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Univariate and multivariate quantiles, probabilistic and statistical approaches; radar applications

Abstract : The description and the estimation of univariate and multivariate models whose underlying distribution is heavy-tailed is a strategic challenge. L-moments have become classical tools alternative to central moments for the description of dispersion, skewness and kurtosis of a univariate heavy-tailed distribution. Indeed, contrary to corresponding central moments, they are well defined since the expectation of the distribution of interest is finite. L-moments can be seen as projections of the quantile function on a family of orthogonal polynomials. First, we will estimate parameters of semi-parametric models defined by constraints on L-moments through divergence methods. We will then propose a generalization of L-moments for multivariate distributions using a multivariate quantile function defined as a transport of the uniform distribution on [0; 1]d and the distribution of interest. As their univariate versions, these multivariate L-moments are adapted for the study of heavy-tailed distributions. We explicitly give their formulations for models with rotational parameters. Finally, we propose M-estimators of the scatter matrix of complex elliptical distributions. The family of these distributions form a multivariate semi-parametric model especially containing heavy-tailed distributions. Specific M-estimators adapted to complex elliptical distribution with an additional assumption of stationarity are proposed. Performances and robustness of introduced estimators are studied. Ground and sea clutters are often modelized by complex elliptical distributions in the field of radar processing. We illustrate performances of detectors built from estimators of the scatter matrix through proposed methods for different radar scenarios.
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https://hal.inria.fr/tel-01129961
Contributor : Alexis Decurninge <>
Submitted on : Wednesday, March 11, 2015 - 7:32:08 PM
Last modification on : Wednesday, December 9, 2020 - 3:09:37 PM
Long-term archiving on: : Monday, April 17, 2017 - 7:38:04 AM

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Alexis Decurninge. Univariate and multivariate quantiles, probabilistic and statistical approaches; radar applications. Statistics [stat]. Université Pierre et Marie Curie, 2015. English. ⟨tel-01129961⟩

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