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Research summary: Most strategic management studies adopt an average-centered view that uses the central tendency to explain between-group variation in performance (i.e., performance differences between business units, firms, industries, and countries). In this study, we explain within-group variation using a variance-centered view that focuses on the peripheral characteristics of performance distributions as defined by skew and heavy tails (i.e., variance and kurtosis). Drawing on performance feedback theory, we hypothesize that successful firms tend to develop a positive skew in their performance distributions, which we call a \"positive skew effect\" in this study, and that heavy tails moderate this effect. Our analysis of the performance of a group of foreign affiliates provides general support for our hypotheses at both the firm and segment (industry and country) levels. Managerial summary: Managers of multi-business firms use various approaches to improve the aggregate performance of their business units. Some expand the range of upper performance outliers (exploration) or reduce the range of lower outliers (downsizing); others improve the performance of current business units (exploitation). We find that firms with superior performance tend to have a balanced mix of the three approaches. We also find that segments (countries and industries) with higher mean performances provide environments that facilitate the entry of productive firms and the exit of unproductive firms and provide environments in which incumbents can further improve their performance by learning from others. We observe that successful firms and segments have a positive skew in their performance distributions, which we call a \"positive skew effect.\"

We look at joint regular variation properties of MA(∞) processes of the form X = (X k , k ∈ Z), where X k = ∑ j=0 ∞ψ j Z k-j and the sequence of random variables (Z i , i ∈ Z) are independent and identically distributed with regularly varying tails. We use the setup of M O -convergence and obtain hidden regular variation properties for X under summability conditions on the constant coefficients (ψ j : j ≥ 0). Our approach emphasizes continuity properties of mappings and produces regular variation in sequence space.

We study polynomial time algorithms for estimating the mean of a heavytailed multivariate random vector. We assume only that the random vector X has finite mean and covariance. In this setting, the radius of confidence intervals achieved by the empirical mean are large compared to the case that X is Gaussian or sub-Gaussian. We offer the first polynomial time algorithm to estimate the mean with sub-Gaussian-size confidence intervals under such mild assumptions. Our algorithm is based on a new semidefinite programming relaxation of a highdimensional median. Previous estimators which assumed only existence of finitely many moments of X either sacrifice sub-Gaussian performance or are only known to be computable via brute-force search procedures requiring time exponential in the dimension.

Extreme value theory is routinely applied to estimate rainfall frequency for several accumulation periods. Typically, it is found that sub‐daily precipitation has power‐type tails, meaning that the probability of observing increasingly large magnitudes decreases as a power law. Physical arguments, however, suggest it should have lighter, stretched exponential, tails. Here, we reconcile these perspectives showing that part of the contradiction is caused by precipitation process heterogeneity. We examine hundreds of sub‐daily precipitation records in the Greater Alpine Area, for which a classification of storms into homogeneous types is available. We find that an apparent heavy‐tail behavior is reported at scales of 1–6 hr, and is explained by the coexistence of stratiform and convective processes, both characterized by stretched exponential tails. Our results challenge the assumptions which justify the use of extreme value theory for sub‐daily precipitation, with important implications for how design values are determined.

Estimation of the covariance matrix has attracted a lot of attention of the statistical research community over the years, partially due to important applications such as principal component analysis. However, frequently used empirical covariance estimator, and its modifications, is very sensitive to the presence of outliers in the data. As P. Huber wrote [Ann. Math. Stat. 35 (1964) 73–101], “… This raises a question which could have been asked already by Gauss, but which was, as far as I know, only raised a few years ago (notably by Tukey): what happens if the true distribution deviates slightly from the assumed normal one? As is now well known, the sample mean then may have a catastrophically bad performance….” Motivated by Tukey’s question,we develop a new estimator of the (element-wise) mean of a random matrix, which includes covariance estimation problem as a special case. Assuming that the entries of a matrix possess only finite second moment, this new estimator admits sub-Gaussian or sub-exponential concentration around the unknown mean in the operator norm. We explain the key ideas behind our construction, and discuss applications to covariance estimation and matrix completion problems.

Regular variation of distributional tails is known to be preserved by various linear transformations of some random structures. An inverse problem for regular variation aims at understanding whether the regular variation of a transformed random object is caused by regular variation of components of the original random structure. In this paper we build on previous work, and derive results in the multivariate case and in situations where regular variation is not restricted to one particular direction or quadrant.

We provide a new computationally efficient class of estimators for risk minimization. We show that these estimators are robust for general statistical models, under varied robustness settings, including in the classical Huber ϵ-contamination model, and in heavy-tailed settings. Our workhorse is a novel robust variant of gradient descent, and we provide conditions under which our gradient descent variant provides accurate estimators in a general convex risk minimization problem. We provide specific consequences of our theory for linear regression and logistic regression and for canonical parameter estimation in an exponential family. These results provide some of the first computationally tractable and provably robust estimators for these canonical statistical models. Finally, we study the empirical performance of our proposed methods on synthetic and real data sets, and we find that our methods convincingly outperform a variety of baselines.

We consider a two-node fluid network with batch arrivals of random size having a heavy-tailed distribution. We are interested in the tail asymptotics for the stationary distribution of a two-dimensional workload process. Tail asymptotics have been well studied for two-dimensional reflecting processes where jumps have either a bounded or an unbounded light-tailed distribution. However, the presence of heavy tails totally changes these asymptotics. Here we focus on the case of strong stability where both nodes release fluid at sufficiently high speeds to minimise their mutual influence. We show that, as in the one-dimensional case, big jumps provide the main cause for workloads to become large, but now they can have multidimensional features. We first find the weak tail asymptotics of an arbitrary directional marginal of the stationary distribution at Poisson arrival epochs. In this analysis, decomposition formulae for the stationary distribution play a key role. Then we employ sample-path arguments to find the exact tail asymptotics of a directional marginal at renewal arrival epochs assuming one-dimensional batch arrivals.

Expectiles define a least squares analogue of quantiles. They are determined by tail expectations rather than tail probabilities. For this reason and many other theoretical and practical merits, expectiles have recently received a lot of attention, especially in actuarial and financial risk management. Their estimation, however, typically requires to consider non-explicit asymmetric least squares estimates rather than the traditional order statistics used for quantile estimation. This makes the study of the tail expectile process a lot harder than that of the standard tail quantile process. Under the challenging model of heavy-tailed distributions, we derive joint weighted Gaussian approximations of the tail empirical expectile and quantile processes. We then use this powerful result to introduce and study new estimators of extreme expectiles and the standard quantile-based expected shortfall, as well as a novel expectile-based form of expected shortfall. Our estimators are built on general weighted combinations of both top order statistics and asymmetric least squares estimates. Some numerical simulations and applications to actuarial and financial data are provided.

We consider least squares estimation in a general nonparametric regression model where the error is allowed to depend on the covariates. The rate of convergence of the least squares estimator (LSE) for the unknown regression function is well studied when the errors are sub-Gaussian. We find upper bounds on the rates of convergence of the LSE when the error has a uniformly bounded conditional variance and has only finitely many moments. Our upper bound on the rate of convergence of the LSE depends on the moment assumptions on the error, the metric entropy of the class of functions involved and the “local” structure of the function class around the truth. We find sufficient conditions on the error distribution under which the rate of the LSE matches the rate of the LSE under sub-Gaussian error. Our results are finite sample and allow for heteroscedastic and heavy-tailed errors.