The problem of efficient estimation of the parameters of the extreme value distribution has not been addressed in the literature. We obtain efficient estimators of the parameters of type I (maximum) extreme value distribution without solving the likelihood equations. This research provides for the first time simple expressions for the elements of the information matrix for type II censoring. We construct efficient estimators of the parameters using linear combinations of order statistics of a random sample drawn from the population. We derive explicit formulas for the information matrix for this problem for type II censoring and construct efficient estimators of the parameters using linear combinations of available order statistics with additional weights to the smallest and largest order statistics. We consider numerical examples to illustrate the applications of the estimators. We also perform an extensive Monte Carlo simulation study to examine the performance of the estimators for different sample sizes.

In this paper, we study the properties of k-omnisequences of length n, defined to be strings of length n that contain all strings of smaller length k embedded as (not necessarily contiguous) subsequences. We start by proving an elementary result that relates our problem to the classical coupon collector problem. After a short survey of relevant results in coupon collection, we focus our attention on the number M of strings (or words) of length k that are not found as subsequences of an n string, showing that there is a gap between the probability threshold for the emergence of an omnisequence and the zero-infinity threshold for E(M).

In this paper, we study the properties of k-omnisequences of length n, defined to be strings of length n that contain all strings of smaller length k embedded as (not necessarily contiguous) subsequences. We start by proving an elementary result that relates our problem to the classical coupon collector problem. After a short survey of relevant results in coupon collection, we focus our attention on the number M of strings (or words) of length k that are not found as subsequences of an n string, showing that there is a gap between the probability threshold for the emergence of an omnisequence and the zero-infinity threshold for E(M).

Neste trabalho é introduzido um novo modelo estatístico para modelar dados limitados no intervalo continuo (0;1). O modelo proposto é construído sob uma transformação de variáveis, onde a variável transformada é resultado da combinação de uma variável com distribuição normal padrão e a função de distribuição acumulada da distribuição valor extremo generalizado. Para o novo modelo são estudadas suas propriedades estruturais. A nova família é estendida para modelos de regressão, onde o modelo é reparametrizado na mediana da variável resposta e este conjuntamente com o parâmetro de dispersão são relacionados com covariáveis através de uma função de ligação. Procedimentos inferênciais são desenvolvidos desde uma perspectiva clássica e bayesiana. A inferência clássica baseia-se na teoria de máxima verossimilhança e a inferência bayesiana no método de Monte Carlo via cadeias de Markov. Além disso estudos de simulação foram realizados para avaliar o desempenho das estimativas clássicas e bayesianas dos parâmetros do modelo. Finalmente um conjunto de dados de câncer colorretal é considerado para mostrar a aplicabilidade do modelo. / In this research a new statistical model is introduced to model data restricted in the continuous interval (0;1). The proposed model is constructed under a transformation of variables, in which the transformed variable is the result of the combination of a variable with standard normal distribution and the cumulative distribution function of the generalized extreme value distribution. For the new model its structural properties are studied. The new family is extended to regression models, in which the model is reparametrized in the median of the response variable and together with the dispersion parameter are related to covariables through a link function. Inferential procedures are developed from a classical and Bayesian perspective. The classical inference is based on the theory of maximum likelihood, and the Bayesian inference is based on the Markov chain Monte Carlo method. In addition, simulation studies were performed to evaluate the performance of the classical and Bayesian estimates of the model parameters. Finally a set of colorectal cancer data is considered to show the applicability of the model

This study consists of developing descriptive, parametric, linear and non-linear statistical models for such natural phenomena as hurricanes, lightning, flooding, red tide and volcanic fallout. In the present study, the focus of research is determining the stochastic nature of phenomena in the environment. These statistical models are necessary to address the variability of nature and the misgivings of the deterministic models, particularly when considering the necessity for man to estimate the occurrence and prepare for the aftermath.The relationship between statistics and physics looking at the correlation between wind speed and pressure versus wind speed and temperature play a significant role in hurricane prediction. Contrary to previous studies, this study indicates that a drop in pressure is a result of the storm and less a cause. It shows that temperature is a key indicator that a storm will form in conjunction with a drop in pressure. This study demonstrates a model that estimates the wind speed within a storm with a high degree of accuracy. With the verified model, we can perform surface response analysis to estimate the conditions under which the wind speed is maximized or minimized. Additional studies introduce a model that estimates the number of lightning strikes dependent on significantly contributing factors such as precipitable water, the temperatures within a column of air and the temperature range. Using extreme value distribution and historical data we can best fit flood stages, and obtain profiling estimate return periods. The natural logarithmic count of Karenia Brevis was used to homogenize the variance and create the base for an index of the magnitude of an outbreak of Red Tide. We have introduced a logistic growth model that addresses the subject behavior as a function of time and characterizes the growth rate of Red Tide. This information can be used to develop strategic plans with respect to the health of citizens and to minimize the economic impact. Studying the bivariate nature of tephra fallout from volcanoes, we analyze the correlation between the northern and eastern directions of a topological map to find the best possible probabilistic characterization of the subject data.

The dissertation focuses on developing statistical models in environmental and life sciences. The Generalized Extreme Value distribution is used to model annual monthly maximum rainfall data from 44 locations in Florida. Time dependence of the rainfall data is incorporated into the model by assuming the location parameter to be a function of time, both linear and quadratic. Estimates and confidence intervals are obtained for return levels of return periods of 10, 20, 50, and 100 years. Locations are grouped into statistical profiles based on their similarities in return level graphs for all locations, and locations within each climatic zone. A family of extreme values distributions is applied to model simulated maximum drug concentration (Cmax) data of an anticoagulant drug. For small samples (n <̲ 100) data exhibited bimodality. The results of investigating a mixture of two extreme value distributions to model such bimodal data using two-parameter Gumbel, Pareto and Weibu ll concluded that a mixture of two Weibull distributions is the only suitable FTSel.For large samples , Cmax data are modeled using the Generalized Extreme Value, Gumbel, Weibull, and Pareto distributions. These results concluded that the Generalized Extreme Value distribution is the only suitable model. A system of random differential equations is used to investigate the drug concentration behavior in a three-compartment pharmacokinetic model which describes coumermycin's disposition. The rate constants used in the differential equations are assumed to have a trivariate distribution, and hence, simulated from the trivariate truncated normal probability distribution. Numerical solutions are developed under different combinations of the covariance structure and the nonrandom initial conditions. We study the dependence effect that such a pharmacokinetic system has among the three compartments as well as the effect of variance in identifying the concentration behavior in each compartment. We identify the time delays in each compartment. We extend these models to incorporate the identified time delays. We provide the graphical display of the time delay effects on the drug concentration behavior as well as the comparison of the deterministic behavior with and without the time delay, and effect of different sets of time delay on deterministic and stochastic behaviors.

The Weibull distribution, an extreme value distribution, is frequently used to model survival, reliability, wind speed, and other data. One reason for this is its flexibility; it can mimic various distributions like the exponential or normal. The two-parameter Weibull has a shape (γ) and scale (β) parameter. Parameter estimation has been an ongoing search to find efficient, unbiased, and minimal variance estimators. Through data analysis and simulation studies, the following three methods of estimation will be discussed and compared: maximum likelihood estimation (MLE), method of moments estimation (MME), and median rank regression (MRR). The analysis of wind speed data from the TW Daniels Experimental Forest are used for this study to test the performance and flexibility of the Weibull distribution.

In this article, we make inferences for the Weibull parameters and propose two test statistics for the comparison of two Weibull distributions based on interval-censored data. However, the distributions of the two statistics are unknown and not easy to obtain, therefore a simulation study is necessary. An urn model in the simulation of interval-censored data was proposed by Lee (1999) to select random intervals. Then we propose a simulation procedure with urn model to obtain approximately the quantiles of the two statistics. We demonstrate an example in AIDS study to illustrate how the tests can be applied to the infection time distributions of AIDS.

The presence of nonlinear dependence and chaos has strong implications for predictive modeling and the analysis of dominant processes in hydrology and climate. Analysis of extremes may aid in developing predictive models in hydro-climatology by giving enhanced understanding of processes driving the extremes and perhaps delineate possible anthropogenic or natural causes. This dissertation develops and utilizes different set of tools for predictive modeling, specifically nonlinear dependence, extreme, and chaos, and tests the viability of these tools on the real data. Commonly used dependence measures, such as linear correlation, cross-correlogram or Kendall's tau, cannot capture the complete dependence structure in data unless the structure is restricted to linear, periodic or monotonic. Mutual information (MI) has been frequently utilized for capturing the complete dependence structure including nonlinear dependence. Since the geophysical data are generally finite and noisy, this dissertation attempts to address a key gap in the literature, specifically, the evaluation of recently proposed MI-estimation methods to choose the best method for capturing nonlinear dependence, particularly in terms of their robustness for short and noisy data. The performance of kernel density estimators (KDE) and k-nearest neighbors (KNN) are the best for 100 data points at high and low noise-to-signal levels, respectively, whereas KNN is the best for 1000 data points consistently across noise levels. One real application of nonlinear dependence based on MI is to capture extrabasinal connections between El Nino-Southern Oscillation (ENSO) and river flows in the tropics and subtropics, specifically the Nile, Amazon, Congo, Parana, and Ganges rivers which reveals 20-70% higher dependence than those suggested so far by linear correlations. For extremes analysis, this dissertation develops a new measure precipitation extremes volatility index (PEVI), which measures the variability of extremes, is defined as the ratio of return levels. Spatio-temporal variability of PEVI, based on the Poisson-generalized Pareto (Poisson-GP) model, is investigated on weekly maxima observations available at 2.5 degree grids for 1940-2004 in South America. From 1965-2004, the PEVI shows increasing trends in few parts of the Amazon basin and the Brazilian highlands, north-west Venezuela including Caracas, north Argentina, Uruguay, Rio De Janeiro, Sao Paulo, Asuncion, and Cayenne. Catingas, few parts of the Brazilian highlands, Sao Paulo and Cayenne experience increasing number of consecutive 2- and 3-days extremes from 1965-2004. This dissertation also addresses the ability to detect the chaotic signal from a finite time series observation of hydrologic systems. Tests with simulated data demonstrate the presence of thresholds, in terms of noise to chaotic-signal and seasonality to chaotic-signal ratios, beyond which the set of currently available tools is not able to detect the chaotic component. Our results indicate that the decomposition of a simulated time series into the corresponding random, seasonal and chaotic components is possible from finite data. Real streamflow data from the Arkansas and Colorado rivers do not exhibit chaos. While a chaotic component can be extracted from the Arkansas data, such a component is either not present or can not be extracted from the Colorado data.

<p>In this thesis are extreme value theory used to estimate the probability that large insuranceclaims are exceeding a certain threshold. The expected claim size, given that the claimhas exceeded a certain limit, are also estimated. Two different models are used for thispurpose. The first model is based on maximum domain of attraction conditions. A Paretodistribution is used in the other model. Different graphical tools are used to check thevalidity for both models. Länsförsäkring Kronoberg has provided us with insurance datato perform the study.Conclusions, which have been drawn, are that both models seem to be valid and theresults from both models are essential equal.</p> / <p>I detta arbete används extremvärdesteori för att uppskatta sannolikheten att stora försäkringsskadoröverträffar en vis nivå. Även den förväntade storleken på skadan, givetatt skadan överstiger ett visst belopp, uppskattas. Två olika modeller används. Den förstamodellen bygger på antagandet att underliggande slumpvariabler tillhör maximat aven extremvärdesfördelning. I den andra modellen används en Pareto fördelning. Olikagrafiska verktyg används för att besluta om modellernas giltighet. För att kunna genomförastudien har Länsförsäkring Kronoberg ställt upp med försäkringsdata.Slutsatser som dras är att båda modellerna verkar vara giltiga och att resultaten ärlikvärdiga.</p>