Chris P. Tsokos

Developments in Nonparametric Density-Estimation

Summary The object of the present study is to summarize recent developments in nonparametric density estimation. The study covers the period of time from 1956 to 1978. Most of the important types of nonparametric density estimations are discussed. These include Parzen or kernel estimators, series estimators, penalized maximum likelihood estimators, and various other types of nonparametric density estimation techniques.

Changes in coral reef communities among the Florida Keys, 1996–2003

Hard coral (Scleractinia and Milleporina) cover data were examined from 37 sites surveyed annually from 1996 to 2003 in the Florida reef tract, USA. Analyses of species numbers and total cover showed that site-to-site differences were generally very much greater than differences among times within sites. There were no significant differences among different geographical areas within the reef tract (Upper, Middle and Lower Keys). Large-scale changes documented included a reduction in species numbers and total cover on both deep and shallow offshore reefs between 1997 and 1999 followed by no recovery in cover, and only scant evidence of any recovery in species numbers by 2003. These changes coincided with bleaching events in 1997 and 1998, and the passage of Hurricane Georges through the Lower Keys in 1998. The lack of recovery among offshore reefs suggests that they were no longer resilient. Multivariate analyses revealed that some sites showed relatively little temporal variation in community composition, essentially random in direction, while others showed relatively large year-on-year changes. There was little evidence of any major region-wide changes affecting assemblage composition, or of any events that had impacted all of the sampling sites in any single year. Instead, different sites exhibited differing patterns of temporal variation, with certain sites displaying greater variation than others. Changes in community composition at some sites are interpreted in the light of knowledge of events at those sites and the relative sensitivities of species to various stressors, such as changes in cover of Acropora palmata and Millepora complanata at Sand Key following the bleaching events and hurricane in 1998, and declines in Montastraea annularis at Smith Shoal following a harmful algal bloom in 2002. For most sites, however, it is impossible to determine the causes of observed variation.

On a stochastic integral equation of the Volterra type

In a very general sense, random or stochastic integral equations arise in such diverse areas as the physical, biological, oceanographic and engineering sciences. The manner in which such equations arise and their importance to various physical phenomena have been investigated by many scientists [1]; however, not until quite recently have attempts been made to develop and unify the theory of stochastic integral equations. The objective of this investigation is the study of a specific class of stochastic integral equations of Volterra type, which seems to be the most general form considered to date. Specifically, we consider a stochastic integral equation of Volterra type of the form

A Quasi-Bayes Estimate of the Failure Intensity of a Reliability-Growth Model

A non-homogeneous Poisson process has empirically been shown to be useful in tracking the reliability growth of a system as it undergoes development. It is of interest to estimate the failure intensity of this model at the time of failure n. The maximum likelihood estimate is known, but it is desirable to have a Bayesian estimate to allow for input of prior information. Since the ordinary Bayes approach appears to be mathematically intractable, a quasi-Bayes approach is taken. The proposed estimate has the qualitative properties one anticipates from the ordinary Bayes estimate, but it is easy to compute. A numerical example illustrates the Bayesian character of the proposed estimate. A simulation study shows that the proposed estimate, when considered in the classical framework, generally has smaller r.m.s. error than the maximum likelihood estimate.

Parameter estimation of the Weibull probability distribution

Newton—Raphsons method plays a fundamental role in the maximum likelihood estimation of the two parameters of the Weibull probability distribution. It is well known that the method depends on the initial point of the iterative process and the iteration does not always converge.

Estimation of the three parameter Weibull probability distribution

The aim of the present paper is to propose an algorithm to easily obtain good estimates of the three parameter Weibull distribution. Our proposed procedure is given in eight steps and it depends on the Simple Iteration Procedure, which always converges, converges fast and does not depend on any conditions, whatsoever, that has been developed by the authors for the two parameter Weibull model. Numerical examples will be given to illustrate the effectiveness of our proposed statistical procedure. Finally, we address the issue of what we lose in characterizing the probabilistic behavior of a certain phenomenon with a two parameter Weibull model when in fact the true characterization calls for a three parameter Weibull model. We use the concept of percentiles, cumulative distribution function and graphical presentations to answer the above questions.

Random integral equations with applications to stochastic systems

General introduction.- Preliminaries.- A random integral equation of the volterra type.- Approximate solutions of the random volterra integral equation.- A stochastic integral equation of the fredholm type with application to systems theory.- Random discrete fredholm and volterra equations.- The stochastic differential systems.- The stochastic differential systems.- The stochastic differential systems with lag time.

A Random Differential Equation Approach to the Probability Distribution of Bod and Do in Streams

In this paper a stochastic model for stream pollution is given which involves a random differential equation of the form \[( * )\qquad \dot {\bf X}( t ) = {\bf A}{\bf X}( t ) + {\bf Y},\quad t\geqq 0,\] where

TIME SERIES ANALYSIS OF WATER POLLUTION DATA

{\bf X}( t )

Existence of a Solution of a Stochastic Integral Equation in Turbulence Theory

is a two-dimensional vector-valued stochastic process with the first component giving the biochemical oxygen demand (BOD) and the second component representing the dissolved oxygen (DO) at distance t downstream from the source of pollution. The fundamental Liouville’s theorem is utilized to obtain the probability distribution of the solution of