Multidimensional Setting Research Articles

Large sample properties of the regression depth induced median

Notions of depth in regression have been introduced and studied in the literature. Regression depth (RD) of Rousseeuw and Hubert (1999), the most famous one, is a direct extension of Tukey location depth (Tukey, 1975) to regression.Like its location counterpart, the most remarkable advantage of the notion of depth in regression is to directly introduce the maximum (or deepest) regression depth estimator (aka depth induced median) for regression parameters in a multi-dimensional setting. Classical questions for the regression depth induced median include (i) is it a consistent estimator (or rather under what sufficient conditions, it is consistent)? and (ii) is there any limiting distribution?Bai and He (1999) (BH99) pioneered an attempt to answer these questions. Under some stringent conditions on (i) the design points, (ii) the conditional distributions of y given xi, and (iii) the error distributions, BH99 proved the strong consistency of the depth induced median. Under another set of conditions, BH99 showed the existence of the limiting distribution of the estimator.This article establishes the strong consistency of the depth induced median without any of the stringent conditions in BH99, and proves the existence of the limiting distribution of the estimator by sufficient conditions and an approach different from BH99.

Functional diversity metrics using kernel density n‐dimensional hypervolumes

Abstract The use of n‐dimensional hypervolumes in trait‐based ecology is rapidly increasing. By representing the functional space of a species or community as a Hutchinsonian niche, the abstract Euclidean space defined by a set of independent axes corresponding to individuals or species traits, these multidimensional techniques show great potential for the advance of functional ecology theory. In the panorama of existing methods for delineating multidimensional spaces, the r package hypervolume (Global Ecology and Biogeography, 23, 2014, 595–609) is currently the most used. However, functions for calculating the standard set of functional diversity (FD) indices—richness, divergence and regularity—have not been developed within the hypervolume framework yet. This gap is delaying its full exploitation in functional ecology, meanwhile preventing the possibility to compare its performance with that of other methods. We develop a set of functions to calculate FD indices based on n‐dimensional hypervolumes, including alpha (richness), beta (and respective components), dispersion, evenness, contribution and originality. Altogether, these indices provide a coherent framework to explore the primary mathematical components of FD within a multidimensional setting. These new functions can work either with hypervolume objects or with raw data (species presence or abundance and their traits) as input data, and are versatile in terms of input parameters and options. These functions are implemented within bat (Biodiversity Assessment Tools), an r package for biodiversity assessments. As a coherent corpus of functional indices based on a common algorithm, it opens the possibility to fully explore the strengths of the Hutchinsonian niche concept in community ecology research.

A Fourier-based Picard-iteration approach for a class of McKean–Vlasov SDEs with Lévy jumps

We consider a prototype class of Lévy-driven stochastic differential equations (SDEs) with McKean–Vlasov (MK–V) interaction in the drift coefficient. It is assumed that the drift coefficient is affine in the state variable, and only measurable in the law of the solution. We study the equivalent functional fixed-point equation for the unknown time-dependent coefficients of the associated linear Markovian SDE. By proving a contraction property for the functional map in a suitable normed space, we infer existence and uniqueness results for the MK–V SDE, and derive a discretized Picard iteration scheme that approximates the law of the solution through its characteristic function. Numerical illustrations show the effectiveness of our method, which appears to be appropriate to handle the multi-dimensional setting.

Convergence in variation for the multidimensional generalized sampling series and applications to smoothing for digital image processing

In this paper we study the problem of the convergence in variation for the generalized sampling series based upon averaged-type kernels in the multidimensional setting. As a crucial tool, we introduce a family of operators of sampling-Kantorovich type for which we prove convergence in L^p on a subspace of L^p(R^N): therefore we obtain the convergence in variation for the multidimensional generalized sampling series by means of a relation between the partial derivatives of such operators acting on an absolutely continuous function f and the sampling-Kantorovich type operators acting on the partial derivatives of f. Applications to digital image processing are also furnished.

Risk functionals with convex level sets

AbstractWe analyze the “convex level sets” (CxLS) property of risk functionals, which is a necessary condition for the notions of elicitability, identifiability, and backtestability, popular in the recent statistics and risk management literature. We put the CxLS property in the multidimensional setting, with a special focus on signed Choquet integrals, a class of risk functionals that are generally not monotone or convex. We obtain two main analytical results in dimension one and dimension two, by characterizing the CxLS property of all one‐dimensional signed Choquet integrals, and that of all two‐dimensional signed Choquet integrals with a quantile component. Using these results, we proceed to show that under some continuity assumption, a comonotonic‐additive coherent risk measure is co‐elicitable with Value‐at‐Risk if and only if it is the corresponding Expected Shortfall. The new findings generalize several results in the recent literature, and partially answer an open question on the characterization of multidimensional elicitability.

Sewing transnational textures of labour regulation: towards an integrative perspective

Labour regulation is often approached either by methodological nationalism or methodological globalism. Main arguments are that labour and its regulation are increasingly dis-embedded and commodified. As an alternative framing we propose a (neo-)institutionalist multi-level and multi-actors approach. Based on literature review and own studies we argue that there is empirical evidence of transnational labour regulation combining different logics of action and institutional contexts. Taking the example of the Bangladesh Accord on Fire and Building Safety, we demonstrate the multi-level, multi-dimensional institutional settings and sketch out basic elements of an integrative perspective on sewing transnational textures of labour regulation.Key termsLabour regulation, globalization, transnationalization, Bangladesh Accord, neo-institutionalism, multi-level multi-actor approach.

Well-posedness of Hamilton–Jacobi equations in population dynamics and applications to large deviations

We prove Freidlin–Wentzell type large deviation principles for various rescaled models in populations dynamics that have immigration and possibly harvesting: birth–death processes, Galton–Watson trees, epidemic SI models, and prey–predator models.The proofs are carried out using a general analytic approach based on the well-posedness of a class of associated Hamilton–Jacobi equations. The notable feature for these Hamilton–Jacobi equations is that the Hamiltonian can be discontinuous at the boundary. We prove a well-posedness result for a large class of Hamilton–Jacobi equations corresponding to one-dimensional models, and give partial results for the multi-dimensional setting.

Measuring Multidimensional Inequality of Opportunity

This paper develops a normative approach to the measurement of ex-ante inequality of opportunity in a multidimensional setting, i.e., when the individual outcome is represented by a multidimensional variable. We characterize three classes of social welfare functions, all endorsing ex ante compensation but each of them reflecting a specific reward principle: (1) utilitarian, (2) agnostic and (3) averse. The first class is implemented via generalized Lorenz Dominance applied to each attribute separately. The agnostic and inequality averse classes are implemented by a welfarist Lorenz ordering, namely, of type-aggregate utilities. In the case of inequality-averse class, utility functions are submodular, hence capturing the dependence between attributes. We also develop normative inequality indices (Atkinson, 1970; Kolm 1969; Sen, 1973) for the classes of welfare functions and study their properties. Finally, we propose an empirical applications of the methods developed in the paper: by using the National Longitudinal Study of Adolescent to Adult Health (Add Health) we evaluate inequality of opportunity in U.S. for the case of three dimensions of individual outcomes, namely, education, health and income.

Copula‐based analysis of multivariate dependence patterns between dimensions of poverty in Europe*

It is widely recognized that poverty is a multidimensional phenomenon involving not only income, but also other aspects such as education or health. In this multidimensional setting, analyzing the dependence between dimensions becomes an important issue, since a high degree of dependence could exacerbate poverty. In this paper, we propose measuring the multivariate dependence between the dimensions of poverty in Europe using copula‐based methods. This approach focuses on the positions of individuals across dimensions, allowing for other types of dependence beyond linear correlation. In particular, we analyze how orthant dependence between the dimensions of the AROPE rate has evolved in the EU‐28 countries between 2008 and 2014 by applying non‐parametric estimates of multivariate copula‐based generalisations of Spearman’s rank correlation coefficient. We find a general increase in the dependence between dimensions, regardless of the coefficient used. Moreover, countries with higher AROPE rates also tend to experiment more dependence between its dimensions.

Multi-dimensional mechanism design via random order contention resolution schemes

Already in 1981 Myerson gave a characterization of the optimal mechanism for a single parameter Bayesian mechanism design. However, till today we have no idea for how such a characterization for the multi-dimensional setting could even look like. Moreover, it wasn't until no that long time ago that we could not develop mechanisms for such setting with any reasonable and provable performance guarantees. The seminal work of [Chawla et al. 2009] on sequential posted pricing mechanisms gave us an approach for approximately solving the Bayesian multi-parameter unit-demand mechanism design problem (BMUMD). The paper left open the question on how to obtain a constant approximation for the matroid setting. Two mathematically beautiful results from combinatorial optimization under uncertainty where devised in order to answer this question. First, Kleinberg and Weinberg in 2011 extended the classical Prophet Inequality result into the matroid setting to give a 2-approximation for BMUMD for a single matroid setting. Second, Feldman, Svensson and Zenklusen in 2016 adapted the Contention Resolution Scheme framework for online settings. We add to this line of work by considering the Contention Resolution Scheme framework in the random order setting. The most impressive implication of this research are the new algorithms for BMUMD which improve the previous results in the multimatroid setting. Although the range of implications of the CR Scheme framework in the random order is reasonably wide, we shall focus in this letter on presenting only the single matroid setting and how it is connected to BMUMD.

Non-catastrophicity in multidimensional convolutional coding

The property of non-catastrophicity of multidimensional convolutional codes is studied. In particular, an algebraic and system-theoretic characterization of non-catastrophicity is offered in the multidimensional setting, and the Massey–Sain classical criterion is extended to this setting.

Harmonic Functions of Random Walks in a Semigroup via Ladder Heights

We investigate harmonic functions and the convergence of the sequence of ratios $$({\mathbb {P}}_x(\tau _\vartheta> n)/{\mathbb {P}}_e(\tau _\vartheta > n))$$ for a random walk on a countable group killed upon the time $$\tau _\vartheta $$ of the first exit from some semigroup with an identity element e. Several results of classical renewal theory for one-dimensional random walk killed at the first exit from the positive half-line are extended to a multi-dimensional setting. For this purpose, an analogue of the ladder height process and the corresponding renewal function V are introduced. The results are applied to multi-dimensional random walks (X(t)) killed upon the times of first exit from a convex cone. Our approach combines large deviation estimates and an extension of Choquet–Deny theory.

Fluctuations of the Propagation Front of a Catalytic Branching Walk

We consider a supercritical catalytic branching random walk (CBRW) on a multidimensional lattice Z^d (d is positive integer). The main subject of study is the behavior of particles cloud in space and time. For CBRW on an integer line, Carmona and Hu (2014) examined the asymptotical behavior of the maximal coordinate M_n of the particles at time n. They proved that M_n/n converges to \mu almost surely (on a set of local non-degeneracy of CBRW), as n tends to infinity, where \mu>0 is a certain constant. Under additional assumption of a single catalyst in CBRW they also investigated the fluctuations of M_n with respect to \mu n, as n grows to infinity. Bulinskaya (2018) extended the strong limit theorem by Carmona and Hu having estimated the rate of the population propagation for the front of a multidimensional CBRW. Now our aim is to analyze fluctuations of the propagation front in CBRW on Z^d. We not only solve the problem in a multidimensional setting but also, treating the case of an arbitrary finite number of catalysts for d = 1, generalize the result by Carmona and Hu with the help of other probabilistic-analytic methods. Keywords and phrases: catalytic branching random walk, supercritical regime, spread of population, propagation front, fluctuations of front.

A New Methodology for the Study of Dynamic Decision-Making with a Continuous Unknown Parameter

A new methodology is presented to solve an important model of dynamic decision-making with a continuous unknown parameter (or state). The methodology centers on the concepts of “continuation-value function” (which gives the expected value-to-go from every possible state under a feasible policy) and “efficient frontier” of such functions in each period. When the model primitives can be described through a family of basis functions, e.g. polynomials, a continuation-value function retains that property and can be fully represented by a basis weight vector. The efficient frontiers of the weight vectors can be constructed through backward induction, which leads to an essential reduction of problem complexity and enables an exact solution for small-sized problems. A set of approximation methods based on the new methodology are developed to tackle larger problems. The methodology is also extended to the multi-dimensional (multi-parameter) setting, which features the important problem of contextual multi-armed bandits with linear expected rewards. We demonstrate that our approximation algorithm for that problem has a clear edge over three benchmark algorithms in the challenging learning environment with many actions and relatively short horizons.

Flexible, boundary adapted, nonparametric methods for the estimation of univariate piecewise-smooth functions

We present and compare some nonparametric estimation methods (wavelet and/or spline-based) designed to recover a one-dimensional piecewise-smooth regression function in both a fixed equidistant or not equidistant design regression model and a random design model. Wavelet methods are known to be very competitive in terms of denoising and compression, due to the simultaneous localization property of a function in time and frequency. However, boundary assumptions, such as periodicity or symmetry, generate bias and artificial wiggles which degrade overall accuracy. Simple methods have been proposed in the literature for reducing the bias at the boundaries. We introduce new ones based on adaptive combinations of two estimators. The underlying idea is to combine a highly accurate method for non-regular functions, e.g., wavelets, with one well behaved at boundaries, e.g., Splines or Local Polynomial. We provide some asymptotic optimal results supporting our approach. All the methods can handle data with a random design. We also sketch some generalization to the multidimensional setting. To study the performance of the proposed approaches we have conducted an extensive set of simulations on synthetic data. An interesting regression analysis of two real data applications using these procedures unambiguously demonstrates their effectiveness.

Price Discovery in Bitcoin: The Impact of Unregulated Markets

We analyse minute-level multi-dimensional information flows within and between bitcoin spot and derivatives. We show that perpetual swaps and futures traded on the unregulated exchanges Huobi, OKEx and BitMEX are much the strongest instruments for bitcoin price discovery and we examine potential determinants of their leadership strength. Prices on the regulated CME bitcoin futures and the US-based spot exchanges react to, rather than lead, price movements on the unregulated exchanges and they may do so relatively slowly. In a multi-dimensional setting including the main price leaders within futures, perpetuals and spot markets, the CME futures have a very minor effect on price discovery, even less than the spot exchanges Bitfinex, Bitstamp and Coinbase. Our findings highlight the persistent problems stemming from inconsistent regulation in bitcoin spot and derivatives markets, including insufficient price stability and lack of resistance to manipulative trading. We conclude that the SEC are correct to maintain such issues as their main concern for bitcoin ETF applications.

Testing One Hypothesis Multiple Times: The Multidimensional Case

The identification of new rare signals in data, the detection of a sudden change in a trend, and the selection of competing models are among the most challenging problems in statistical practice. These challenges can be tackled using a test of hypothesis where a nuisance parameter is present only under the alternative, and a computationally efficient solution can be obtained by the “testing one hypothesis multiple times” (TOHM) method. In the one-dimensional setting, a fine discretization of the space of the non identifiable parameter is specified, and a global p-value is obtained by approximating the distribution of the supremum of the resulting stochastic process. In this article, we propose a computationally efficient inferential tool to perform TOHM in the multidimensional setting. Here, the approximations of interest typically involve the expected Euler characteristics (EC) of the excursion set of the underlying random field. We introduce a simple algorithm to compute the EC in multiple dimensions and for arbitrarily large significance levels. This leads to an highly generalizable computational tool to perform hypothesis testing under nonstandard regularity conditions. Supplementary materials for this article are available online.

An Inherent Difficulty in the Aggregation of Multidimensional Data

In the field of information fusion, the problem of data aggregation has been formalized as an order-preserving process that builds upon the property of monotonicity. However, fields such as computational statistics, data analysis, and geometry usually emphasize the role of equivariances to various geometrical transformations in aggregation processes. Admittedly, if we consider a unidimensional data fusion task, both requirements are often compatible with each other. Nevertheless, in this paper, we show that, in the multidimensional setting, the only idempotent functions that are monotone and orthogonal equivariant are the over-simplistic weighted centroids. Even more, this result still holds after replacing monotonicity and orthogonal equivariance by the weaker property of orthomonotonicity. This implies that the aforementioned approaches to the aggregation of multidimensional data are irreconcilable, and that, if a weighted centroid is to be avoided, we must choose between monotonicity and a desirable behavior with regard to orthogonal transformations.

Exact sampling for some multi-dimensional queueing models with renewal input

AbstractUsing a result of Blanchet and Wallwater (2015) for exactly simulating the maximum of a negative drift random walk queue endowed with independent and identically distributed (i.i.d.) increments, we extend it to a multi-dimensional setting and then we give a new algorithm for simulating exactly the stationary distribution of a first-in–first-out (FIFO) multi-server queue in which the arrival process is a general renewal process and the service times are i.i.d.: the FIFO GI/GI/c queue with $ 2 \leq c \lt \infty$ . Our method utilizes dominated coupling from the past (DCFP) as well as the random assignment (RA) discipline, and complements the earlier work in which Poisson arrivals were assumed, such as the recent work of Connor and Kendall (2015). We also consider the models in continuous time, and show that with mild further assumptions, the exact simulation of those stationary distributions can also be achieved. We also give, using our FIFO algorithm, a new exact simulation algorithm for the stationary distribution of the infinite server case, the GI/GI/ $\infty$ model. Finally, we even show how to handle fork–join queues, in which each arriving customer brings c jobs, one for each server.

Exponential convergence of solutions for random Hamilton–Jacobi equations

We show that for a family of randomly kicked Hamilton–Jacobi equations on the torus, almost surely, the solution of an initial value problem converges exponentially fast to the unique stationary solution. Combined with the earlier results of the authors, this completes the program in the multi-dimensional setting started by E, Khanin, Mazel and Sinai in the one-dimensional case.