Mathematics

We consider the problem of designing experiments for the comparison of two regression curves describing the relation between a predictor and a response in two groups, where the data between and within the group may be dependent. In order to derive efficient designs we use results from stochastic analysis to identify the best linear unbiased estimator (BLUE) in a corresponding continuous time model. It is demonstrated that in general simultaneous estimation using the data from both groups yields more precise results than estimation of the parameters separately in the two groups. Using the BLUE from simultaneous estimation, we then construct an efficient linear estimator for finite sample size by minimizing the mean squared error between the optimal solution in the continuous time model and its discrete approximation with respect to the weights (of the linear estimator). Finally, the optimal design points are determined by minimizing the maximal width of a simultaneous confidence band for the difference of the two regression functions. The advantages of the new approach are illustrated by means of a simulation study, where it is shown that the use of the optimal designs yields substantially narrower confidence bands than the application of uniform designs.

In the present paper, personalized treatment means choosing the best treatment for a patient while taking into account certain relevant personal covariate values. We study the design of trials whose goal is to find the best treatment for a given patient with given covariates. We assume that the subjects in the trial represent a random sample from the population, and consider the allocation, possibly with randomization, of these subjects to the different treatment groups in a way that depends on their covariates. We derive approximately optimal allocations, aiming to minimize expected regret, assuming that future patients will arrive from the same population as the trial subjects. We find that, for the case of two treatments, an approximately optimal allocation design does not depend on the value of the covariates but only on the variances of the responses. In contrast, for the case of three treatments the optimal allocation design does depend on the covariates as we show for specific scenarios. Another finding is that the optimal allocation can vary a lot as a function of the sample size, and that randomized allocations are relevant for relatively small samples, and may not be needed for very large studies.

We study the problem of online network change point detection. In this setting, a collection of independent Bernoulli networks is collected sequentially, and the underlying distributions change when a change point occurs. The goal is to detect the change point as quickly as possible, if it exists, subject to a constraint on the number or probability of false alarms. In this paper, on the detection delay, we establish a minimax lower bound and two upper bounds based on NP-hard algorithms and polynomial-time algorithms, i.e., detection delay ???????????????????????�log(1/α) max{ r 2 /n,1} κ 2 0 n? , ?�log(?/α) max{ r 2 /n,log(r)} κ 2 0 n? , ?�log(?/α) r κ 2 0 n? , with NP-hard algorithms, with polynomial-time algorithms, where κ 0 ,n,?,r and α are the normalised jump size, network size, entrywise sparsity, rank sparsity and the overall Type-I error upper bound. All the model parameters are allowed to vary as ? , the location of the change point, diverges. The polynomial-time algorithms are novel procedures that we propose in this paper, designed for quick detection under two different forms of Type-I error control. The first is based on controlling the overall probability of a false alarm when there are no change points, and the second is based on specifying a lower bound on the expected time of the first false alarm. Extensive experiments show that, under different scenarios and the aforementioned forms of Type-I error control, our proposed approaches outperform state-of-the-art methods.

We consider the problem of robust mean and location estimation w.r.t. any pseudo-norm of the form x??R d ?�||x| | S = sup v?�S <v,x> where S is any symmetric subset of R d . We show that the deviation-optimal minimax subgaussian rate for confidence 1?��?is max( l ??( Σ 1/2 S) N ??????, sup v?�S || Σ 1/2 v| | 2 log(1/δ) N ????????????????) where l ??( Σ 1/2 S) is the Gaussian mean width of Σ 1/2 S and Σ the covariance of the data (in the benchmark i.i.d. Gaussian case). This improves the entropic minimax lower bound from [Lugosi and Mendelson, 2019] and closes the gap characterized by Sudakov's inequality between the entropy and the Gaussian mean width for this problem. This shows that the right statistical complexity measure for the mean estimation problem is the Gaussian mean width. We also show that this rate can be achieved by a solution to a convex optimization problem in the adversarial and L 2 heavy-tailed setup by considering minimum of some Fenchel-Legendre transforms constructed using the Median-of-means principle. We finally show that this rate may also be achieved in situations where there is not even a first moment but a location parameter exists.

We prove some efficient inference results concerning estimation of a Ornstein-Uhlenbeck regression model, which is driven by a non-Gaussian stable Levy process and where the output process is observed at high-frequency over a fixed time period. Local asymptotics for the likelihood function is presented, followed by a way to construct an asymptotically efficient estimator through a suboptimal yet very simple preliminary estimator, which enables us to bypass not only numerical optimization of the likelihood function, but also the multiple-root problem.

Sequential change-point detection for graphs is a fundamental problem for streaming network data types and has wide applications in social networks and power systems. Given fixed vertices and a sequence of random graphs, the objective is to detect the change-point where the underlying distribution of the random graph changes. In particular, we focus on the local change that only affects a subgraph. We adopt the classical Erdos-Renyi model and revisit the generalized likelihood ratio (GLR) detection procedure. The scan statistic is computed by sequentially estimating the most-likely subgraph where the change happens. We provide theoretical analysis for the asymptotic optimality of the proposed procedure based on the GLR framework. We demonstrate the efficiency of our detection algorithm using simulations.

A generic out-of-sample error estimate is proposed for robust M -estimators regularized with a convex penalty in high-dimensional linear regression where (X,y) is observed and p,n are of the same order. If ψ is the derivative of the robust data-fitting loss ρ , the estimate depends on the observed data only through the quantities ψ ^ =ψ(y−X β ^ ) , X ⊤ ψ ^ and the derivatives (∂/∂y) ψ ^ and (∂/∂y)X β ^ for fixed X . The out-of-sample error estimate enjoys a relative error of order n −1/2 in a linear model with Gaussian covariates and independent noise, either non-asymptotically when p/n≤γ or asymptotically in the high-dimensional asymptotic regime p/n→ γ ′ ∈(0,∞) . General differentiable loss functions ρ are allowed provided that ψ= ρ ′ is 1-Lipschitz. The validity of the out-of-sample error estimate holds either under a strong convexity assumption, or for the ℓ 1 -penalized Huber M-estimator if the number of corrupted observations and sparsity of the true β are bounded from above by s ∗ n for some small enough constant s ∗ ∈(0,1) independent of n,p . For the square loss and in the absence of corruption in the response, the results additionally yield n −1/2 -consistent estimates of the noise variance and of the generalization error. This generalizes, to arbitrary convex penalty, estimates that were previously known for the Lasso.

The research is about a systematic investigation on the following issues. First, we construct different outcome regression-based estimators for conditional average treatment effect under, respectively, true (oracle), parametric, nonparametric and semiparametric dimension reduction structure. Second, according to the corresponding asymptotic variance functions, we answer the following questions when supposing the models are correctly specified: what is the asymptotic efficiency ranking about the four estimators in general? how is the efficiency related to the affiliation of the given covariates in the set of arguments of the regression functions? what do the roles of bandwidth and kernel function selections play for the estimation efficiency; and in which scenarios should the estimator under semiparametric dimension reduction regression structure be used in practice? As a by-product, the results show that any outcome regression-based estimation should be asymptotically more efficient than any inverse probability weighting-based estimation. All these results give a relatively complete picture of the outcome regression-based estimation such that the theoretical conclusions could provide guidance for practical use when more than one estimations can be applied to the same problem. Several simulation studies are conducted to examine the performances of these estimators in finite sample cases and a real dataset is analyzed for illustration.

We study the problem of outlier robust high-dimensional mean estimation under a finite covariance assumption, and more broadly under finite low-degree moment assumptions. We consider a standard stability condition from the recent robust statistics literature and prove that, except with exponentially small failure probability, there exists a large fraction of the inliers satisfying this condition. As a corollary, it follows that a number of recently developed algorithms for robust mean estimation, including iterative filtering and non-convex gradient descent, give optimal error estimators with (near-)subgaussian rates. Previous analyses of these algorithms gave significantly suboptimal rates. As a corollary of our approach, we obtain the first computationally efficient algorithm with subgaussian rate for outlier-robust mean estimation in the strong contamination model under a finite covariance assumption.

Datasets displaying temporal dependencies abound in science and engineering applications, with Markov models representing a simplified and popular view of the temporal dependence structure. In this paper, we consider Bayesian settings that place prior distributions over the parameters of the transition kernel of a Markov model, and seeks to characterize the resulting, typically intractable, posterior distributions. We present a PAC-Bayesian analysis of variational Bayes (VB) approximations to tempered Bayesian posterior distributions, bounding the model risk of the VB approximations. Tempered posteriors are known to be robust to model misspecification, and their variational approximations do not suffer the usual problems of over confident approximations. Our results tie the risk bounds to the mixing and ergodic properties of the Markov data generating model. We illustrate the PAC-Bayes bounds through a number of example Markov models, and also consider the situation where the Markov model is misspecified.