peaks-over-threshold frameworks in extreme value analysis. The asymptotic normality of extreme U-statistics based on location-scale invariant kernels is established. Although the asymptotic variance coincides with the one of the Hájek projection, the proof goes beyond considering the first term in Hoeffding’s variance decomposition. We propose a kernel depending on the three highest order statistics leading to a location-scale invariant estimator of the extreme value index resembling the Pickands estimator. This extreme Pickands U-estimator is asymptotically normal and its finite-sample performance is competitive with that of the pseudo-maximum likelihood estimator.

This work considers the problem of modified Portmanteau tests for testing the adequacy of FARIMA models under the assumption that the errors are uncorrelated but not necessarily independent (i.e. weak FARIMA). We first study the joint distribution of the least squares estimator and the noise empirical autocovariances. We then derive the asymptotic distribution of residual empirical autocovariances and autocorrelations. We deduce the asymptotic distribution of the Ljung-Box (or Box-Pierce) modified Portmanteau statistics for weak FARIMA models. We also propose another method based on a self-normalization approach to test the adequacy of FARIMA models. Finally some simulation studies are presented to corroborate our theoretical work. An application to the Standard & Poor’s 500 and Nikkei returns also illustrates the practical relevance of our theoretical results.

This manuscript makes two contributions to the field of change-point detection. In a general change-point setting, we provide a generic algorithm for aggregating local homogeneity tests into an estimator of change-points in a time series. Interestingly, we establish that the error rates of the collection of tests directly translate into detection properties of the change-point estimator. This generic scheme is then applied to various problems including covariance change-point detection, nonparametric change-point detection and sparse multivariate mean change-point detection. For the latter, we derive minimax optimal rates that are adaptive to the unknown sparsity and to the distance between change-points when the noise is Gaussian. For sub-Gaussian noise, we introduce a variant that is optimal in almost all sparsity regimes.

A widely adopted semiparametric model for analyzing panel count data is a proportional mean model, which may be deemed inappropriate when the proportionality assumption is violated. Motivated by the popular accelerated failure time model that relaxes such assumption, we investigate accelerated mean models for semiparametric regression analysis of panel count data. For estimation of bundled parameters, we develop a sieve least squares estimation procedure, which is robust in the sense that no distributional assumption is required for the underlying recurrent event process. Overcoming the theoretical challenges from bundled parameters, we establish the consistency and convergence rate of the proposed estimators, and derive the asymptotic normality of both the finite-dimensional estimator and the functionals of the infinite-dimensional estimator. Simulation studies demonstrate promising performances of the proposed approach, and an application to a skin cancer chemoprevention trial yields some new findings.

To maximize clinical benefit, clinicians routinely tailor treatment to the individual characteristics of each patient, where individualized treatment rules are needed and are of significant research interest to statisticians. In the covariate-adjusted randomization clinical trial with many covariates, we model the treatment effect with an unspecified function of a single index of the covariates and leave the baseline response completely arbitrary. We devise a class of estimators to consistently estimate the treatment effect function and its associated index while bypassing the estimation of the baseline response, which is subject to the curse of dimensionality. We further develop inference tools to identify predictive covariates and isolate effective treatment region. The usefulness of the methods is demonstrated in both simulations and a clinical data example.

In this paper, we develop tools for statistical inference on replicated realizations of spatiotemporal processes that are locally time-harmonizable. Our method estimates both the rescaled spatial time-varying Loève-spectrum and the spatial time-varying dual-frequency coherence function under realistic modeling assumptions. We construct confidence intervals for these parameters of interest using the Circular Block Bootstrap method and prove its consistency. We illustrate the application of our methodology on a dataset arising from an experiment in neuropsychology. From EEG recordings, our method allows studying the dynamic functional connectivity within the brain associated to visual working memory performance.

Intractable generative models, or simulators, are models for which the likelihood is unavailable but sampling is possible. Most approaches to parameter inference in this setting require the computation of some discrepancy between the data and the generative model. This is for example the case for minimum distance estimation and approximate Bayesian computation. These approaches require simulating a high number of realisations from the model for different parameter values, which can be a significant challenge when simulating is an expensive operation. In this paper, we propose to enhance this approach by enforcing “sample diversity” in simulations of our models. This will be implemented through the use of quasi-Monte Carlo (QMC) point sets. Our key results are sample complexity bounds which demonstrate that, under smoothness conditions on the generator, QMC can significantly reduce the number of samples required to obtain a given level of accuracy when using three of the most common discrepancies: the maximum mean discrepancy, the Wasserstein distance, and the Sinkhorn divergence. This is complemented by a simulation study which highlights that an improved accuracy is sometimes also possible in some settings which are not covered by the theory.

We derive quantitative bounds on the rate of convergence in $L^1$ Wasserstein distance of general M-estimators, with an almost sharp (up to a logarithmic term) behavior in the number of observations. We focus on situations where the estimator does not have an explicit expression as a function of the data. The general method may be applied even in situations where the observations are not independent. Our main application is a rate of convergence for cross validation estimation of covariance parameters of Gaussian processes.

Multiple heterogeneous data sources are becoming increasingly available for statistical analyses in the era of big data. As an important example in finite-population inference, we develop a unified framework of the test-and-pool approach to general parameter estimation by combining gold-standard probability and non-probability samples. We focus on the case when the study variable is observed in both datasets for estimating the target parameters, and each contains other auxiliary variables. Utilizing the probability design, we conduct a pretest procedure to determine the comparability of the non-probability data with the probability data and decide whether or not to leverage the non-probability data in a pooled analysis. When the probability and non-probability data are comparable, our approach combines both data for efficient estimation. Otherwise, we retain only the probability data for estimation. We also characterize the asymptotic distribution of the proposed test-and-pool estimator under a local alternative and provide a data-adaptive procedure to select the critical tuning parameters that target the smallest mean square error of the test-and-pool estimator. Lastly, to deal with the non-regularity of the test-and-pool estimator, we construct a robust confidence interval that has a good finite-sample coverage property.

This work is motivated by the study of local protein structure, which is defined by two variable dihedral angles that take values from probability distributions on the flat torus. Our goal is to provide the space $P({\mathbb{R}}^2∕{\mathbb{Z}}^2)$ with a metric that quantifies local structural modifications due to changes in the protein sequence, and to define associated two-sample goodness-of-fit testing approaches. Due to its adaptability to the geometry of the underlying space, we focus on the Wasserstein distance as a metric between distributions.

We extend existing results of the theory of Optimal Transport to the d-dimensional flat torus ${\mathbb{T}}^d={\mathbb{R}}^d∕{\mathbb{Z}}^d$, in particular a Central Limit Theorem for the fluctuations of the empirical optimal transport cost. Moreover, we propose different approaches for two-sample goodness-of-fit testing for the one and two-dimensional case, based on the Wasserstein distance. We prove their validity and consistency. We provide an implementation of these tests in R. Their performance is assessed by numerical experiments on synthetic data and illustrated by an application to protein structure data.

We consider the problem of estimating latent positions in a one-dimensional torus from pairwise affinities. The observed affinity between a pair of items is modeled as a noisy observation of a function $f(x_i^{\ast },x_j^{\ast })$ of the latent positions $x_i^{\ast },x_j^{\ast }$ of the two items on the torus. The affinity function f is unknown, and it is only assumed to fulfill some shape constraints ensuring that $f(x,y)$ is large when the distance between x and y is small, and vice-versa. This non-parametric modeling offers a good flexibility to fit data. We introduce an estimation procedure that provably localizes all the latent positions with a maximum error of the order of $\sqrt{log(n)∕n}$, with high-probability. This rate is proven to be minimax optimal. A computationally efficient variant of the procedure is also analyzed under some more restrictive assumptions. Our general results can be instantiated to the problem of statistical seriation, leading to new bounds for the maximum error in the ordering.

Generalized Lasso has received much attention for sparse learning in statistics and machine learning, since it extends many existing sparse learning methods and provides estimators with flexible sparsity structures. When the design matrix is rank-deficient, however, the objective function of generalized Lasso may not be strictly convex, in which case multiple solutions exist. In practice, a small $l_2$ penalty is added to the objective function to make it strictly convex but its validity is largely unknown. In this paper, we study the solutions set of the G-Lasso problem with a rank-deficient design matrix. The main contributions of this paper are (1) to develop a solution path algorithm to find an exact solution even when the design matrix is rank-deficient and (2) to characterize the solutions set by a linear subspace with linear inequalities. Using the solution path algorithm and the characterization of the solutions set, we explain how to find a solution with specific properties, such as one with the least $l_{\infty }$ norm. By analyzing simulated and real data sets, we illustrate that there exist quite different multiple solutions and finding a solution with a specific property is important to have better accuracy as well as better interpretability. In addition, we show that the approach of adding a small $l_2$ penalty to the objective function is sometimes numerically unstable and the resulting estimator is inaccurate.

The training of high-dimensional regression models on comparably sparse data is an important yet complicated topic, especially when there are many more model parameters than observations in the data. From a Bayesian perspective, inference in such cases can be achieved with the help of shrinkage prior distributions, at least for generalized linear models. However, real-world data usually possess multilevel structures, such as repeated measurements or natural groupings of individuals, which existing shrinkage priors are not built to deal with.

We generalize and extend one of these priors, the R2D2 prior by Zhang et al. (2020), to linear multilevel models leading to what we call the R2D2M2 prior. The proposed prior enables both local and global shrinkage of the model parameters. It comes with interpretable hyperparameters, which we show to be intrinsically related to vital properties of the prior, such as rates of concentration around the origin, tail behavior, and amount of shrinkage the prior exerts.

We offer guidelines on how to select the prior’s hyperparameters by deriving shrinkage factors and measuring the effective number of non-zero model coefficients. Hence, the user can readily evaluate and interpret the amount of shrinkage implied by a specific choice of hyperparameters.

Finally, we perform extensive experiments on simulated and real data, showing that our inference procedure for the prior is well calibrated, has desirable global and local regularization properties and enables the reliable and interpretable estimation of much more complex Bayesian multilevel models than was previously possible.

Large-scale networks are commonly encountered in practice. To study the network interaction between different nodes of large-scale networks, the spatial autoregressive (SAR) model has been popularly employed. Despite its popularity, the estimation of a SAR model on large-scale networks remains very challenging. On the one hand, due to policy limitations or high collection costs, it is often impossible for independent researchers to observe or collect all network information. On the other hand, even if the entire network is accessible, estimating the SAR model using the quasi-maximum likelihood estimator (QMLE) could be computationally infeasible due to its high computational cost. To address these challenges, we propose here a subnetwork estimation method based on QMLE for the SAR model. By using appropriate sampling methods, a subnetwork, consisting of a much-reduced number of nodes, can be constructed. Subsequently, the standard QMLE can be computed by treating the sampled subnetwork as if it were the entire network. This leads to a significant reduction in information collection and model computation costs, which increases the practical feasibility of the effort. Theoretically, we show that the subnetwork-based QMLE is consistent and asymptotically normal under appropriate regularity conditions. Extensive simulation studies, based on both simulated and real network structures, are presented.