Direct estimation of differential networks under high-dimensional nonparanormal graphical models

In genomics, it is often of interest to study the structural change of a genetic network between two phenotypes. Under Gaussian graphical models, the problem can be transformed to estimating the difference between two precision matrices, and several approaches have been recently developed for this task such as joint graphical lasso and fused graphical lasso. However, the multivariate Gaussian assumptions made in the existing approaches are often violated in reality. For instance, most RNA-Seq data follow non-Gaussian distributions even after log-transformation or other variance-stabilizing transformations. In this work, we consider the problem of directly estimating differential networks under a flexible semiparametric model, namely the nonparanormal graphical model, where the random variables are assumed to follow a multivariate Gaussian distribution after a set of monotonically increasing transformations. We propose to use a novel rank-based estimator to directly estimate the differential network, together with a parametric simplex algorithm for fast implementation. Theoretical properties of the new estimator are established under a high-dimensional setting where $p$ grows with $n$ almost exponentially fast. In particular, we show that the proposed estimator is consistent in both parameter estimation and support recovery. Both synthetic data and real genomic data are used to illustrate the promise of the new approach. The Canadian Journal of Statistics 48: 187–203; 2020 © 2019 Statistical Society of Canada     

Graphon estimation via nearest-neighbour algorithm and two-dimensional fused-lasso denoising

We propose a class of methods for graphon estimation based on exploiting connections with nonparametric regression. The idea is to construct an ordering of the nodes in the network, similar in spirit to Chan & Airoldi (2014). However, rather than considering orderings based only on the empirical degree as in Chan & Airoldi (2014), we use the nearest-neighbour algorithm which is an approximative solution to the travelling salesman problem. This algorithm in turn can handle general distances $\widehat{d}$ between the nodes, allowing us to incorporate rich information from the network. Once an ordering is constructed, we formulate a two-dimensional-grid graph-denoising problem that we solve through fused-lasso regularization. For particular choices of the metric $\widehat{d}$, we show that the corresponding two-step estimator can attain competitive rates when the true model is the stochastic block model, and when the underlying graphon is piecewise Hölder or has bounded variation.     

Two-dimensional projection uniformity for space-filling designs

We investigate a space-filling criterion based on $L_2$-type discrepancies, namely the uniform projection criterion, aiming at improving designs' two-dimensional projection uniformity. Under a general reproducing kernel, we establish a formula for the uniform projection criterion function, which builds a connection between rows and columns of the design. For the commonly used discrepancies, we further use this formula to represent the two-dimensional projection uniformity in terms of the $L_p$-distances of U-type designs. These results generalize existing works and reveal new links between the two seemingly unrelated criteria of projection uniformity and the maximin $L_p$-distance for U-type designs. We also apply the obtained results to study several families of space-filling designs with appealing projection uniformity. Because of good projected space-filling properties, these designs are well adapted for computer experiments, especially for the case where not all the input factors are active.     

Decompounding discrete distributions: A nonparametric Bayesian approach

Suppose that a compound Poisson process is observed discretely in time and assume that its jump distribution is supported on the set of natural numbers. In this paper we propose a nonparametric Bayesian approach to estimate the intensity of the underlying Poisson process and the distribution of the jumps. We provide a Markov chain Monte Carlo scheme for obtaining samples from the posterior. We apply our method on both simulated and real data examples, and compare its performance with the frequentist plug-in estimator proposed by Buchmann and Grübel. On a theoretical side, we study the posterior from the frequentist point of view and prove that as the sample size n→∞, it contracts around the “true,” data-generating parameters at rate $1/\sqrt{n}$, up to a $\log n$ factor.     

Variable selection and estimation for high-dimensional spatial autoregressive models

Spatial regression models are important tools for many scientific disciplines including economics, business, and social science. In this article, we investigate postmodel selection estimators that apply least squares estimation to the model selected by penalized estimation in high-dimensional regression models with spatial autoregressive errors. We show that by separating the model selection and estimation process, the postmodel selection estimator performs at least as well as the simultaneous variable selection and estimation method in terms of the rate of convergence. Moreover, under perfect model selection, the ℓ₂ rate of convergence is the oracle rate of $\sqrt{s/n}$, compared with the convergence rate of $\text{◂√▸}\sqrt{s\log p/n}$ in the general case. Here, n is the sample size and p, s are the model dimension and number of significant covariates, respectively. We further provide the convergence rate of the estimation error in the form of $\sup$ norm, and ideally the rate can reach as fast as $\text{◂√▸}\sqrt{\log s/n}$.     

Stochastic functional estimates in longitudinal models with interval-censored anchoring events

Timelines of longitudinal studies are often anchored by specific events. In the absence of the fully observed anchoring event times, the study timeline becomes undefined, and the traditional longitudinal analysis loses its temporal reference. In this paper, we considered an analytical situation where the anchoring events are interval censored. We demonstrated that by expressing the regression parameter estimators as stochastic functionals of a plug-in estimate of the unknown anchoring event time distribution, the standard longitudinal models could be extended to accommodate the situation of less well-defined timelines. We showed that for a broad class of longitudinal models, the functional parameter estimates are consistent and asymptotically normally distributed with a $\sqrt{n}$ convergence rate under mild regularity conditions. Applying the developed theory to linear mixed-effects models, we further proposed a hybrid computational procedure that combines the strengths of the Fisher's scoring method and the expectation-expectation (EM) algorithm for model parameter estimation. We conducted a simulation study to validate the asymptotic properties and to assess the finite sample performance of the proposed method. A real data example was used to illustrate the proposed method. The method fills in a gap in the existing longitudinal analysis methodology for data with less well-defined timelines.     

Combining evidence from clinical trials in conditional or accelerated approval

Conditional (European Medicines Agency) or accelerated (U.S. Food and Drug Administration) approval of drugs allows earlier access to promising new treatments that address unmet medical needs. Certain post-marketing requirements must typically be met in order to obtain full approval, such as conducting a new post-market clinical trial. We study the applicability of the recently developed harmonic mean ${\chi }^2$-test to this conditional or accelerated approval framework. The proposed approach can be used both to support the design of the post-market trial and the analysis of the combined evidence provided by both trials. Other methods considered are the two-trials rule, Fisher's criterion and Stouffer's method. In contrast to some of the traditional methods, the harmonic mean ${\chi }^2$-test always requires a post-market clinical trial. If the $p$-value from the pre-market clinical trial is $\ll 0.025$, a smaller sample size for the post-market clinical trial is needed than with the two-trials rule. For illustration, we apply the harmonic mean ${\chi }^2$-test to a drug which received conditional (and later full) market licensing by the EMA. A simulation study is conducted to study the operating characteristics of the harmonic mean ${\chi }^2$-test and two-trials rule in more detail. We finally investigate the applicability of these two methods to compute the power at interim of an ongoing post-market trial. These results are expected to aid in the design and assessment of the required post-market studies in terms of the level of evidence required for full approval.     

Asymptotic approximation of the likelihood of stationary determinantal point processes

Continuous determinantal point processes (DPPs) are a class of repulsive point processes on ${ℝ}^d$ with many statistical applications. Although an explicit expression of their density is known, it is too complicated to be used directly for maximum likelihood estimation. In the stationary case, an approximation using Fourier series has been suggested, but it is limited to rectangular observation windows and no theoretical results support it. In this contribution, we investigate a different way to approximate the likelihood by looking at its asymptotic behavior when the observation window grows toward ${ℝ}^d$. This new approximation is not limited to rectangular windows, is faster to compute than the previous one, does not require any tuning parameter, and some theoretical justifications are provided. It moreover provides an explicit formula for estimating the asymptotic variance of the associated estimator. The performances are assessed in a simulation study on standard parametric models on ${ℝ}^d$ and compare favorably to common alternative estimation methods for continuous DPPs.     

Daisee: Adaptive importance sampling by balancing exploration and exploitation

We study adaptive importance sampling (AIS) as an online learning problem and argue for the importance of the trade-off between exploration and exploitation in this adaptation. Borrowing ideas from the online learning literature, we propose Daisee, a partition-based AIS algorithm. We further introduce a notion of regret for AIS and show that Daisee has $𝒪(\sqrt{T}{(\log T)}^{\frac{3}{4}})$ cumulative pseudo-regret, where $T$ is the number of iterations. We then extend Daisee to adaptively learn a hierarchical partitioning of the sample space for more efficient sampling and confirm the performance of both algorithms empirically.     

Selection of linear mixed-effects models for clustered data

We consider model selection for linear mixed-effects models with clustered structure, where conditional Kullback–Leibler (CKL) loss is applied to measure the efficiency of the selection. We estimate the CKL loss by substituting the empirical best linear unbiased predictors (EBLUPs) into random effects with model parameters estimated by maximum likelihood. Although the BLUP approach is commonly used in predicting random effects and future observations, selecting random effects to achieve asymptotic loss efficiency concerning CKL loss is challenging and has not been well studied. In this paper, we propose addressing this difficulty using a conditional generalized information criterion (CGIC) with two tuning parameters. We further consider a challenging but practically relevant situation where the number, $m$, of clusters does not go to infinity with the sample size. Hence the random-effects variances are not consistently estimable. We show that via a novel decomposition of the CKL risk, the CGIC achieves consistency and asymptotic loss efficiency, whether $m$ is fixed or increases to infinity with the sample size. We also conduct numerical experiments to illustrate the theoretical findings.     

On the robustness to outliers of the Student-t process

The theory of Bayesian robustness modeling uses heavy-tailed distributions to resolve conflicts of information by rejecting automatically the outlying information in favor of the other sources of information. In particular, the Student's-t process is a natural alternative to the Gaussian process when the data might carry atypical information. Several works attest to the robustness of the Student $t$ process, however, the studies are mostly guided by intuition and focused mostly on the computational aspects rather than the mathematical properties of the involved distributions. This work uses the theory of regular variation to address the robustness of the Student $t$ process in the context of nonlinear regression, that is, the behavior of the posterior distribution in the presence of outliers in the inputs, in the outputs, or in both sources of information. In all these cases, under certain conditions, it is shown that the posterior distribution tends to a quantity that does not depend on the atypical information, then, for every case, the limiting posterior distribution as the outliers tend to infinity is provided. The impact of outliers on the predictive posterior distribution is also addressed. The theory is illustrated with a few simulated examples.     

Alone,together: On the benefits of Bayesian borrowing in a meta-analytic setting

It is common practice to use hierarchical Bayesian model for the informing of a pediatric randomized controlled trial (RCT) by adult data, using a prespecified borrowing fraction parameter (BFP). This implicitly assumes that the BFP is intuitive and corresponds to the degree of similarity between the populations. Generalizing this model to any $K\ge 1$ historical studies, naturally leads to empirical Bayes meta-analysis. In this paper we calculate the Bayesian BFPs and study the factors that drive them. We prove that simultaneous mean squared error reduction relative to an uninformed model is always achievable through application of this model. Power and sample size calculations for a future RCT, designed to be informed by multiple external RCTs, are also provided. Potential applications include inference on treatment efficacy from independent trials involving either heterogeneous patient populations or different therapies from a common class.     

Sparse concordance-based ordinal classification

Ordinal classification is an important area in statistical machine learning, where labels exhibit a natural order. One of the major goals in ordinal classification is to correctly predict the relative order of instances. We develop a novel concordance-based approach to ordinal classification, where a concordance function is introduced and a penalized smoothed method for optimization is designed. Variable selection using the $L_1$ penalty is incorporated for sparsity considerations. Within the set of classification rules that maximize the concordance function, we find optimal thresholds to predict labels by minimizing a loss function. After building the classifier, we derive nonparametric estimation of class conditional probabilities. The asymptotic properties of the estimators as well as the variable selection consistency are established. Extensive simulations and real data applications show the robustness and advantage of the proposed method in terms of classification accuracy, compared with other existing methods.     

A fiducial approach for testing the non-inferiority of proportion difference in matched-pairs design

The non-inferiority of one treatment/drug to another is a common and important issue in medical and pharmaceutical fields. This study explored a fiducial approach for testing the non-inferiority of proportion difference in matched-pairs design. Approximate tests constructed using fiducial quantities with a combination of different parameters were proposed. Four simulation studies were employed to compare the performance of fiducial tests by comparing their type I errors and powers. The results showed that fiducial quantities with parameter $0.6\le w_1\le 0.8$ performed satisfactorily from small to large samples. Therefore, the fiducial tests could be recommended for practical applications. The recommended fiducial tests might be a competitive alternative to other available tests. Three real data sets were analyzed to illustrate the proposed methods were competitive or even better than other tests.     

Grenander functionals and Cauchy's formula

Let ${\widehat{f}}_n$ be the nonparametric maximum likelihood estimator of a decreasing density. Grenander characterized this as the left‐continuous slope of the least concave majorant of the empirical distribution function. For a sample from the uniform distribution, the asymptotic distribution of the L₂‐distance of the Grenander estimator to the uniform density was derived in an article by Groeneboom and Pyke by using a representation of the Grenander estimator in terms of conditioned Poisson and gamma random variables. This representation was also used in an article by Groeneboom and Lopuhaä to prove a central limit result of Sparre Andersen on the number of jumps of the Grenander estimator. Here we extend this to the proof of the main result on the L₂‐distance of the Grenander estimator to the uniform density and also prove a similar asymptotic normality results for the entropy functional. Cauchy's formula and saddle point methods are the main tools in our development.     

Asymptotic properties of the maximum smoothed partial likelihood estimator in the change-plane Cox model

The change-plane Cox model is a popular tool for the subgroup analysis of survival data. Despite the rich literature on this model, there has been limited investigation into the asymptotic properties of the estimators of the finite-dimensional parameter. Particularly, the convergence rate, not to mention the asymptotic distribution, has not been fully characterized for the general model where classification is based on multiple covariates. To bridge this theoretical gap, this study proposes a maximum smoothed partial likelihood estimator and establishes the following asymptotic properties. First, it shows that the convergence rate for the classification parameter can be arbitrarily close to $n^{-1}$ up to a logarithmic factor under a certain condition on covariates and the choice of tuning parameter. Given this convergence rate result, it also establishes the asymptotic normality for the regression parameter.