Sorted L-One Penalized Estimator (SLOPE) is a relatively new convex optimization procedure for selecting predictors in large data bases. Contrary to LASSO, SLOPE has been proved to be asymptotically minimax in the context of sparse high-dimensional generalized linear models. Additionally, in case when the design matrix is orthogonal, SLOPE with the sequence of tuning parameters $λ^{BH}$, corresponding to the sequence of decaying thresholds for the Benjamini-Hochberg multiple testing correction, provably controls False Discovery Rate in the multiple regression model. In this article we provide new asymptotic results on the properties of SLOPE when the elements of the design matrix are iid random variables from the Gaussian distribution. Specifically, we provide the conditions, under which the asymptotic FDR of SLOPE based on the sequence $λ^{BH}$ converges to zero and the power converges to 1. We illustrate our theoretical asymptotic results with extensive simulation study. We also provide precise formulas describing FDR of SLOPE under different loss functions, which sets the stage for future results on the model selection properties of SLOPE and its extensions.
University of Pennsylvania（宾夕法尼亚大学）；Rutgers University（罗格斯大学）；Columbia University（哥伦比亚大学）
SLOPE is a relatively new convex optimization procedure for high-dimensional linear regression via the sorted l1 penalty: the larger the rank of the fitted coefficient, the larger the penalty. This non-separable penalty renders many existing techniques invalid or inconclusive in analyzing the SLOPE solution. In this paper, we develop an asymptotically exact characterization of the SLOPE solution under Gaussian random designs through solving the SLOPE problem using approximate message passing (AMP). This algorithmic approach allows us to approximate the SLOPE solution via the much more amenable…
Asymptotic Theory for Graphical SLOPE: Precision Estimation and Pattern Convergence
Lund University（隆德大学）；“Kore” University of Enna（恩纳“科雷”大学）；La Francaise Systematic Asset Management GmbH（法国系统资产管理有限责任公司）；University of Trento（特伦托大学）；University of Wroclaw（弗罗茨瓦夫大学）
Statistical inference for large data panels is omnipresent in modern economic applications. An important benefit of panel analysis is the possibility to reduce noise and thus to guarantee stable inference by intersectional pooling. However, it is wellknown that pooling can lead to a biased analysis if individual heterogeneity is too strong. In classical linear panel models, this trade-off concerns the homogeneity of slope parameters, and a large body of tests has been developed to validate this assumption. Yet, such tests can detect inconsiderable deviations from slope homogeneity, discouragin…
We build penalized least-squares estimators using the slope heuristic and resampling penalties. We prove oracle inequalities for the selected estimator with leading constant asymptotically equal to 1. We compare the practical performances of these methods in a short simulation study.
We construct uniform and point-wise asymptotic confidence sets for the single edge in an otherwise smooth image function which are based on rotated differences of two one-sided kernel estimators. Using methods from M-estimation, we show consistency of the estimators of location, slope and height of the edge function and develop a uniform linearization of the contrast process. The uniform confidence bands then rely on a Gaussian approximation of the score process together with anti-concentration results for suprema of Gaussian processes, while point-wise bands are based on asymptotic normality.…
In this paper, we propose a class of low-rank panel quantile regression models which allow for unobserved slope heterogeneity over both individuals and time. We estimate the heterogeneous intercept and slope matrices via nuclear norm regularization followed by sample splitting, row- and column-wise quantile regressions and debiasing. We show that the estimators of the factors and factor loadings associated with the intercept and slope matrices are asymptotically normally distributed. In addition, we develop two specification tests: one for the null hypothesis that the slope coefficient is a co…
This paper studies a class of exponential family models whose canonical parameters are specified as linear functionals of an unknown infinite-dimensional slope function. The optimal minimax rates of convergence for slope function estimation are established. The estimators that achieve the optimal rates are constructed by constrained maximum likelihood estimation with parameters whose dimension grows with sample size. A change-of-measure argument, inspired by Le Cam's theory of asymptotic equivalence, is used to eliminate the bias caused by the non-linearity of exponential family models.
In sparse linear regression, the SLOPE estimator generalizes LASSO by penalizing different coordinates of the estimate according to their magnitudes. In this paper, we present a precise performance characterization of SLOPE in the asymptotic regime where the number of unknown parameters grows in proportion to the number of observations. Our asymptotic characterization enables us to derive the fundamental limits of SLOPE in both estimation and variable selection settings. We also provide a computational feasible way to optimally design the regularizing sequences such that the fundamental limits…
The problems of outliers detection and robust regression in a high-dimensional setting are fundamental in statistics, and have numerous applications. Following a recent set of works providing methods for simultaneous robust regression and outliers detection, we consider in this paper a model of linear regression with individual intercepts, in a high-dimensional setting. We introduce a new procedure for simultaneous estimation of the linear regression coefficients and intercepts, using two dedicated sorted-$\ell_1$ penalizations, also called SLOPE. We develop a complete theory for this problem:…
The simple linear model $$Y_i = α+ β\, x_i + ε_i \qquad i=1,2, \ldots,N \geq 2$$ is considered, where the $x_i$'s are given constants and $ε_1, ε_2 , \ldots, ε_N$ are iid with continuous distribution function $F$. An estimator of $β$ is proposed, based on Gini's rank association coefficient $G(\underline y;b)$ and defined as $\tildeβ = \frac 12 \, \left\{ \sup (b: G(\underline y;b) >0) + \right. $ $ \left. \inf (b: G(\underline y;b) <0) \right\}.$ The properties of $\tildeβ$ and of the related confidence interval are studied. Some comparisons are given, in terms of asymptotic relative efficien…
We introduce a general approach to prove oracle properties in context tree selection. The results derive from a concentration condition that is verified, for example, by mixing processes. Moreover, we show the superiority of the oracle approach from a non-asymptotic point of view in simulations where the classical BIC estimator has nice oracle properties even when it does not recover the source. Our second objective is to extend the slope algorithm of \cite{AM08} to context tree estimation. The algorithm gives a practical way to evaluate the leading constant in front of the penalties. We study…
In this paper we extend the work of Owen (2007) by deriving a second order expansion for the slope parameter in logistic regression, when the size of the majority class is unbounded and the minority class is finite. More precisely, we demonstrate that the second order term converges to a normal distribution and explicitly compute its variance, which surprisingly once again depends only on the mean of the minority class points and not their arrangement under mild regularity assumptions. In the case that the majority class is normally distributed, we illustrate that the variance of the the limit…
Point estimators may not exist, need not be unique, and their distributions are not parameter invariant. Generalized estimators provide distributions that are parameter invariant, unique, and exist when point estimates do not. Comparing point estimators using variance is less useful when estimators are biased. A squared slope $Λ$ is defined that can be used to compare both point and generalized estimators and is unaffected by bias. Fisher information $I$ and variance are fundamentally different quantities: the latter is defined at a distribution that need not belong to a family, while the form…
Sorted L-One Penalized Estimation is a relatively new convex optimization procedure which allows for adaptive selection of regressors under sparse high dimensional designs. Here we extend the idea of SLOPE to deal with the situation when one aims at selecting whole groups of explanatory variables instead of single regressors. This approach is particularly useful when variables in the same group are strongly correlated and thus true predictors are difficult to distinguish from their correlated "neighbors"'. We formulate the respective convex optimization problem, gSLOPE (group SLOPE), and propo…
We describe the asymptotic properties of the edge-triangle exponential random graph model as the natural parameters diverge along straight lines. We show that as we continuously vary the slopes of these lines, a typical graph drawn from this model exhibits quantized behavior, jumping from one complete multipartite graph to another, and the jumps happen precisely at the normal lines of a polyhedral set with infinitely many facets. As a result, we provide a complete description of all asymptotic extremal behaviors of the model.
