r packages for robust asymptotic statistics
play

R-Packages for Robust Asymptotic Statistics Dr. Matthias Kohl Chair - PowerPoint PPT Presentation

Nov 01, 2022 •310 likes •665 views

Robust Asymptotic Statistics Exponential Families Regression-Type Models R-Packages for Robust Asymptotic Statistics Dr. Matthias Kohl Chair for Stochastics joint work with Dr. Peter Ruckdeschel Fraunhofer ITWM useR! The R User

  1. Robust Asymptotic Statistics Exponential Families Regression-Type Models R-Packages for Robust Asymptotic Statistics Dr. Matthias Kohl Chair for Stochastics joint work with Dr. Peter Ruckdeschel Fraunhofer ITWM useR! – The R User Conference 2008 Dortmund August 12 Dr. Matthias Kohl R-Packages for Robust Asymptotic Statistics

  2. Robust Asymptotic Statistics Exponential Families Regression-Type Models Outline Robust Asymptotic Statistics 1 Exponential Families 2 Regression-Type Models 3 Dr. Matthias Kohl R-Packages for Robust Asymptotic Statistics

  3. Robust Asymptotic Statistics Exponential Families Regression-Type Models Outline Robust Asymptotic Statistics 1 Exponential Families 2 Regression-Type Models 3 Dr. Matthias Kohl R-Packages for Robust Asymptotic Statistics

  4. Robust Asymptotic Statistics Exponential Families Regression-Type Models Setup I Ideal model: L 2 -differentiable parametric family of probability measures, parameter space: Θ ⊂ R k (open) Estimator class: asymptotically linear estimators (ALEs) S n n S n ( x 1 , . . . , x n ) = θ + 1 � ψ θ ( x i ) + R n n i =1 x 1 , . . . , x n : sample ψ θ : influence curve/function (IC) at θ ∈ Θ R n : asymptotically negligible remainder E.g. as. normal M-, L-, R-, S- and MD-estimators Dr. Matthias Kohl R-Packages for Robust Asymptotic Statistics

  5. Robust Asymptotic Statistics Exponential Families Regression-Type Models Setup I Ideal model: L 2 -differentiable parametric family of probability measures, parameter space: Θ ⊂ R k (open) Estimator class: asymptotically linear estimators (ALEs) S n n S n ( x 1 , . . . , x n ) = θ + 1 � ψ θ ( x i ) + R n n i =1 x 1 , . . . , x n : sample ψ θ : influence curve/function (IC) at θ ∈ Θ R n : asymptotically negligible remainder E.g. as. normal M-, L-, R-, S- and MD-estimators Dr. Matthias Kohl R-Packages for Robust Asymptotic Statistics

  6. Robust Asymptotic Statistics Exponential Families Regression-Type Models Setup II Infinitesimal neighborhood: deviations (gross errors, outliers, etc.) from the ideal model P θ of form r d ∗ ( P θ , Q ) = √ n =: r n Q ∈ M 1 M 1 : set of all probability measures d ∗ : some distance or pseudo-distance r : radius in [0 , √ n ] E.g. Tukey’s gross error model Q = (1 − r n ) P θ + r n H n H n ∈ M 1 Dr. Matthias Kohl R-Packages for Robust Asymptotic Statistics

  7. Robust Asymptotic Statistics Exponential Families Regression-Type Models Optimally robust ALEs Optimization problem: � � G asBias ( S n ) , asVar ( S n ) = min! G : positive, convex, strictly increasing in both args asBias ( S n ): some function of ψ θ (IC) asVar ( S n ): some function of ψ θ (IC) Hence: minimum is taken over all ICs ψ θ Optimal solutions: Rieder (1994) [3], Ruckdeschel and Rieder (2004) [10], Kohl (2005) [2] Unknown radius: radius-minimax estimator; cf. Rieder et al. (2008) [8] Dr. Matthias Kohl R-Packages for Robust Asymptotic Statistics

  8. Robust Asymptotic Statistics Exponential Families Regression-Type Models Optimally robust ALEs Optimization problem: � � G asBias ( S n ) , asVar ( S n ) = min! G : positive, convex, strictly increasing in both args asBias ( S n ): some function of ψ θ (IC) asVar ( S n ): some function of ψ θ (IC) Hence: minimum is taken over all ICs ψ θ Optimal solutions: Rieder (1994) [3], Ruckdeschel and Rieder (2004) [10], Kohl (2005) [2] Unknown radius: radius-minimax estimator; cf. Rieder et al. (2008) [8] Dr. Matthias Kohl R-Packages for Robust Asymptotic Statistics

  9. Robust Asymptotic Statistics Exponential Families Regression-Type Models Optimally robust ALEs Optimization problem: � � G asBias ( S n ) , asVar ( S n ) = min! G : positive, convex, strictly increasing in both args asBias ( S n ): some function of ψ θ (IC) asVar ( S n ): some function of ψ θ (IC) Hence: minimum is taken over all ICs ψ θ Optimal solutions: Rieder (1994) [3], Ruckdeschel and Rieder (2004) [10], Kohl (2005) [2] Unknown radius: radius-minimax estimator; cf. Rieder et al. (2008) [8] Dr. Matthias Kohl R-Packages for Robust Asymptotic Statistics

  10. Robust Asymptotic Statistics Exponential Families Regression-Type Models Optimally robust estimation Possible steps to compute an optimally robust estimator: 1 Decide on ideal model, neighborhood and risk 2 Try to find a rough estimate for the amount r n ∈ [0 , 1] of gross errors such that r n ∈ [ r n , r n ]. 3 Choose and evaluate appropriate initial estimate; e.g., Kolmogorov or Cram´ er von Mises MD-estimator 4 Estimate the parameter(s) of interest by means of the corresponding radius-minimax estimator (cf. Rieder et al. (2008) [8]) using a k -step ( k ≥ 1) construction. Dr. Matthias Kohl R-Packages for Robust Asymptotic Statistics

  11. Robust Asymptotic Statistics Exponential Families Regression-Type Models Optimally robust estimation Possible steps to compute an optimally robust estimator: 1 Decide on ideal model, neighborhood and risk 2 Try to find a rough estimate for the amount r n ∈ [0 , 1] of gross errors such that r n ∈ [ r n , r n ]. 3 Choose and evaluate appropriate initial estimate; e.g., Kolmogorov or Cram´ er von Mises MD-estimator 4 Estimate the parameter(s) of interest by means of the corresponding radius-minimax estimator (cf. Rieder et al. (2008) [8]) using a k -step ( k ≥ 1) construction. Dr. Matthias Kohl R-Packages for Robust Asymptotic Statistics

  12. Robust Asymptotic Statistics Exponential Families Regression-Type Models Optimally robust estimation Possible steps to compute an optimally robust estimator: 1 Decide on ideal model, neighborhood and risk 2 Try to find a rough estimate for the amount r n ∈ [0 , 1] of gross errors such that r n ∈ [ r n , r n ]. 3 Choose and evaluate appropriate initial estimate; e.g., Kolmogorov or Cram´ er von Mises MD-estimator 4 Estimate the parameter(s) of interest by means of the corresponding radius-minimax estimator (cf. Rieder et al. (2008) [8]) using a k -step ( k ≥ 1) construction. Dr. Matthias Kohl R-Packages for Robust Asymptotic Statistics

  13. Robust Asymptotic Statistics Exponential Families Regression-Type Models Optimally robust estimation Possible steps to compute an optimally robust estimator: 1 Decide on ideal model, neighborhood and risk 2 Try to find a rough estimate for the amount r n ∈ [0 , 1] of gross errors such that r n ∈ [ r n , r n ]. 3 Choose and evaluate appropriate initial estimate; e.g., Kolmogorov or Cram´ er von Mises MD-estimator 4 Estimate the parameter(s) of interest by means of the corresponding radius-minimax estimator (cf. Rieder et al. (2008) [8]) using a k -step ( k ≥ 1) construction. Dr. Matthias Kohl R-Packages for Robust Asymptotic Statistics

  14. Robust Asymptotic Statistics Exponential Families Regression-Type Models Outline Robust Asymptotic Statistics 1 Exponential Families 2 Regression-Type Models 3 Dr. Matthias Kohl R-Packages for Robust Asymptotic Statistics

  15. Robust Asymptotic Statistics Exponential Families Regression-Type Models Some examples Normal (Gaussian): location and scale Binomial: probability of success Poisson: positive mean Gamma: shape and scale Gumbel: location and scale all smoothly parameterized exponential families of full rank Approach also works for other smoothly parametrized families! Dr. Matthias Kohl R-Packages for Robust Asymptotic Statistics

  16. Robust Asymptotic Statistics Exponential Families Regression-Type Models Some examples Normal (Gaussian): location and scale Binomial: probability of success Poisson: positive mean Gamma: shape and scale Gumbel: location and scale all smoothly parameterized exponential families of full rank Approach also works for other smoothly parametrized families! Dr. Matthias Kohl R-Packages for Robust Asymptotic Statistics

  17. Robust Asymptotic Statistics Exponential Families Regression-Type Models Some examples Normal (Gaussian): location and scale Binomial: probability of success Poisson: positive mean Gamma: shape and scale Gumbel: location and scale all smoothly parameterized exponential families of full rank Approach also works for other smoothly parametrized families! Dr. Matthias Kohl R-Packages for Robust Asymptotic Statistics

  18. Robust Asymptotic Statistics Exponential Families Regression-Type Models Basic R-Packages distr: S4-classes for distributions. distrEx: Functionals on distributions. RandVar: S4-classes and methods for random variables. distrMod: S4-classes for parametric families of probability measures, minimum distance (MD) estimators. RobAStBase: S4-classes for ICs and infinitesimal neighborhoods. cf. Ruckdeschel et al. (2006) [9], Kohl (2005) [2], http://r-forge.r-project.org/projects/distr/ , http://r-forge.r-project.org/projects/robast/ Dr. Matthias Kohl R-Packages for Robust Asymptotic Statistics

  19. Robust Asymptotic Statistics Exponential Families Regression-Type Models Basic R-Packages distr: S4-classes for distributions. distrEx: Functionals on distributions. RandVar: S4-classes and methods for random variables. distrMod: S4-classes for parametric families of probability measures, minimum distance (MD) estimators. RobAStBase: S4-classes for ICs and infinitesimal neighborhoods. cf. Ruckdeschel et al. (2006) [9], Kohl (2005) [2], http://r-forge.r-project.org/projects/distr/ , http://r-forge.r-project.org/projects/robast/ Dr. Matthias Kohl R-Packages for Robust Asymptotic Statistics

Download Presentation
Download Policy: The content available on the website is offered to you 'AS IS' for your personal information and use only. It cannot be commercialized, licensed, or distributed on other websites without prior consent from the author. To download a presentation, simply click this link. If you encounter any difficulties during the download process, it's possible that the publisher has removed the file from their server.

Recommend

"RobExtremes" Robust Extreme Value Statistics Outline a New Member in the

"RobExtremes" Robust Extreme Value Statistics Outline a New Member in the

"RobExtremes" Robust Extreme Value Statistics Outline a New Member in the RobASt-Family of R Packages Introduction Project Robust Risk Estimation UseR! 2013 Infrastructure of R packages Packages for distributions: distr

377 views • 8 slides
Asymptotic Behaviour of the Quadratic Knapsack Problem Joachim Schauer Department of Statistics

Asymptotic Behaviour of the Quadratic Knapsack Problem Joachim Schauer Department of Statistics

Asymptotic Behaviour of the Quadratic Knapsack Problem Asymptotic Behaviour of the Quadratic Knapsack Problem Joachim Schauer Department of Statistics and Operations Research University of Graz, Austria AGTAC - Koper 17.06.2015 Asymptotic

445 views • 33 slides
How Robust are Thresholds for Community Detection? Ankur Moitra (MIT) Robust Statistics Summer

How Robust are Thresholds for Community Detection? Ankur Moitra (MIT) Robust Statistics Summer

How Robust are Thresholds for Community Detection? Ankur Moitra (MIT) Robust Statistics Summer School Let me tell you a story about the success of belief propagation and statistical physics THE STOCHASTIC BLOCK MODEL Introduced by Holland,

1.65k views • 119 slides
Local Asymptotic Normality in Quantum Statistics M d lin Gu School of

Local Asymptotic Normality in Quantum Statistics M d lin Gu School of

Local Asymptotic Normality in Quantum Statistics M d lin Gu School of Mathematical Sciences University of Nottingham Richard Gill (Leiden) Jonas Kahn (Paris XI) Bas Janssens (Utrecht) Anna Jencova (Bratislava) Luc Bouten

560 views • 42 slides
Statistics Asymptotic Theory Shiu-Sheng Chen Department of Economics National Taiwan University

Statistics Asymptotic Theory Shiu-Sheng Chen Department of Economics National Taiwan University

Statistics Asymptotic Theory Shiu-Sheng Chen Department of Economics National Taiwan University Fall 2019 Shiu-Sheng Chen (NTU Econ) Statistics Fall 2019 1 / 28 Asymptotic Theory: Motivation Asymptotic theory (or large sample theory) aims

1.07k views • 28 slides
Asymptotic Theory for Statistics of Geometric Structures Joe Yukich Universidad Carlos III,

Asymptotic Theory for Statistics of Geometric Structures Joe Yukich Universidad Carlos III,

Asymptotic Theory for Statistics of Geometric Structures Joe Yukich Universidad Carlos III, October 11, 2019 Joe Yukich Asymptotic Theory for Statistics of Geometric Structures October 11, 2019 1 / 26 Introduction X R d random finite

823 views • 56 slides
Discussion: Robust Sparse Quadratic Discrimination Han Xiao Department of Statistics &

Discussion: Robust Sparse Quadratic Discrimination Han Xiao Department of Statistics &

Discussion: Robust Sparse Quadratic Discrimination Han Xiao Department of Statistics & Biostatistics Rutgers University 2014 Rutgers Statistics Symposium Statistics and the Century of Data May 02 2014 High Dimensional Classification

558 views • 13 slides
Robust Statistics and Generative Adversarial Networks Yuan YAO HKUST 1 Chao GAO Jiyi LIU

Robust Statistics and Generative Adversarial Networks Yuan YAO HKUST 1 Chao GAO Jiyi LIU

Robust Statistics and Generative Adversarial Networks Yuan YAO HKUST 1 Chao GAO Jiyi LIU Weizhi ZHU U. Chicago Yale U. HKUST 2 Deep Learning is Notoriously Not Robust! Imperceivable adversarial examples are ubiquitous to fail

833 views • 72 slides
Asymptotic normality of numbers of observations near order statistics from stationary processes

Asymptotic normality of numbers of observations near order statistics from stationary processes

Asymptotic normality of numbers of observations near order statistics from stationary processes Krzysztof Jasiski Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Toru December 1, 2016 Krzysztof Jasiski

537 views • 25 slides
Part 3 Robust Bayesian statistics & applications in reliability networks by Gero Walter 69

Part 3 Robust Bayesian statistics & applications in reliability networks by Gero Walter 69

Tuesday 9:00-12:30 Part 3 Robust Bayesian statistics & applications in reliability networks by Gero Walter 69 Robust Bayesian statistics & applications in reliability networks Outline Robust Bayesian Analysis (9am) Why The

1.31k views • 77 slides
Robust Statistics and Generative Adversarial Networks Yuan YAO HKUST Chao Gao (Chicago) Jiyu

Robust Statistics and Generative Adversarial Networks Yuan YAO HKUST Chao Gao (Chicago) Jiyu

Robust Statistics and Generative Adversarial Networks Yuan YAO HKUST Chao Gao (Chicago) Jiyu Liu (Yale) Weizhi Zhu (HKUST) Deep Learning is Notoriously Not Robust! Imperceivable adversarial examples are ubiquitous to fail neural networks

1.8k views • 133 slides
Robust Scene Categorization by Learning Image Statistics in Context for BBC rushes Jan van

Robust Scene Categorization by Learning Image Statistics in Context for BBC rushes Jan van

Robust Scene Categorization by Learning Image Statistics in Context for BBC rushes Jan van Gemert,Jan-Mark Geusebroek, Cees Snoek, Dennis Koelma, Cor Veenman, Frank Seinstra, Marcel Worring and Arnold Smeulders Intelligent Sensory Information

315 views • 12 slides
Stat 5101 Lecture Slides Deck 6 Charles J. Geyer School of Statistics University of Minnesota

Stat 5101 Lecture Slides Deck 6 Charles J. Geyer School of Statistics University of Minnesota

Stat 5101 Lecture Slides Deck 6 Charles J. Geyer School of Statistics University of Minnesota 1 Asymptotic Approximation The last big subject in probability theory is asymptotic approxi- mation, also called asymptotics, also called large

1.09k views • 108 slides
Stat 5101 Lecture Slides Deck 7 Charles J. Geyer School of Statistics University of Minnesota

Stat 5101 Lecture Slides Deck 7 Charles J. Geyer School of Statistics University of Minnesota

Stat 5101 Lecture Slides Deck 7 Charles J. Geyer School of Statistics University of Minnesota 1 Asymptotic Approximation The last big subject in probability theory is asymptotic approxi- mation, also called asymptotics, also called large

1.61k views • 108 slides
Connections with other combinatorial families Asymptotic analysis of statistics on maps Lessons

Connections with other combinatorial families Asymptotic analysis of statistics on maps Lessons

Connections with other combinatorial families Asymptotic analysis of statistics on maps Lessons to learn

1.22k views • 109 slides
Connections with other combinatorial families Asymptotic analysis of statistics on maps Lessons

Connections with other combinatorial families Asymptotic analysis of statistics on maps Lessons

Connections with other combinatorial families Asymptotic analysis of statistics on maps Lessons to learn

1.27k views • 108 slides
Dynamic Robust Utility P Beissner* F Maccheroni # M Marinacci # S Mukerji @ * ANU # U Bocconi and @

Dynamic Robust Utility P Beissner* F Maccheroni # M Marinacci # S Mukerji @ * ANU # U Bocconi and @

Dynamic Robust Utility Dynamic Robust Utility P Beissner* F Maccheroni # M Marinacci # S Mukerji @ * ANU # U Bocconi and @ QMU Robustness in Economics and Econometrics Conference BFI - University of Chicago April 2019 Dynamic Robust Utility

546 views • 36 slides
Web Crawling Introduction to Information Retrieval INF 141 Donald J. Patterson Content adapted

Web Crawling Introduction to Information Retrieval INF 141 Donald J. Patterson Content adapted

Web Crawling Introduction to Information Retrieval INF 141 Donald J. Patterson Content adapted from Hinrich Schtze http://www.informationretrieval.org Web Crawlers Robust Crawling A Robust Crawl Architecture Doc. Finger- Robots.txt URL

393 views • 25 slides
Rapid, Robust, and Reliable Blind Deconvolution via Nonconvex Optimization Shuyang Ling

Rapid, Robust, and Reliable Blind Deconvolution via Nonconvex Optimization Shuyang Ling

Rapid, Robust, and Reliable Blind Deconvolution via Nonconvex Optimization Shuyang Ling Department of Mathematics, UC Davis Oct.18th, 2016 Shuyang Ling (UC Davis) 16w5136, Oaxaca, Mexico Oct.18th, 2016 1 / 31 Acknowledgements Research in

399 views • 35 slides
ROBUST LOCATION AND SCALE ESTIMATION WITH CENSORED OUTCOMES Jerome H. Friedman Stanford

ROBUST LOCATION AND SCALE ESTIMATION WITH CENSORED OUTCOMES Jerome H. Friedman Stanford

ROBUST LOCATION AND SCALE ESTIMATION WITH CENSORED OUTCOMES Jerome H. Friedman Stanford University MACHINE LEARNING y = F ( x , z ) y = outcome variable x = ( x 1 , x p ) observed predictor variables z = ( z 1 , z 2 , ) other

739 views • 53 slides
Trade-off between Efficiency and Robustness Doctoral Colloqium @ SenSys18, Shenzhen Robert

Trade-off between Efficiency and Robustness Doctoral Colloqium @ SenSys18, Shenzhen Robert

Institute of Operating Systems and Computer Networks E ffi ciency E ffi ciency vs Robustness = Trade-o ff Robustness Energy Budget Trade-off between Efficiency and Robustness Doctoral Colloqium @ SenSys18, Shenzhen Robert Hartung,

680 views • 19 slides
Robust Flows over Time: Models and Complexity Results Corinna Gottschalk 1 , Arie Koster 1 ,

Robust Flows over Time: Models and Complexity Results Corinna Gottschalk 1 , Arie Koster 1 ,

Robust Flows over Time: Models and Complexity Results Corinna Gottschalk 1 , Arie Koster 1 , Frauke Liers 2 , Britta Peis 1 , Daniel Schmand 1 , Andreas Wierz 1 1 RWTH Aachen University, 2 Universitt Erlangen-Nrnberg January 2016, Aussois

581 views • 37 slides
Key-Robustness for Cryptographic Primitives ie 1 R azvan Ros 1 ENS, CNRS, INRIA & PSL

Key-Robustness for Cryptographic Primitives ie 1 R azvan Ros 1 ENS, CNRS, INRIA & PSL

Key-Robustness for Cryptographic Primitives ie 1 R azvan Ros 1 ENS, CNRS, INRIA & PSL Research University, Paris, France ECRYPT-NET Summer School, Crete, Greece 12 th October 2017 R azvan Ros ie Key-Robustness for Cryptographic

949 views • 48 slides
What Is Robust Code? Robust code A style of programming that

What Is Robust Code? Robust code A style of programming that

What Is Robust Code? Robust code A style of programming that prevents abnormal termina8on or unexpected ac8ons Handles bad input gracefully Detects

433 views • 12 slides

More recommend