Miracle-dz/UniFAR-FTUs huggingface dataset:
Asymptotic Bayes optimality under sparsity for generally distributed effect sizes under the alternative

Recent results concerning asymptotic Bayes-optimality under sparsity (ABOS) of multiple testing procedures are extended to fairly generally distributed effect sizes under the alternative. An asymptotic framework is considered where both the number of tests m and the sample size m go to infinity, while the fraction p of true alternatives converges to zero. It is shown that under mild restrictions on the loss function nontrivial asymptotic inference is possible only if n increases to infinity at least at the rate of log m. Based on this assumption precise conditions are given under which the Bonferroni correction with nominal Family Wise Error Rate (FWER) level alpha and the Benjamini- Hochberg procedure (BH) at FDR level alpha are asymptotically optimal. When n is proportional to log m then alpha can remain fixed, whereas when n increases to infinity at a quicker rate, then alpha has to converge to zero roughly like n^(-1/2). Under these conditions the Bonferroni correction is ABOS in case of extreme sparsity, while BH adapts well to the unknown level of sparsity. In the second part of this article these optimality results are carried over to model selection in the context of multiple regression with orthogonal regressors. Several modifications of Bayesian Information Criterion are considered, controlling either FWER or FDR, and conditions are provided under which these selection criteria are ABOS. Finally the performance of these criteria is examined in a brief simulation study.
On the generic uniform uniqueness of the LASSO estimator

The LASSO is a variable subset selection procedure in statistical linear regression based on $\ell_1$ penalization of the least-squares operator. Uniqueness of the LASSO is an important issue, especially for the study of the LASSO path. The goal of the present paper is to provide a generic sufficient condition on the design matrix for the LASSO minimizer to be unique. Unlike previous works on the question of uniqueness, our condition only depends on the design matrix. Our study is based on a general position condition on the design matrix which holds with probability one for most experimental models.
A Donsker Theorem for L\'evy Measures

Given $n$ equidistant realisations of a L\'evy process $(L_t,\,t\ge 0)$, a natural estimator $\hat N_n$ for the distribution function $N$ of the L\'evy measure is constructed. Under a polynomial decay restriction on the characteristic function $\phi$, a Donsker-type theorem is proved, that is, a functional central limit theorem for the process $\sqrt n (\hat N_n -N)$ in the space of bounded functions away from zero. The limit distribution is a generalised Brownian bridge process with bounded and continuous sample paths whose covariance structure depends on the Fourier-integral operator ${\cal F}^{-1}[1/\phi(-\cdot)]$. The class of L\'evy processes covered includes several relevant examples such as compound Poisson, Gamma and self-decomposable processes. Main ideas in the proof include establishing pseudo-locality of the Fourier-integral operator and recent techniques from smoothed empirical processes.
Stein couplings for normal approximation

In this article we propose a general framework for normal approximation using Stein's method. We introduce the new concept of Stein couplings and we show that it lies at the heart of popular approaches such as the local approach, exchangeable pairs, size biasing and many other approaches. We prove several theorems with which normal approximation for the Wasserstein and Kolmogorov metrics becomes routine once a Stein coupling is found. To illustrate the versatility of our framework we give applications in Hoeffding's combinatorial central limit theorem, functionals in the classic occupancy scheme, neighbourhood statistics of point patterns with fixed number of points and functionals of the components of randomly chosen vertices of sub-critical Erdos-Renyi random graphs. In all these cases, we use new, non-standard couplings.
Controlling the False Discovery Rate via Knockoff for High Dimensional Ising Model Variable Selection

In high dimensional data analysis, it is important to effectively control the fraction of false discoveries and ensure large enough power for variable selection. In a lot of contemporary data applications, a large set of covariates are discrete variables. In this paper we propose Ising knockoff (IKF) for variable selection in high dimensional regression with discrete covariates. Under mild conditions, we show that the false discovery rate (FDR) is controlled under a target level in a finite sample if the underlying Ising model is known, and the FDR is asymptotically controlled even when the parameters of the Ising model is estimated. We also provide theoretical results on the power for our proposed method. Using simulations and a genome science data set, we show that IKF has higher power than existing knockoff procedures mostly tailored for continuous covariate distributions.
Fast Convergence Rate of Multiple Kernel Learning with Elastic-net Regularization

We investigate the learning rate of multiple kernel leaning (MKL) with elastic-net regularization, which consists of an $\ell_1$-regularizer for inducing the sparsity and an $\ell_2$-regularizer for controlling the smoothness. We focus on a sparse setting where the total number of kernels is large but the number of non-zero components of the ground truth is relatively small, and prove that elastic-net MKL achieves the minimax learning rate on the $\ell_2$-mixed-norm ball. Our bound is sharper than the convergence rates ever shown, and has a property that the smoother the truth is, the faster the convergence rate is.
Causal inference under over-simplified longitudinal causal models

Abstract Many causal models of interest in epidemiology involve longitudinal exposures, confounders and mediators. However, repeated measurements are not always available or used in practice, leading analysts to overlook the time-varying nature of exposures and work under over-simplified causal models. Our objective is to assess whether – and how – causal effects identified under such misspecified causal models relates to true causal effects of interest. We derive sufficient conditions ensuring that the quantities estimated in practice under over-simplified causal models can be expressed as weighted averages of longitudinal causal effects of interest. Unsurprisingly, these sufficient conditions are very restrictive, and our results state that the quantities estimated in practice should be interpreted with caution in general, as they usually do not relate to any longitudinal causal effect of interest. Our simulations further illustrate that the bias between the quantities estimated in practice and the weighted averages of longitudinal causal effects of interest can be substantial. Overall, our results confirm the need for repeated measurements to conduct proper analyses and/or the development of sensitivity analyses when they are not available.
Controlling FDR while highlighting selected discoveries

Often modern scientific investigations start by testing a very large number of hypotheses in an effort to comprehensively mine the data for possible discoveries. Multiplicity adjustment strategies are employed to ensure replicability of the results of this broad search. Furthermore, in many cases, discoveries are subject to a second round of selection, where researchers identify the rejected hypotheses that better represent distinct and interpretable findings for reporting and follow-up. For example, in genetic studies, one DNA variant is often chosen to represent a group of neighboring polymorphisms, all apparently associated to a trait of interest. Unfortunately the guarantees of false discovery rate (FDR) control that might be true for the initial set of findings do not translate to this subset, possibly leading to an inflation of FDR in the reported discoveries. To guarantee valid inference, we introduce Focused BH, a multiple testing procedure that allows the researcher to curate rejections by subsetting or prioritizing them according to pre-specified but possibly data-dependent rules (filters). Focused BH assures FDR control on the selected discoveries under a range of assumptions on the filter and the p-value dependency structure; simulations illustrate that this is obtained without substantial power loss and that the procedure is robust to violations of our theoretical assumptions.
A local stochastic Lipschitz condition with application to Lasso for high dimensional generalized linear models

For regularized estimation, the upper tail behavior of the random Lipschitz coefficient associated with empirical loss functions is known to play an important role in the error bound of Lasso for high dimensional generalized linear models. The upper tail behavior is known for linear models but much less so for nonlinear models. We establish exponential type inequalities for the upper tail of the coefficient and illustrate an application of the results to Lasso likelihood estimation for high dimensional generalized linear models. AMS 2010 subject classification. 62G08; 60E15.
Statistical applications of the multivariate skew normal distribution

Azzalini and Dalla Valle have recently discussed the multivariate skew normal distribution which extends the class of normal distributions by the addition of a shape parameter. The first part of the present paper examines further probabilistic properties of the distribution, with special emphasis on aspects of statistical relevance. Inferential and other statistical issues are discussed in the following part, with applications to some multivariate statistics problems, illustrated by numerical examples. Finally, a further extension is described which introduces a skewing factor of an elliptical density.
On the heavy-tailedness of Student's t-statistic

Let {X-i}(i >= 1) be an i.i. d. sequence of random variables and define, for n >= 2, T-n={(n-1/2 (sigma) over capn-1Sn)(0), ((sigma) over capn > 0,)((sigma) over capn=0,) with S-n = Sigma(n .
Azar y Aritmetica

Let omega(n) be the number of prime divisors of an integer n. Let n be an integer taken at random between 1 and N. What can be said about the value then taken by omega(n)? What is its expected value? What is its distribution in the limit? What is the probability that omega(n) will deviate greatly from its expected value? We will study these questions as an introduction to probabilistic number theory. We treat several central topics in probabilistic number theory without assuming previous knowledge of the area. Neither measure theory nor complex analysis are assumed. In the exercises, among other topics, we develop some of the bases of sieve theory as an application of probabilistic ideas. ----- Sea omega(n) el numero de divisores primos de un entero n. Sea n un entero tomado al azar entre 1 y N. Que se puede decir del valor que entonces tomara' omega(n)? Cual es su esperanza? Cual es su distribucion en el limite? Cual es la probabilidad que omega(n) tome valores que se alejen mucho de su esperanza? Estudiamos estas preguntas a guisa de introduccion a la teoria de numeros probabilistica. Trataremos varios topicos centrales de la teoria de probabilidades sin suponer conocimientos previos en el area. No asumiremos ni teoria de la medida ni analisis complejo. En los ejercicios, entre otros topicos, se desarrollaran las bases de la teoria de cribas como una aplicacion de ideas probabilisticas.
Recovering Direct Effects in Genetics: A Comparison

In genetics it is often of interest to discover single nucleotide polymorphisms (SNPs) that are directly related to a disease, rather than just being associated with it. Few methods exist, however, addressing this so-called `true sparsity recovery' issue. In a thorough simulation study, we show that for moderate or low correlation between predictors, lasso-based methods perform well at true sparsity recovery, despite not being specifically designed for this purpose. For large correlations, however, more specialised methods are needed. Stability selection and direct effect testing perform well in all situations, including when the correlation is large.
Comment. Support Vector Machines with Applications.

Trevor Hastie is Professor, Department of Statistics, Stanford University, Stanford, California 94305, USA (e-mail: hastie @ stanford.edu). Ji Zhu is Assistant Professor, Department of Statistics, University of Michigan, Ann Arbor, Michigan 48109, USA (e-mail: jizhu @ umich. edu). reproducing kernel Hubert space Mk (RKHS) gener ated by K{-, ) (see Burges, 1998; Evgeniou, Pontil and Poggio, 2000; and Wahba, 1999, for details). Suppose the positive definite kernel K(-,-) has a (possibly finite) eigenexpansion,
A note on Metropolis-Hasting for sampling across mixed spaces

We are concerned with the Metropolis-Hastings algorithm for sampling across so-called mixed spaces. The most important example of a mixed space is a space that consists of real vectors of different lengths. Sampling within this space is called trans dimensional sampling and enjoys an enormous popularity across many statistical communities. However, the ubiquitous literature is completely divided over the abilities of the Metropolis-Hastings algorithm in these scenarios. This lead to the success of the reversible jump algorithm as the quasi gold standard. Unfortunately, the actual contribution of reversible jump is consistently misunderstood. Thus, this paper gives a dense overview of the theory that builds the mathematical foundations of sampling within mixed spaces, discusses a changepoint example and finally clears up any misunderstandings.
Tail Behavior of Randomly Weighted Sums

Let {X t , t ≥ 1} be a sequence of identically distributed and pairwise asymptotically independent random variables with regularly varying tails, and let {Θ t , t ≥ 1} be a sequence of positive random variables independent of the sequence {X t , t ≥ 1}. We will discuss the tail probabilities and almost-sure convergence of X (∞) = ∑ t=1 ∞Θ t X t + (where X + = max{0, X}) and max1≤k<∞∑ t=1 k Θ t X t , and provide some sufficient conditions motivated by Denisov and Zwart (2007) as alternatives to the usual moment conditions. In particular, we illustrate how the conditions on the slowly varying function involved in the tail probability of X 1 help to control the tail behavior of the randomly weighted sums. Note that, the above results allow us to choose X 1, X 2,… as independent and identically distributed positive random variables. If X 1 has a regularly varying tail of index -α, where α > 0, and if {Θ t , t ≥ 1} is a positive sequence of random variables independent of {X t }, then it is known – which can also be obtained from the sufficient conditions in this article – that, under some appropriate moment conditions on {Θ t , t ≥ 1}, X (∞) = ∑ t=1 ∞Θ t X t converges with probability 1 and has a regularly varying tail of index -α. Motivated by the converse problems in Jacobsen, Mikosch, Rosiński and Samorodnitsky (2009) we ask the question: if X (∞) has a regularly varying tail then does X 1 have a regularly varying tail under some appropriate conditions? We obtain appropriate sufficient moment conditions, including the nonvanishing Mellin transform of ∑ t=1 ∞Θ t along some vertical line in the complex plane, so that the above is true. We also show that the condition on the Mellin transform cannot be dropped.
Bayesian model choice and information criteria in sparse generalized linear models

We consider Bayesian model selection in generalized linear models that are high-dimensional, with the number of covariates p being large relative to the sample size n, but sparse in that the number of active covariates is small compared to p. Treating the covariates as random and adopting an asymptotic scenario in which p increases with n, we show that Bayesian model selection using certain priors on the set of models is asymptotically equivalent to selecting a model using an extended Bayesian information criterion. Moreover, we prove that the smallest true model is selected by either of these methods with probability tending to one. Having addressed random covariates, we are also able to give a consistency result for pseudo-likelihood approaches to high-dimensional sparse graphical modeling. Experiments on real data demonstrate good performance of the extended Bayesian information criterion for regression and for graphical models.
Optimal arrangements of hyperplanes for multiclass classification.

In this paper, we present a novel approach to construct multiclass classifiers by means of arrangements of hyperplanes. We propose different mixed integer (linear and non linear) programming formulations for the problem using extensions of widely used measures for misclassifying observations where the \textit{kernel trick} can be adapted to be applicable. Some dimensionality reductions and variable fixing strategies are also developed for these models. An extensive battery of experiments has been run which reveal the powerfulness of our proposal as compared with other previously proposed methodologies.
From EM to Data Augmentation: The Emergence of MCMC Bayesian Computation in the 1980s

It was known from Metropolis et al. [J. Chem. Phys. 21 (1953) 1087--1092] that one can sample from a distribution by performing Monte Carlo simulation from a Markov chain whose equilibrium distribution is equal to the target distribution. However, it took several decades before the statistical community embraced Markov chain Monte Carlo (MCMC) as a general computational tool in Bayesian inference. The usual reasons that are advanced to explain why statisticians were slow to catch on to the method include lack of computing power and unfamiliarity with the early dynamic Monte Carlo papers in the statistical physics literature. We argue that there was a deeper reason, namely, that the structure of problems in the statistical mechanics and those in the standard statistical literature are different. To make the methods usable in standard Bayesian problems, one had to exploit the power that comes from the introduction of judiciously chosen auxiliary variables and collective moves. This paper examines the development in the critical period 1980--1990, when the ideas of Markov chain simulation from the statistical physics literature and the latent variable formulation in maximum likelihood computation (i.e., EM algorithm) came together to spark the widespread application of MCMC methods in Bayesian computation.
A Comparison of Two Proximity Catch Digraph Families in Testing Spatial Clustering

We consider two parametrized random digraph families, namely, proportional-edge and central similarity proximity catch digraphs (PCDs) and compare the performance of these two PCD families in testing spatial point patterns. These PCD families are based on relative positions of data points from two classes and the relative density of the PCDs is used as a statistic for testing segregation and association against complete spatial randomness. When scaled properly, the relative density of a PCD is a U-statistic. We extend the distribution of the relative density of central similarity PCDs for expansion parameter being larger than one. We compare the asymptotic distribution of the statistic for the two PCD families, using the standard central limit theory of U-statistics. We compare finite sample performance of the tests by Monte Carlo simulations and prove the consistency of the tests under the alternatives. The asymptotic performance of the tests under the alternatives is assessed by Pitman's asymptotic efficiency. We find the optimal expansion parameters of the PCDs for testing each of the segregation and association alternatives in finite samples and in the limit. We demonstrate that in terms of empirical power (i.e., for finite samples) relative density of central similarity PCD has better performance (which occurs for expansion parameter values larger than one) under segregation alternative, while relative density of proportional-edge PCD has better performance under association alternative. The methods are illustrated in a real-life example from plant ecology.
Multivariate Evolutionary Analyses in Astrophysics

The large amount of data on galaxies, up to higher and higher redshifts, asks for sophisticated statistical approaches to build adequate classifications. Multivariate cluster analyses, that compare objects for their global similarities, are still confidential in astrophysics, probably because their results are somewhat difficult to interpret. We believe that the missing key is the unavoidable characteristics in our Universe: evolution. Our approach, known as Astrocladistics, is based on the evolutionary nature of both galaxies and their properties. It gathers objects according to their "histories" and establishes an evolutionary scenario among groups of objects. In this presentation, I show two recent results on globular clusters and earlytype galaxies to illustrate how the evolutionary concepts of Astrocladistics can also be useful for multivariate analyses such as K-means Cluster Analysis.
A measure of skewness for testing departures from normality

We propose a new skewness test statistic for normality based on the Pearson measure of skewness. We obtain asymptotic first four moments of the null distribution for this statistic by using a computer algebra system and its normalizing transformation based on the Johnson $S_{U}$ system. Finally the performance of the proposed statistic is shown by comparing the powers of several skewness test statistics against some alternative hypotheses.
Infinitely exchangeable random graphs generated from a Poisson point process on monotone sets and applications to cluster analysis for networks

We construct an infinitely exchangeable process on the set $\cate$ of subsets of the power set of the natural numbers $\mathbb{N}$ via a Poisson point process with mean measure $\Lambda$ on the power set of $\mathbb{N}$. Each $E\in\cate$ has a least monotone cover in $\catf$, the collection of monotone subsets of $\cate$, and every monotone subset maps to an undirected graph $G\in\catg$, the space of undirected graphs with vertex set $\mathbb{N}$. We show a natural mapping $\cate\rightarrow\catf\rightarrow\catg$ which induces an infinitely exchangeable measure on the projective system $\catg^{\rest}$ of graphs $\catg$ under permutation and restriction mappings given an infinitely exchangeable family of measures on the projective system $\cate^{\rest}$ of subsets with permutation and restriction maps. We show potential connections of this process to applications in cluster analysis, machine learning, classification and Bayesian inference.