Statistical Distribution Theory

Exercises and solutions for a second year statistics course.

Uma solução da equação difusão-advecção com o termo contragradiente

Neste trabalho, apresenta­se uma solução para equação de difusão­advecção considerando o  termo  contragradiente  que  é um termo  adicional. Esse termo  adicional  contém informações  sobre a assimetria, escala de tempo Lagrangeana e velocidade turbulenta vertical. A solução  da equação foi obtida pela utilização da técnica de Transformada de Laplace, considerando a  Camada  Limite  Planetária  (CLP)  como  um  sistema  de  multicamadas.  Os  parâmetros  turbulentos foram derivados da teoria de difusão estatística de Taylor, combinada com a teoria  da similaridade. Assim, são apresentadas simulações para diferentes valores de assimetria, o  que  propiciou  a  obtenção  de  uma  concentração  de  contaminantes  em  diferentes  alturas,  em  uma  camada  limite  convectiva.  A  avaliação  do  desempenho  do  modelo,  considerando  a  assimetria  no  processo  de  dispersão  de  poluentes  atmosféricos, foi realizada  através  de  um  experimento  de  tanque  convectivo  tradicional.  Nesse  experimento,  o  termo  contragradiente  influenciou a concentração de poluentes para uma camada limite convectiva. Entretanto, com  as  parametrizações  utilizadas,  o  modelo  não  conseguiu  captar  de  forma  eficiente  o  comportamento da concentração em pontos mais distantes da fonte.

Effects of algal turbidity on foraging and antipredator behaviour of the three-spined stickleback (Gasterosteus aculeatus)

The aim of this thesis was to examine how aquatic organisms, such as fish, behave in an altered environmental condition. Many species of fish use vision as their primary tool to gain information about their surrounding environment. The visual conditions of aquatic habitats are often altered as a result of anthropogenic disturbance, such as eutrophication that initiates algal turbidity. In general, turbidity reduces the visibility and can be hypothesized to have an influence on the behaviour of fish. I used the three-spined stickleback (Gasterosteus aculeatus) as a model species and conducted four studies in the laboratory to test how algal turbidity affects its behaviour. In this thesis, two major behavioural aspects are discussed. The first is antipredator behaviour. In study I, the combined effects of turbidity and shoot density on habitat choice (shelter vs open) behaviour was tested on a group of sticklebacks (20 fish) in the presence and absence of piscivorous perch (Perca fluviatilis). In study II, I examined the behavioural responses of feeding sticklebacks when they were exposed to the sudden appearance of an avian predator (the silhouette of a common tern, Sterna hirundo). The study was done in turbid and clear water using three different groups sizes (1, 3 and 6 fish). The second aspect is foraging behaviour. Study III & IV focused on the effects of algal turbidity on the foraging performance of sticklebacks. In study III, I conducted two separate experiments to examine the effects of turbidity on prey consumption and prey choice of sticklebacks. In this experiment turbidity levels and the proportion of large and small prey (Daphnia spp.) were manipulated. In study IV, I studied whether a group of six sticklebacks can distribute themselves according to food input at two feeding stations in a way that provided each fish with the same amount of food in clear and turbid water. I also observed whether the fish can follow changes in resource distribution between the foraging patches. My results indicate an overall influence of algal turbidity on the antipredator and foraging behaviour of sticklebacks. In the presence of a potential predator, the use of the sheltered habitat was more pronounced at higher turbidity. Besides this, sticklebacks reduced their activity levels with predator presence at higher turbidity and shoot density levels, suggesting a possible antipredator adaptation to avoid a predator. When exposed to a sudden appearance of an avian predator, sticklebacks showed a weaker antipredator response in turbid water, which suggests that turbidity degrades the risk assessment capabilities of sticklebacks. I found an effect of group size but not turbidity in the proportion of sticklebacks that fled to the shelter area, which indicates that sticklebacks are able to communicate among group members at the experimental turbidity levels. I found an overall negative effect of turbidity on food intake. Both turbidity and changes in the proportion of prey sizes played a significant role in a stickleback’s prey selection. At lower turbidity levels (clear <1 and 5 NTU) sticklebacks showed preferences for large prey, whereas in more turbid conditions and when the proportion of large to small prey increased sticklebacks became increasingly random in their prey selection. Finally, my results showed that groups of sticklebacks disperse themselves between feeding stations according to the reward ratios following the predictions of the ideal free distribution theory. However, they took a significantly longer time to reach the equilibrium distribution in turbid water than in clear water. In addition, they showed a slower response to changes in resource distribution in a turbid environment. These findings suggest that turbidity interferes with the information transfer among group foragers. It is important to understand that aquatic animals are often exposed to a degraded environment. The findings of this thesis suggest that algal turbidity negatively affects their behavioural performance. The results also shed light on the underlying behavioural strategies of sticklebacks in turbid conditions that might help them adapt to an altered environmental situation and increase their survival. In conclusion, I hold that although algal turbidity has detrimental effects on the antipredator and foraging behaviour of sticklebacks, their behavioural adjustment might help them adapt to a changing environment.

Concomitants of Order Statistics from Morgenstern Family

The study deals with the distribution theory and applications of concomitants from the Morgenstern family of bivariate distributions.The Morgenstern system of distributions include all cumulative distributions of the form FX,Y(X,Y)=FX(X) FY(Y)[1+α(1-FX(X))(1-FY(Y))], -1≤α≤1.The system provides a very general expression of a bivariate distributions from which members can be derived by substituting expressions of any desired set of marginal distributions.It is a brief description of the basic distribution theory and a quick review of the existing literature.The Morgenstern family considered in the present study provides a very general expression of a bivariate distribution from which several members can be derived by substituting expressions of any desired set of marginal distributions.Order statistics play a very important role in statistical theory and practice and accordingly a remarkably large body of literature has been devoted to its study.It helps to develop special methods of statistical inference,which are valid with respect to a broad class of distributions.The present study deals with the general distribution theory of Mk, [r: m] and Mk, [r: m] from the Morgenstern family of distributions and discuss some applications in inference, estimation of the parameter of the marginal variable Y in the Morgestern type uniform distributions.

Some bivariate life time models in discrete time"

The term reliability of an equipment or device is often meant to indicate the probability that it carries out the functions expected of it adequately or without failure and within specified performance limits at a given age for a desired mission time when put to use under the designated application and operating environmental stress. A broad classification of the approaches employed in relation to reliability studies can be made as probabilistic and deterministic, where the main interest in the former is to device tools and methods to identify the random mechanism governing the failure process through a proper statistical frame work, while the latter addresses the question of finding the causes of failure and steps to reduce individual failures thereby enhancing reliability. In the probabilistic attitude to which the present study subscribes to, the concept of life distribution, a mathematical idealisation that describes the failure times, is fundamental and a basic question a reliability analyst has to settle is the form of the life distribution. It is for no other reason that a major share of the literature on the mathematical theory of reliability is focussed on methods of arriving at reasonable models of failure times and in showing the failure patterns that induce such models. The application of the methodology of life time distributions is not confined to the assesment of endurance of equipments and systems only, but ranges over a wide variety of scientific investigations where the word life time may not refer to the length of life in the literal sense, but can be concieved in its most general form as a non-negative random variable. Thus the tools developed in connection with modelling life time data have found applications in other areas of research such as actuarial science, engineering, biomedical sciences, economics, extreme value theory etc.

Reliability of timber bridges using nailed plywood box beams

The Nailed Box Beam structural efficiency is directly dependent of the flange-web joint behavior, which determines the partial composition of the section, as the displacement between elements reduces the effective rigidity of the section and changes the stress distribution and the total displacement of the section. This work discusses the use of Nailed Plywood Box Beams in small span timber bridges, focusing on the reliability of the beam element. It is presented the results of tests carried out in 21 full scale Nailed Plywood Box Beams. The analysis of maximum load tests results shows that it presents a normal distribution, permitting the characteristic values calculation as the normal distribution theory specifies. The reliability of those elements was analyzed focusing on a timber bridge design, to estimate the failure probability in function of the load level.

Some Properties of Mittag-Leffler Functions and Matrix-Variate Analogues: A Statistical Perspective

Mathematical Subject Classification 2010:26A33, 33E99, 15A52, 62E15.

Special Functions and Pathways for Problems in Astrophysics: An Essay in Honor of A.M. Mathai

The paper provides a review of A.M. Mathai's applications of the theory of special functions, particularly generalized hypergeometric functions, to problems in stellar physics and formation of structure in the Universe and to questions related to reaction, diffusion, and reaction-diffusion models. The essay also highlights Mathai's recent work on entropic, distributional, and differential pathways to basic concepts in statistical mechanics, making use of his earlier research results in information and statistical distribution theory. The results presented in the essay cover a period of time in Mathai's research from 1982 to 2008 and are all related to the thematic area of the gravitationally stabilized solar fusion reactor and fractional reaction-diffusion, taking into account concepts of non-extensive statistical mechanics. The time period referred to above coincides also with Mathai's exceptional contributions to the establishment and operation of the Centre for Mathematical Sciences, India, as well as the holding of the United Nations (UN)/European Space Agency (ESA)/National Aeronautics and Space Administration (NASA) of the United States/ Japanese Aerospace Exploration Agency (JAXA) Workshops on basic space science and the International Heliophysical Year 2007, around the world. Professor Mathai's contributions to the latter, since 1991, are a testimony for his social con-science applied to international scientific activity.

Matrix-Variate Statistical Distributions and Fractional Calculus

MSC 2010: 15A15, 15A52, 33C60, 33E12, 44A20, 62E15 Dedicated to Professor R. Gorenflo on the occasion of his 80th birthday

Keeping Variables within Bounds: Using Information between Observations

This research develops an econometric framework to analyze time series processes with bounds. The framework is general enough that it can incorporate several different kinds of bounding information that constrain continuous-time stochastic processes between discretely-sampled observations. It applies to situations in which the process is known to remain within an interval between observations, by way of either a known constraint or through the observation of extreme realizations of the process. The main statistical technique employs the theory of maximum likelihood estimation. This approach leads to the development of the asymptotic distribution theory for the estimation of the parameters in bounded diffusion models. The results of this analysis present several implications for empirical research. The advantages are realized in the form of efficiency gains, bias reduction and in the flexibility of model specification. A bias arises in the presence of bounding information that is ignored, while it is mitigated within this framework. An efficiency gain arises, in the sense that the statistical methods make use of conditioning information, as revealed by the bounds. Further, the specification of an econometric model can be uncoupled from the restriction to the bounds, leaving the researcher free to model the process near the bound in a way that avoids bias from misspecification. One byproduct of the improvements in model specification is that the more precise model estimation exposes other sources of misspecification. Some processes reveal themselves to be unlikely candidates for a given diffusion model, once the observations are analyzed in combination with the bounding information. A closer inspection of the theoretical foundation behind diffusion models leads to a more general specification of the model. This approach is used to produce a set of algorithms to make the model computationally feasible and more widely applicable. Finally, the modeling framework is applied to a series of interest rates, which, for several years, have been constrained by the lower bound of zero. The estimates from a series of diffusion models suggest a substantial difference in estimation results between models that ignore bounds and the framework that takes bounding information into consideration.