Graduation Year

2013

Document Type

Dissertation

Degree

Ph.D.

Degree Granting Department

Mathematics and Statistics

Major Professor

Christos P. Tsokos

Keywords

Algorithms, Inference, Modeling, Optimization, Reliability, Survival

Abstract

Many of the open problems of current interest in probability and statistics involve complicated data

sets that do not satisfy the strong assumptions of being independent and identically distributed. Often,

the samples are known only empirically, and making assumptions about underlying parametric

distributions is not warranted by the insufficient information available. Under such circumstances,

the usual Fisher or parametric Bayes approaches cannot be used to model the data or make predictions.

However, this situation is quite often encountered in some of the main challenges facing statistical,

data-driven studies of climate change, clinical studies, or financial markets, to name a few.

We propose a novel approach, based on large deviations theory, convex optimization, and recent

results on surrogate loss functions for classifier-type problems, that can be used in order to estimate

the probability of large deviations for complicated data. This may include, for instance, highdimensional

data, highly-correlated data, or very sparse data.

The thesis introduces the new approach, reviews the current known theoretical results, and then

presents a number of numerical explorations meant to quantify how far the approximation of survival

functions via large deviations principle can be taken, once we leave the limitations imposed

by the existing theoretical results.

The explorations are encouraging, indicating that indeed the new approximation scheme may

be very efficient and can be used under much more general conditions than those warranted by the

current theoretical thresholds.

After applying the new methodology to two important contemporary problems (atmospheric

CO2 data and El Ni~no/La Ni~na phenomena), we conclude with a summary outline of possible further

research.

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