Some statistical results in high-dimensional dependence modeling

Research output: ThesisPhD Thesis - Research external, graduation externalAcademic

Abstract

This thesis can be divided into three parts. In the first part, we study adaptivity to the noise level in the high-dimensional linear regression framework. We prove that two square-root estimators attains the minimax rates of estimation and prediction. We show that a corresponding median-of-means version can still attains the same optimal rates while being robust to outliers in the data.

The second part is devoted to the analysis of several conditional dependence models. We propose some tests of the simplifying assumption that a conditional copula is constant with respect to its conditioning event, and prove the consistency of a semiparametric bootstrap scheme.
If the conditional copula is not constant with respect to the conditional event, then it can be modelled using the corresponding Kendall's tau. We study the estimation of this conditional dependence parameter using 3 different approaches : kernel techniques, regression-type models and classification algorithms.

The last part regroups two different topics in inference. We review and propose estimators for regular conditional functionals using U-statistics. Finally, we study the construction and the theoretical properties of confidence intervals for ratios of means under different sets of assumptions and paradigms.
Original languageEnglish
Awarding Institution
  • ENSAE ParisTech
Supervisors/Advisors
  • Fermanian, Jean-David, Supervisor
  • Tsybakov, Alexandre, Co-Supervisor
Award date15 May 2019
Publication statusPublished - 2019
Externally publishedYes

Fingerprint

High-dimensional
Modeling
Linear regression
Statistics
Copula
Minimax Rate
Kendall's tau
Estimator
U-statistics
Optimal Rates
Adaptivity
Classification Algorithm
Square root
Conditioning
Bootstrap
Outlier
Confidence interval
Regression
Paradigm
kernel

Keywords

  • Conditional copula
  • high-dimensional statistics
  • conditional distribution

Cite this

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title = "Some statistical results in high-dimensional dependence modeling",
abstract = "This thesis can be divided into three parts. In the first part, we study adaptivity to the noise level in the high-dimensional linear regression framework. We prove that two square-root estimators attains the minimax rates of estimation and prediction. We show that a corresponding median-of-means version can still attains the same optimal rates while being robust to outliers in the data.The second part is devoted to the analysis of several conditional dependence models. We propose some tests of the simplifying assumption that a conditional copula is constant with respect to its conditioning event, and prove the consistency of a semiparametric bootstrap scheme.If the conditional copula is not constant with respect to the conditional event, then it can be modelled using the corresponding Kendall's tau. We study the estimation of this conditional dependence parameter using 3 different approaches : kernel techniques, regression-type models and classification algorithms.The last part regroups two different topics in inference. We review and propose estimators for regular conditional functionals using U-statistics. Finally, we study the construction and the theoretical properties of confidence intervals for ratios of means under different sets of assumptions and paradigms.",
keywords = "Conditional copula, high-dimensional statistics, conditional distribution",
author = "Alexis Derumigny",
year = "2019",
language = "English",
school = "ENSAE ParisTech",

}

Some statistical results in high-dimensional dependence modeling. / Derumigny, Alexis.

2019. 304 p.

Research output: ThesisPhD Thesis - Research external, graduation externalAcademic

TY - THES

T1 - Some statistical results in high-dimensional dependence modeling

AU - Derumigny, Alexis

PY - 2019

Y1 - 2019

N2 - This thesis can be divided into three parts. In the first part, we study adaptivity to the noise level in the high-dimensional linear regression framework. We prove that two square-root estimators attains the minimax rates of estimation and prediction. We show that a corresponding median-of-means version can still attains the same optimal rates while being robust to outliers in the data.The second part is devoted to the analysis of several conditional dependence models. We propose some tests of the simplifying assumption that a conditional copula is constant with respect to its conditioning event, and prove the consistency of a semiparametric bootstrap scheme.If the conditional copula is not constant with respect to the conditional event, then it can be modelled using the corresponding Kendall's tau. We study the estimation of this conditional dependence parameter using 3 different approaches : kernel techniques, regression-type models and classification algorithms.The last part regroups two different topics in inference. We review and propose estimators for regular conditional functionals using U-statistics. Finally, we study the construction and the theoretical properties of confidence intervals for ratios of means under different sets of assumptions and paradigms.

AB - This thesis can be divided into three parts. In the first part, we study adaptivity to the noise level in the high-dimensional linear regression framework. We prove that two square-root estimators attains the minimax rates of estimation and prediction. We show that a corresponding median-of-means version can still attains the same optimal rates while being robust to outliers in the data.The second part is devoted to the analysis of several conditional dependence models. We propose some tests of the simplifying assumption that a conditional copula is constant with respect to its conditioning event, and prove the consistency of a semiparametric bootstrap scheme.If the conditional copula is not constant with respect to the conditional event, then it can be modelled using the corresponding Kendall's tau. We study the estimation of this conditional dependence parameter using 3 different approaches : kernel techniques, regression-type models and classification algorithms.The last part regroups two different topics in inference. We review and propose estimators for regular conditional functionals using U-statistics. Finally, we study the construction and the theoretical properties of confidence intervals for ratios of means under different sets of assumptions and paradigms.

KW - Conditional copula

KW - high-dimensional statistics

KW - conditional distribution

M3 - PhD Thesis - Research external, graduation external

ER -