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Skewness and elongation are two factors that directly determine the shape of a probability distribution. Thus, to obtain a flexible distribution it is always desirable that the parameters of the distribution directly determine the skewness and elongation. To meet this purpose, Tukey (1977) introduced a family of distributions called g-and-h family (gh family) based on a transformation of the standard normal variable where g and h determine the skewness and the elongation, respectively. The gh family of distributions was extensively studied by Hoaglin (1985) and Martinez and Iglewicz (1984). For its flexibility in shape He and Raghunathan (2006) have used this distribution for multiple imputations. Because of the complex nature of this family of distributions, it is not possible to have an explicit mathematical form of the density function and the estimates of the parameters g and h fully depend on extensive numerical computations.In this study, we have developed algorithms to numerically compute the density functions. We present algorithms to obtain the estimates of g and h using method of moments, quantile method and maximum likelihood method. We analyze the performance of each method and compare them using simulation technique. Finally, we study some special cases of gh family and their properties. / Department of Mathematical Sciences

In modern day engineering, the designer has become more and more dependent on computer simulation. Oftentimes, computational cost and convergence accuracy accompany these simulations to reach global solutions for engineering design problems causes traditional optimization techniques to perform poorly. To overcome these issues nontraditional optimization algorithms based region elimination and space exploration are introduced. Approximation models, which are also known as metamodels or surrogate models, are used to explore and give more information about the design space that needs to be explored. Usually the approximation models are constructed in the promising regions where global solutions are expected to exist. The approximation models imitate the original expensive function, black-box function, and contribute towards getting comparably acceptable solutions with fewer resources and at low computation cost. The primary contributions of this dissertation are associated with the development of new methods for exploring the design space for large scale computer simulations. Primarily, the proposed design space exploration procedure uses a hierarchical partitioning method to help mitigate the curse of dimensionality often associated with the analysis of large scale systems. The research presented in this dissertation focuses on introducing new optimization algorithms based on metamodeling techniques that alleviate the burden of the computation cost associated with complex engineering design problems. Three new global optimization algorithms were introduced in this dissertation, Approximated Unimodal Region Elimination (AUMRE), Space Exploration and Unimodal Region Elimination (SEUMRE), and Mixed Surrogate Space Exploration (MSSE) for computation intensive and black-box engineering design optimization problems. In these algorithms, the design space was divided into many subspaces and the search was focused on the most promising regions to reach global solutions with the resources available and with less computation cost. Metamodeling techniques such as Response Surface Method (RSM), Radial Basis Function (RBF), and Kriging (KRG) are introduced and used in this work. RSM has been used because of its advantages such as being easy to construct, understand and implement. Also due to its smoothing capability, it allows quick convergence of noisy functions in the optimization. RBF has the advantage of smoothing data and interpolating them. KRG metamodels can provide accurate predictions of highly nonlinear or irregular behaviours. These features in metamodeling techniques have contributed largely towards obtaining comparably accurate global solutions besides reducing the computation cost and resources. Many multi-objective optimization algorithms, specifically those used for engineering problems and applications involve expensive fitness evaluations. In this dissertation, a new multi-objective global optimization algorithm for black-box functions is also introduced and tested on benchmark test problems and real life engineering applications. Finally, the new proposed global optimization algorithms were tested using benchmark global optimization test problems to reveal their pros and cons. A comparison with other well known and recently introduced global optimization algorithms were carried out to highlight the proposed methods’ advantages and strength points. In addition, a number of practical examples of global optimization in industrial designs were used and optimized to further test these new algorithms. These practical examples include the design optimization of automotive Magnetorheological Brake Design and the design optimization of two-mode hybrid powertrains for new hybrid vehicles. It is shown that the proposed optimization algorithms based on metamodeling techniques comparably provide global solutions with the added benefits of fewer function calls and the ability to efficiently visualize the design space. / Graduate

A one-dimensional transient mathematical model which can predict simultaneous movement of water and reactive solutes through homogeneous soil systems under saturated/unsaturated conditions is developed. The physically-based numerical model, called PESTFADE, considers the interactive processes/mechanisms such as mass flow, plant uptake, adsorption/desorption, dispersion, volatilization, chemical/microbial degradation and runoff in the simulation. / The PESTFADE model employs SWACROP, a model developed in the Netherlands, to simulate transient water flow in the unsaturated zone; evaluates non-equilibrium sorption in macropores, analyzes soil heat flow to to model microbial degradation, calculates pesticide partitioning in runoff/sediment as affected by agricultural management practices, and describes first order degradation and sorption kinetics. The governing partial differential equation describing the various processes is solved numerically via the Numerical Method of Lines (NMOL) technique, and the computer programs are written in FORTRAN 77. The resulting computer code (PESTFADE), is run on a microcomputer and has been implemented for interactive simulation on IBM PC or compatible microcomputers. / The model was tested and validated using actual data measured from field plot experiments involving herbicide atrazine which was post-emergently applied in a corn field on a loam soil. Various analytical solutions were used to check the accuracy of the different components of PESTFADE, and parametric sensitivity analyses were performed to determine how the model output reacts to changes in some selected input parameters. / Results indicate that model predictions generally agreed with measured concentrations of atrazine and compared closely with the analytical solutions. Moreover, model performance tests showed that predicted values are within acceptable ranges of model accuracy and bound of experimental uncertainties. It was also found that the model is very sensitive to degradation rate constant (k), sorption coefficient (K$ sb{ rm d})$ and soil temperature and slightly sensitive to management practice (CN) and sorption site fraction (F). Finally, the various field scenarios and pathways for non-point source contamination evaluated in the study have demonstrated the wide applicability and flexibility of PESTFADE.

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