This thesis describes the Delta-Normal method of computing Value-at-Risk. The advantages and disadvantages of the Delta-Normal method compared to the Historical and Monte Carlo method of computing Value-at-Risk are discussed. The Delta-Normal method of computing Value-at-Risk is compared with the Historical Simulation method of Value-at-Risk using an implementation of portfolio consisting of ten stocks for 400 time intervals. Based on the normality of the distribution of the portfolio risk factors, Delta-Normal would be suitable if the distribution is normal and Historical Simulation method of calculating Value-at-Risk would be ideally suited if the distribution is non-normal.

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1979. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND SCIENCE. / Vita. / Bibliography: leaves 364-373. / by Robert Buchanan Washburn, Jr. / Ph.D.

The first part of this thesis considers the residual empirical process of a nearly unstable long-memory time series. Chan and Ling [8] showed that the usual limit distribution of the Kolmogorov-Smirnov test statistics does not hold when the characteristic polynomial of the unstable autoregressive model has a unit root. A key question of interest is what happens when this model has a near unit root, that is, when it is nearly non-stationary. In this thesis, it is established that the statistics proposed by Chan and Ling can be extended. The limit distribution is expressed as a functional of an Orenstein-Uhlenbeck process that is driven by a fractional Brownian motion. This result extends and generalizes Chan and Ling's results to a nearly non-stationary long-memory time series. / The second part of the thesis investigates the weak convergence of weighted sums of random variables that are functionals of moving aver- age processes. A non-central limit theorem is established in which the Wiener integrals with respect to the Hermite processes appear as the limit. As an application of the non-central limit theorem, we examine the asymptotic theory of least squares estimators (LSE) for a nearly unstable AR(1) model when the innovation sequences are functionals of moving average processes. It is shown that the limit distribution of the LSE appears as functionals of the Ornstein-Uhlenbeck processes driven by Hermite processes. / Liu, Weiwei. / Adviser: Chan Ngai Hang. / Source: Dissertation Abstracts International, Volume: 73-01, Section: B, page: . / Thesis (Ph.D.)--Chinese University of Hong Kong, 2009. / Includes bibliographical references (leaves 60-67). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [201-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstract also in Chinese.

This dissertation is motivated by the problem of uncertainty and sensitivity in business- class models such as the carbon emission abatement policy model featured in this work. Uncertain model inputs are represented by numerical random variables and a computational methodology is developed to numerically compute business-class models as if sharp inputs were given. A new description for correlation of random variables is presented that arises spontaneously within a numerical model. Methods of numerically computing correlated random variables are implemented in software and represented. The major contribution of this work is a methodology for the numerical computation of models under uncertainty that expresses no preference for unlikelihood of model input combinations. The methodology presented here serves a sharp contrast to traditional Monte Carlo methods that implicitly equate likelihood of model input values with importance of results. The new methodology herein shifts the computational burden from likelihood of inputs to resolution of input space.

by A. S. Willsky, N. R. Sandell. / Grants AFOSR-77-3281B and ONR-N00014-76-C-0346. / Bibliography: leaf 34.

Thesis (M. Phil.)--University of Hong Kong, 2009. / Includes bibliographical references (leaves 82-87). Also available in print.

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The research of this thesis deals with the derivation of optimum designs for linear mixed models. The problem of constructing optimal designs for linear mixed models is very broad. Thus the thesis is mainly focused on the design theory for random coefficient regression models which are a special case of the linear mixed model. Specifically, the major objective of the thesis is to construct optimal designs for the simple linear and the quadratic regression models with a random intercept algebraically. A second objective is to investigate the nature of optimal designs for the simple linear random coefficient regression model numerically. In all models time is considered as an explanatory variable and its values are assumed to belong the set {a, 1, ... , k}. Two sets of individual designs, designs with non-repeated time points comprising up to k + 1 distinct time points and designs with repeated time points comprising up to k + 1 time points not necessarily distinct, are used in the thesis. In the first case there are 2k+ - 1 individual designs while in the second case there are ( 2 2k k+ 1 ) - 1 such designs. The problems of constructing population designs, which allocate weights to the individual designs in such a way that the information associated with the model parameters is in some sense maximized and the variances associated with the mean responses at a given vector of time points are in some sense minimized, are addressed. In particular D- and V-optimal designs are discussed. A geometric approach is introduced to confirm the global optimality of D- and V-optimal designs for the simple linear regression model with a random intercept. It is shown that for the simple linear regression model with a random intercept these optimal designs are robust to the choice of the variance ratio. A comparison of these optimal designs over the sets of individual designs with repeated and non-repeated points for that model is also made and indicates that the D- and V-optimal iii population designs based on the individual designs with repeated points are more efficient than the corresponding optimal population designs with non-repeated points. Except for the one-point case, D- and V-optimal population designs change with the values of the variance ratio for the quadratic regression model with a random intercept. Further numerical results show that the D-optimal designs for the random coefficient models are dependent on the choice of variance components. / Thesis (Ph.D.) - University of KwaZulu-Natal, Pietermaritzburg, 2004.