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ERROR REDUCTION IN PARAMETER ESTIMATION WITH CASE STUDIES IN RISK MEASUREMENT AND PORTFOLIO OPTIMIZATION

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dc.contributor.advisor Rachev, Svetlozar T. en_US
dc.contributor.advisor Glimm, James en_US
dc.contributor.author Zhou, Xiaoping en_US
dc.contributor.other Department of Applied Mathematics and Statistics. en_US
dc.date.accessioned 2017-09-20T16:50:47Z
dc.date.available 2017-09-20T16:50:47Z
dc.date.issued 2013-12-01 en_US
dc.identifier.uri http://hdl.handle.net/11401/76604 en_US
dc.description 149 pg. en_US
dc.description.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. en_US
dc.description.sponsorship This work is sponsored by the Stony Brook University Graduate School in compliance with the requirements for completion of degree. en_US
dc.format Monograph en_US
dc.format.medium Electronic Resource en_US
dc.language.iso en_US en_US
dc.publisher The Graduate School, Stony Brook University: Stony Brook, NY. en_US
dc.subject.lcsh Applied mathematics en_US
dc.subject.other Bayesian estimation, Covariance estimation, Operational risk, Parameter estimation, Portfolio optimization en_US
dc.title ERROR REDUCTION IN PARAMETER ESTIMATION WITH CASE STUDIES IN RISK MEASUREMENT AND PORTFOLIO OPTIMIZATION en_US
dc.type Dissertation en_US
dc.mimetype Application/PDF en_US
dc.contributor.committeemember Xing, Haipeng en_US
dc.contributor.committeemember Smith, Noah. en_US


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