RATE OF CONVERGENCE OF MAXIMUM LIKELIHOOD ESTIMATORS UNDER RELAXED SMOOTHNESS CONDITIONS ON THE LIKELIHOOD FUNCTION

This report reviews literature on the rate of convergence of maximum likelihood estimators and establishes a Central Limit Theorem, which yields an O(1/sqrt(n)) rate of convergence of the maximum likelihood estimator under somewhat relaxed smoothness conditions. These conditions include the existence of a one-sided derivative in θ of the pdf, compared to up to three that are classically required. A verification through simulation is included in the end of the report.

Probabilistic-based hurricane risk assessment and mitigation considering the potential impacts of climate change

Studies are suggesting that hurricane hazard patterns (e.g. intensity and frequency) may change as a consequence of the changing global climate. As hurricane patterns change, it can be expected that hurricane damage risks and costs may change as a result. This indicates the necessity to develop hurricane risk assessment models that are capable of accounting for changing hurricane hazard patterns, and develop hurricane mitigation and climatic adaptation strategies. This thesis proposes a comprehensive hurricane risk assessment and mitigation strategies that account for a changing global climate and that has the ability of being adapted to various types of infrastructure including residential buildings and power distribution poles. The framework includes hurricane wind field models, hurricane surge height models and hurricane vulnerability models to estimate damage risks due to hurricane wind speed, hurricane frequency, and hurricane-induced storm surge and accounts for the timedependant properties of these parameters as a result of climate change. The research then implements median insured house values, discount rates, housing inventory, etc. to estimate hurricane damage costs to residential construction. The framework was also adapted to timber distribution poles to assess the impacts climate change may have on timber distribution pole failure. This research finds that climate change may have a significant impact on the hurricane damage risks and damage costs of residential construction and timber distribution poles. In an effort to reduce damage costs, this research develops mitigation/adaptation strategies for residential construction and timber distribution poles. The costeffectiveness of these adaptation/mitigation strategies are evaluated through the use of a Life-Cycle Cost (LCC) analysis. In addition, a scenario-based analysis of mitigation strategies for timber distribution poles is included. For both residential construction and timber distribution poles, adaptation/mitigation measures were found to reduce damage costs. Finally, the research develops the Coastal Community Social Vulnerability Index (CCSVI) to include the social vulnerability of a region to hurricane hazards within this hurricane risk assessment. This index quantifies the social vulnerability of a region, by combining various social characteristics of a region with time-dependant parameters of hurricanes (i.e. hurricane wind and hurricane-induced storm surge). Climate change was found to have an impact on the CCSVI (i.e. climate change may have an impact on the social vulnerability of hurricane-prone regions).

Problems in the classical calculus of variations

This dissertation concerns convergence analysis for nonparametric problems in the calculus of variations and sufficient conditions for weak local minimizer of a functional for both nonparametric and parametric problems. Newton's method in infinite-dimensional space is proved to be well-defined and converges quadratically to a weak local minimizer of a functional subject to certain boundary conditions. Sufficient conditions for global converges are proposed and a well-defined algorithm based on those conditions is presented and proved to converge. Finite element discretization is employed to achieve an implementable line-search-based quasi-Newton algorithm and a proof of convergence of the discretization of the algorithm is included. This work also proposes sufficient conditions for weak local minimizer without using the language of conjugate points. The form of new conditions is consistent with the ones in finite-dimensional case. It is believed that the new form of sufficient conditions will lead to simpler approaches to verify an extremal as local minimizer for well-known problems in calculus of variations.

Spatial and Temporal Evolution of the Volcanics and Sediments of the Kenya Rift

New volumetric and mass flux estimates have been calculated for the Kenya Rift. Spatial and temporal histories for volcanic eruptions, lacustrine deposition, and hominin fossil sites are presented, aided by the compilation of a new digital geologic map. Distribution of volcanism over time indicates several periods of southward expansion followed by relative positional stasis. Volcanism occurs throughout the activated rift length, with no obvious abandonment as the rift system migrated. The main exception is a period of volcanic concentration around 10 Ma, when activity was constrained within 2° of the equator. Volumes derived from seismic data indicate a total volume of c. 310,000 km3 (2.47 x 1010 kg/yr ), which is significantly more than the map-derived volumes found here or published previously. Map-based estimates are likely affected by a bias against recognizing small volume events in the older record. Such events are, however, the main driver of erupted volume over the last 5 Ma. A technique developed here to counter this bias results in convergence of the two volume estimation techniques. Relative erupted composition over time is variable. Overall, the erupted material has a mafic to silicic ratio of 0.9:1. Basalts are distinctly more common in the Turkana region, which previously experienced Mesozoic rifting. Despite the near equal ratio of mafic to silicic products, the Kenya Rift otherwise fits the definition of a SLIP. It is proposed that the compositions would better fit the published definition if the Turkana region was not twice-rifted. Lacustrine sedimentation post-dates initial volcanism by about 5 million years, and follows the same volcanic trends, showing south and eastward migration over time. This sedimentation delay is likely related to timing of fault displacements. Evidence of hominin habitation is distinctly abundant in the northern and southern sections of the Kenya Rift, but there is an observed gap in the equatorial rift between 4 and 0.5 million years ago. After 0.5 Ma, sites appear to progress towards the equator. The pattern and timing of hominid site distributions suggests that the equatorial gap in habitation may be the result of active volcanic avoidance.