HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci-entific ... more HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et a ̀ la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
This paper introduces rank-based tests for the cointegrating rank in an Error Correction Model wi... more This paper introduces rank-based tests for the cointegrating rank in an Error Correction Model with i.i.d. elliptical innovations. The tests are asymptotically distribution-free, and their validity does not depend on the actual distribution of the innovations. This result holds despite the fact that, depending on the alternatives considered, the model exhibits a non-standard Locally Asymptotically Brownian Functional (LABF) and Locally Asymptotically Mixed Normal (LAMN) local structure—a structure which we completely characterize. Our tests, which have the general form of Lagrange multiplier tests, depend on a reference density that can freely be chosen, and thus is not restricted to be Gaussian as in traditional quasi-likelihood procedures. Moreover, appropriate choices of the reference density are achieving the semiparametric efficiency bounds. Simulations show that our asymptotic analysis provides an accurate approximation to finite-sample behavior. Our results are based on an ex...
Univariate concepts as quantile and distribution functions involving ranks and signs, do not cano... more Univariate concepts as quantile and distribution functions involving ranks and signs, do not canonically extend to $\mathbb{R}^d, d\geq 2$. Palliating that has generated an abundant literature. Chapter 1 shows that, unlike the many definitions that have been proposed so far, the measure transportation-based ones introduced in Chernozhukov et al. (2017) enjoy all the properties that make univariate quantiles and ranks successful tools for semiparametric statistical inference. We therefore propose a new center-outward definition of multivariate distribution and quantile functions, along with their empirical counterparts, for which we obtain a Glivenko-Cantelli result. Our approach is geometric and, contrary to the Monge-Kantorovich one in Chernozhukov et al. (2017), does not require any moment assumptions. The resulting ranks and signs are strictly distribution-free, and maximal invariant under the action of a data-driven class of (order-preserving) transformations generating the fami...
Journal of the American Statistical Association, 2022
Extending rank-based inference to a multivariate setting such as multiple-output regression or MA... more Extending rank-based inference to a multivariate setting such as multiple-output regression or MANOVA with unspecified d-dimensional error density has remained an open problem for more than half a century. None of the many solutions proposed so far is enjoying the combination of distribution-freeness and efficiency that makes rank-based inference a successful tool in the univariate setting. A concept of center-outward multivariate ranks and signs based on measure transportation ideas has been introduced recently. Center-outward ranks and signs are not only distribution-free but achieve in dimension d > 1 the (essential) maximal ancillarity property of traditional univariate ranks, hence carry all the “distribution-free information" available in the sample. We derive here the Hajek representation and asymptotic normality results required in the construction of center-outward rank tests for multiple-output regression and MANOVA. When based on appropriate spherical scores, thes...
Locally asymptotically optimal (in the Hajek-Le Cam sense) pseudo-Gaussian and rank-based procedu... more Locally asymptotically optimal (in the Hajek-Le Cam sense) pseudo-Gaussian and rank-based procedures for detecting randomness of coefficients in linear regression models are proposed. The parametric and semiparametric efficiency properties of those procedures are investigated. Their asymptotic relative efficiencies (with respect to the pseudo-Gaussian procedure) turns out to be be huge under heavy-tailed and skewed densities, stressing the importance of an adequate choice of scores. Simulations demonstrate the excellent finite-sample performances of a class of rank-based procedures based on data-driven scores.
Rank correlations have found many innovative applications in the last decade. In particular,suita... more Rank correlations have found many innovative applications in the last decade. In particular,suitable versions of rank correlations have been used for consistent tests of independence between pairs of random variables. The use of ranks is especially appealing for continuous data as tests become distribution-free. However, the traditional concept of ranks relies on ordering data and is, thus, tied to univariate observations. As a result it has long remained unclear how one may construct distribution-free yet consistent tests of independence between multivariate random vectors. This is the problem we address in this paper, in which we lay out a general framework for designing dependence measures that give tests of multivariate independence that are not only consistent and distribution-free but which we also prove to be statistically efficient. Our framework leverages the recently introduced concept of center-outward ranks and signs, a multivariate generalization of traditional ranks, a...
Annual Review of Statistics and Its Application, 2021
Unlike the real line, the real space, in dimension d ≥ 2, is not canonically ordered. As a conseq... more Unlike the real line, the real space, in dimension d ≥ 2, is not canonically ordered. As a consequence, extending to a multivariate context fundamental univariate statistical tools such as quantiles, signs, and ranks is anything but obvious. Tentative definitions have been proposed in the literature but do not enjoy the basic properties (e.g., distribution-freeness of ranks, their independence with respect to the order statistic, their independence with respect to signs) they are expected to satisfy. Based on measure transportation ideas, new concepts of distribution and quantile functions, ranks, and signs have been proposed recently that, unlike previous attempts, do satisfy these properties. These ranks, signs, and quantiles have been used, quite successfully, in several inference problems and have triggered, in a short span of time, a number of applications: fully distribution-free testing for multiple-output regression, MANOVA, and VAR models; R-estimation for VARMA parameters;...
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci-entific ... more HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et a ̀ la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
This paper introduces rank-based tests for the cointegrating rank in an Error Correction Model wi... more This paper introduces rank-based tests for the cointegrating rank in an Error Correction Model with i.i.d. elliptical innovations. The tests are asymptotically distribution-free, and their validity does not depend on the actual distribution of the innovations. This result holds despite the fact that, depending on the alternatives considered, the model exhibits a non-standard Locally Asymptotically Brownian Functional (LABF) and Locally Asymptotically Mixed Normal (LAMN) local structure—a structure which we completely characterize. Our tests, which have the general form of Lagrange multiplier tests, depend on a reference density that can freely be chosen, and thus is not restricted to be Gaussian as in traditional quasi-likelihood procedures. Moreover, appropriate choices of the reference density are achieving the semiparametric efficiency bounds. Simulations show that our asymptotic analysis provides an accurate approximation to finite-sample behavior. Our results are based on an ex...
Univariate concepts as quantile and distribution functions involving ranks and signs, do not cano... more Univariate concepts as quantile and distribution functions involving ranks and signs, do not canonically extend to $\mathbb{R}^d, d\geq 2$. Palliating that has generated an abundant literature. Chapter 1 shows that, unlike the many definitions that have been proposed so far, the measure transportation-based ones introduced in Chernozhukov et al. (2017) enjoy all the properties that make univariate quantiles and ranks successful tools for semiparametric statistical inference. We therefore propose a new center-outward definition of multivariate distribution and quantile functions, along with their empirical counterparts, for which we obtain a Glivenko-Cantelli result. Our approach is geometric and, contrary to the Monge-Kantorovich one in Chernozhukov et al. (2017), does not require any moment assumptions. The resulting ranks and signs are strictly distribution-free, and maximal invariant under the action of a data-driven class of (order-preserving) transformations generating the fami...
Journal of the American Statistical Association, 2022
Extending rank-based inference to a multivariate setting such as multiple-output regression or MA... more Extending rank-based inference to a multivariate setting such as multiple-output regression or MANOVA with unspecified d-dimensional error density has remained an open problem for more than half a century. None of the many solutions proposed so far is enjoying the combination of distribution-freeness and efficiency that makes rank-based inference a successful tool in the univariate setting. A concept of center-outward multivariate ranks and signs based on measure transportation ideas has been introduced recently. Center-outward ranks and signs are not only distribution-free but achieve in dimension d > 1 the (essential) maximal ancillarity property of traditional univariate ranks, hence carry all the “distribution-free information" available in the sample. We derive here the Hajek representation and asymptotic normality results required in the construction of center-outward rank tests for multiple-output regression and MANOVA. When based on appropriate spherical scores, thes...
Locally asymptotically optimal (in the Hajek-Le Cam sense) pseudo-Gaussian and rank-based procedu... more Locally asymptotically optimal (in the Hajek-Le Cam sense) pseudo-Gaussian and rank-based procedures for detecting randomness of coefficients in linear regression models are proposed. The parametric and semiparametric efficiency properties of those procedures are investigated. Their asymptotic relative efficiencies (with respect to the pseudo-Gaussian procedure) turns out to be be huge under heavy-tailed and skewed densities, stressing the importance of an adequate choice of scores. Simulations demonstrate the excellent finite-sample performances of a class of rank-based procedures based on data-driven scores.
Rank correlations have found many innovative applications in the last decade. In particular,suita... more Rank correlations have found many innovative applications in the last decade. In particular,suitable versions of rank correlations have been used for consistent tests of independence between pairs of random variables. The use of ranks is especially appealing for continuous data as tests become distribution-free. However, the traditional concept of ranks relies on ordering data and is, thus, tied to univariate observations. As a result it has long remained unclear how one may construct distribution-free yet consistent tests of independence between multivariate random vectors. This is the problem we address in this paper, in which we lay out a general framework for designing dependence measures that give tests of multivariate independence that are not only consistent and distribution-free but which we also prove to be statistically efficient. Our framework leverages the recently introduced concept of center-outward ranks and signs, a multivariate generalization of traditional ranks, a...
Annual Review of Statistics and Its Application, 2021
Unlike the real line, the real space, in dimension d ≥ 2, is not canonically ordered. As a conseq... more Unlike the real line, the real space, in dimension d ≥ 2, is not canonically ordered. As a consequence, extending to a multivariate context fundamental univariate statistical tools such as quantiles, signs, and ranks is anything but obvious. Tentative definitions have been proposed in the literature but do not enjoy the basic properties (e.g., distribution-freeness of ranks, their independence with respect to the order statistic, their independence with respect to signs) they are expected to satisfy. Based on measure transportation ideas, new concepts of distribution and quantile functions, ranks, and signs have been proposed recently that, unlike previous attempts, do satisfy these properties. These ranks, signs, and quantiles have been used, quite successfully, in several inference problems and have triggered, in a short span of time, a number of applications: fully distribution-free testing for multiple-output regression, MANOVA, and VAR models; R-estimation for VARMA parameters;...
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