Sufficient Dimension Reduction for Normal Models

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  • Topic: Variance, Maximum likelihood, Singular value decomposition
  • Pages : 172 (44877 words )
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  • Published : February 9, 2013
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SUFFICIENT DIMENSION REDUCTION BASED ON NORMAL AND WISHART INVERSE MODELS

A THESIS SUBMITTED TO THE FACULTY OF THE GRADUATE SCHOOL OF THE UNIVERSITY OF MINNESOTA BY

LILIANA FORZANI

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

R. DENNIS COOK, Advisor

December, 2007

c Liliana Forzani 2007

UNIVERSITY OF MINNESOTA

This is to certify that I have examined this copy of a doctoral thesis by

Liliana Forzani

and have found that it is complete and satisfactory in all respects, and that any and all revisions required by the final examining committee have been made.

R. Dennis Cook Name of Faculty Adviser

Signature of Faculty Adviser

Date

GRADUATE SCHOOL

0.1

Acknowledgments

I would like to thanks my thesis advisor, Dennis Cook. He always welcomed my questions and doubts (and they were not so few). His insights, contagious enthusiasm and intensity were a big part of what made possible for me to complete this work. I hope to carry some of these attributes with me in my professional life, as I know I will carry all the knowledge he taught me. ¡ Gracias ! Thanks (orderless) to Carlos, who introduced me to the data world and shared with me so many principal component analysis conversations... to Professor Eaton for teaching me, among other things, to resist the temptation to take the derivative every time there is a function to maximize... to John, Pedro, Roberto and Yongwu, who were always willing to have discussions (even about statistics)... to Marcela who generously gave me the bird-plane-car data... to Li Bing for giving me the code for directional reduction A to Aaron, who was always there for a graphical, L TEX or statistics question...

to Marco who in order to share the joy I was having doing this thesis made the beautiful mathematical drawing that is in this thesis... to Davide, for the coffee, the food and being the alma of the fourth floor... to my dear friends from IMAL (Instituto de Matem´tica Aplicada ”Litoral”), a who are still waiting for me... to my little friends, Bruno, Gaby, Leo, Manon, Palito, who together with Marco are the smiles of my life outside the academics...(and their parents too) to Edu, for making this journey with me... to Nancy, she knows why...

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To Bera, Edu, Javier and Marco ....

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Contents
0.1 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i ii 1 5 5 11 14 18 30 31 33 37 38 39 40 41 46

Dedication 1 Introduction 2 Models and their attached reductions 2.1 2.2 2.3 2.4 Normal reductive models with constant variance . . . . . . . . . . . Normal reductive models with non-constant variance . . . . . . . . . Wishart reductive models . . . . . . . . . . . . . . . . . . . . . . . . A look of functional data analysis and dimension reduction . . . . .

3 Normal inverse models with constant variance: the general case 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 Estimation under model PFC∆ . . . . . . . . . . . . . . . . . . . .

Relationships with other methods . . . . . . . . . . . . . . . . . . . . Structured ∆ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Inference about d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Testing predictors . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Robustness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4 Normal inverse models with constant variance: mean and variance connection 4.1 4.2 4.3 4.4 4.5 Variations within model (2.4) . . . . . . . . . . . . . . . . . . . . . . Relationships with other methods . . . . . . . . . . . . . . . . . . . . Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Simulations and data analysis illustrations . . . . . . . . . . . . . . . Discussion . . . . . . . . . . . . . . . . . ....
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