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

Haymen猜測:對任意的超越半純函數 f(z)，f'(z)f(z)^n 取所有值無窮多次，其中至多只有一個例外值。這個著名的猜測，大部分的情形已被證明是正確的。另外，Hayman 證明 f'(z)-af(z)^n 取所有有限值無窮多次 ，其中 a 為一複數且 n≧5 的正整數。在本篇論文裡，我們將探討以小函數為係數的半純函數微分多項式之值分佈問題。並將Hayman的結果推廣至 f^{k}(z)f(z)^n 與 f^{k}(z)-af(z)^n 的情形。同時，我們也證明一些 A類半純函數與其導數的值分佈結果。 / A famous conjecture of Hayman says that if f(z) is a transcendental meromorphic function, then f'(z)f(z)^n assumes all finite values except possibly zero infinitely often. The conjecture was solved in most cases. Another result of Hayman says that f'(z)-af(z)^n, where n≧5 and a is a complex number, assumes all finite values infinitely often. In this thesis, we will study the value distribution of some differential polynomial in a meromorphic function with small functions as coefficents. In fact, we will generalize Hayman's results to the cases f^(k)(z)f(z)^n and f^(k)(z)-af(z)^n. Also, the value distribution of meromorphic functions of class A with their derivatives are obtained.

Thesis (Ph. D.)--University of Joensuu, 2006. / Includes bibliographical references (p. 25-27).

在這篇論文裡，我們利用值分佈的理論來探討半純函數的共值與唯一性的問題，本文包含了以下的結果：將Jank與Terglane有關三個A類中的半純函數唯一性的結果推廣到任意q個半純函數的情形；證明了C. C. Yang的一個猜測；建構了一類半純函數恰有兩個虧值，而且算出它們的虧格；將 Nevanlinna 五個值的定理推廣至兩個半純函數部分共值的情形；探討純函數 與其導數的共值問題；最後，證明了兩個半純函數共四個值且重數皆不同的定 理。 / In this thesis, we study the sharing value problems and the uniqueness problems of meromorphic functions in the theory of value distribution. In fact, this thesis contains the following results: We generalize a unicity condition of three meromorphic functions given by Jank and Terglane in class A to the case of arbitrary q meromorphic functoins. An elementary proof of a conjecture of C. C. Yang is provided. We construct a class of meromorphic functions with exact two deficient values and their deficiencies are explicitly computed. We generalize the Nevanlinna's five-value theorem to the cases that two meromorphic functions partially share either five or more values, or five or more small functions. In each case, we formulate a way to measure how far these two meromorphic functions are from sharing either values or small functions, and use this measurement to prove a uniqueness theorem. Also, we prove some uniqueness theorems on entire functions that share a pair of values (a,-a) with their derivatives, which are reformulations of some important results about uniqueness of entire functions that share values with their derivatives. Finally, we prove that if two distinct non-constant meromorphic functions $f$ and $g$ share four distinct values a_1, a_2, a_3, a_4 DM such that each a_i-point is either a (p,q)-fold or (q,p)-fold point of f and g, then (p,q) is either (1,2) or (1,3) and f, g are in some particular forms.

Univ., Diss.--Joensuu, 2006.

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.