Type
Text
Type
Dissertation
Advisor
Rachev, Svetlozar T. | Xing, Haipeng | Glimm, James | Smith, Noah.
Date
2013-12-01
Keywords
Applied mathematics | Bayesian estimation, Covariance estimation, Operational risk, Parameter estimation, Portfolio optimization
Department
Department of Applied Mathematics and Statistics.
Language
en_US
Source
This work is sponsored by the Stony Brook University Graduate School in compliance with the requirements for completion of degree.
Identifier
http://hdl.handle.net/11401/76604
Publisher
The Graduate School, Stony Brook University: Stony Brook, NY.
Format
application/pdf
Abstract
Parameter estimation is an important step in probabilistic and statistical modeling. Maximum likelihood estimators (MLE) are the classical parameter estimators. However, in the real world, for small samples or ill-posed problems maximum likelihood estimates may be difficult to find through direct numerical optimization, and are unstable and sensitive to outliers. In this dissertation, we study two popular problems in financial risk management and portfolio selection. The first problem is operational risk modeling. Data insufficiency and reporting threshold are two main difficulties in estimating the loss severity distributions in operational risk modeling. We investigate four methods including MLE, expectation-maximization (EM), penalized likelihood estimator (PLE), and Bayesian method. Without expert information, Jeffreys' priors for truncated distributions are used for the Bayesian method. Using the popular lognormal distribution as an example, we provide an extensive simulation study to demonstrate the superiority of Bayesian method in reducing parameter estimation errors for truncated distributions with small sample size. In addition, we apply the methods to actual operational loss data from a European bank using the log-normal and log-gamma distributions. The stability and credibility of the parameters are improved compared to MLE, and the granularity of units of measures for operational risk modeling are improved. The second problem is a classical high dimensional problem---covariance estimation. Sample covariance is known to be a poor input when the sample size is relative small compared to the dimension. There is a vast literature that suggests factor models, shrinkage methods, random matrix theory (RMT) approaches, Bayesian methods, and regularization methods for dealing this problem. We consider an interesting case where there is a natural ordering among the random variables. We study a smooth monotone regularization approach, which is often useful for fixed-income instruments and options when the instruments have a natural ordering (e.g. | by maturities, strikes, or ratings). We analyze the performance of smooth monotone covariance in reducing various statistical distances and improving optimal portfolio selection. We also extend its use in non-Gaussian cases by incorporating various robust covariance estimates for elliptical distributions. Finally, we provide two empirical examples where the smooth monotone covariance improves the out-of-sample covariance prediction and portfolio optimization. | 149 pages
Recommended Citation
Zhou, Xiaoping, "ERROR REDUCTION IN PARAMETER ESTIMATION WITH CASE STUDIES IN RISK MEASUREMENT AND PORTFOLIO OPTIMIZATION" (2013). Stony Brook Theses and Dissertations Collection, 2006-2020 (closed to submissions). 2495.
https://commons.library.stonybrook.edu/stony-brook-theses-and-dissertations-collection/2495