Efficient random variable generation: ratio of uniforms and polar rejection sampling

Monte Carlo techniques, which require the generation of samples from some target density, are often the only alternative for performing Bayesian inference. Two classic sampling techniques to draw independent samples are the ratio of uniforms (RoU) and rejection sampling (RS). An efficient sampling algorithm is proposed combining the RoU and polar RS (i.e. RS inside a sector of a circle using polar coordinates). Its efficiency is shown in drawing samples from truncated Cauchy and Gaussian random variables, which have many important applications in signal processing and communications. RESUMEN. Método eficiente para generar algunas variables aleatorias de uso común en procesado de señal y comunicaciones (por ejemplo, Gaussianas o Cauchy truncadas) mediante la combinación de dos técnicas: "ratio of uniforms" y "rejection sampling".

On Queues with Interruptions and Repeat or Resumption of Service

This thesis entitled' On Queues with Interruptions and Repeat or Resumption of Service' introduces several new concepts into queues with service interruption. It is divided into Seven chapters including an introductory chapter. The following are keywords that we use in this thesis: Phase type (PH) distribution, Markovian Arrival Process (MAP), Geometric Distribution, Service Interruption, First in First out (FIFO), threshold random variable and Super threshold random variable. In the second chapter we introduce a new concept called the 'threshold random variable' which competes with interruption time to decide whether to repeat or resume the interrupted service after removal of interruptions. This notion generalizes the work reported so far in queues with service interruptions. In chapter 3 we introduce the concept of what is called 'Super threshold clock' (a random variable) which keeps track of the total interruption time of a customer during his service except when it is realized before completion of interruption in some cases to be discussed in this thesis and in other cases it exactly measures the duration of all interruptions put together. The Super threshold clock is OIl whenever the service is interrupted and is deactivated when service is rendered. Throughout this thesis the first in first out service discipline is followed except for priority queues.

Disordered ensembles of random matrices

It is shown that the families of generalized matrix ensembles recently considered which give rise to an orthogonal invariant stable Levy ensemble can be generated by the simple procedure of dividing Gaussian matrices by a random variable. The nonergodicity of this kind of disordered ensembles is investigated. It is shown that the same procedure applied to random graphs gives rise to a family that interpolates between the Erdos-Renyi and the scale free models.

Wave-height hazard analysis in Eastern Coast of Spain : Bayesian approach using generalized Pareto distribution

Standard practice of wave-height hazard analysis often pays little attention to the uncertainty of assessed return periods and occurrence probabilities. This fact favors the opinion that, when large events happen, the hazard assessment should change accordingly. However, uncertainty of the hazard estimates is normally able to hide the effect of those large events. This is illustrated using data from the Mediterranean coast of Spain, where the last years have been extremely disastrous. Thus, it is possible to compare the hazard assessment based on data previous to those years with the analysis including them. With our approach, no significant change is detected when the statistical uncertainty is taken into account. The hazard analysis is carried out with a standard model. Time-occurrence of events is assumed Poisson distributed. The wave-height of each event is modelled as a random variable which upper tail follows a Generalized Pareto Distribution (GPD). Moreover, wave-heights are assumed independent from event to event and also independent of their occurrence in time. A threshold for excesses is assessed empirically. The other three parameters (Poisson rate, shape and scale parameters of GPD) are jointly estimated using Bayes' theorem. Prior distribution accounts for physical features of ocean waves in the Mediterranean sea and experience with these phenomena. Posterior distribution of the parameters allows to obtain posterior distributions of other derived parameters like occurrence probabilities and return periods. Predictives are also available. Computations are carried out using the program BGPE v2.0

Queues with inter- ruption in Random/ Markovian Environment

In many situations probability models are more realistic than deterministic models. Several phenomena occurring in physics are studied as random phenomena changing with time and space. Stochastic processes originated from the needs of physicists.Let X(t) be a random variable where t is a parameter assuming values from the set T. Then the collection of random variables {X(t), t ∈ T} is called a stochastic process. We denote the state of the process at time t by X(t) and the collection of all possible values X(t) can assume, is called state space

Una aproximación de la variable aleatoria a procesos de toma de decisión que implican condiciones de riesgo e incertidumbre

La variable aleatoria es una función matemática que permite asignar valores numéricos a cada uno de los posibles resultados obtenidos en un evento de naturaleza aleatoria. Si el número de estos resultados se puede contar, se tiene un conjunto discreto; por el contrario, cuando el número de resultados es infinito y no se puede contar, se tiene un conjunto continuo. El objetivo de la variable aleatoria es permitir adelantar estudios probabilísticos y estadísticos a partir del establecimiento de una asignación numérica a través de la cual se identifiquen cada uno de los resultados que pueden ser obtenidos en el desarrollo de un evento determinado. El valor esperado y la varianza son los parámetros por medio de los cuales es posible caracterizar el comportamiento de los datos reunidos en el desarrollo de una situación experimental; el valor esperado permite establecer el valor sobre el cual se centra la distribución de la probabilidad, mientras que la varianza proporciona información acerca de la manera como se distribuyen los datos obtenidos. Adicionalmente, las distribuciones de probabilidad son funciones numéricas asociadas a la variable aleatoria que describen la asignación de probabilidad para cada uno de los elementos del espacio muestral y se caracterizan por ser un conjunto de parámetros que establecen su comportamiento funcional, es decir, cada uno de los parámetros propios de la distribución suministra información del experimento aleatorio al que se asocia. El documento se cierra con una aproximación de la variable aleatoria a procesos de toma de decisión que implican condiciones de riesgo e incertidumbre.

Preventive maintenance models with random maintenance quality

In real-world environments it is usually difficult to specify the quality of a preventive maintenance (PM) action precisely. This uncertainty makes it problematic to optimise maintenance policy.-This problem is tackled in this paper by assuming that the-quality of a PM action is a random variable following a probability distribution. Two frequently studied PM models, a failure rate PM model and an age reduction PM model, are investigated. The optimal PM policies are presented and optimised. Numerical examples are also given.

Random-walk-based characterisation of multiple access interference in a DS/SSMA system

The problem of calculating the probability of error in a DS/SSMA system has been extensively studied for more than two decades. When random sequences are employed some conditioning must be done before the application of the central limit theorem is attempted, leading to a Gaussian distribution. The authors seek to characterise the multiple access interference as a random-walk with a random number of steps, for random and deterministic sequences. Using results from random-walk theory, they model the interference as a K-distributed random variable and use it to calculate the probability of error in the form of a series, for a DS/SSMA system with a coherent correlation receiver and BPSK modulation under Gaussian noise. The asymptotic properties of the proposed distribution agree with other analyses. This is, to the best of the authors' knowledge, the first attempt to propose a non-Gaussian distribution for the interference. The modelling can be extended to consider multipath fading and general modulation

A Note on Large Deviations for the Stable Marriage of Poisson and Lebesgue with Random Appetites

Let IaS,a"e (d) be a set of centers chosen according to a Poisson point process in a"e (d) . Let psi be an allocation of a"e (d) to I in the sense of the Gale-Shapley marriage problem, with the additional feature that every center xi aI has an appetite given by a nonnegative random variable alpha. Generalizing some previous results, we study large deviations for the distance of a typical point xaa"e (d) to its center psi(x)aI, subject to some restrictions on the moments of alpha.

Modelos de mapas simpléticos para o movimento de deriva elétrica com efeitos de raio de Larmor finito

Mapas simpléticos têm sido amplamente utilizados para modelar o transporte caótico em plasmas e fluidos. Neste trabalho, propomos três tipos de mapas simpléticos que descrevem o movimento de deriva elétrica em plasmas magnetizados. Efeitos de raio de Larmor finito são incluídos em cada um dos mapas. No limite do raio de Larmor tendendo a zero, o mapa com frequência monotônica se reduz ao mapa de Chirikov-Taylor, e, nos casos com frequência não-monotônica, os mapas se reduzem ao mapa padrão não-twist. Mostramos como o raio de Larmor finito pode levar à supressão de caos, modificar a topologia do espaço de fases e a robustez de barreiras de transporte. Um método baseado na contagem dos tempos de recorrência é proposto para analisar a influência do raio de Larmor sobre os parâmetros críticos que definem a quebra de barreiras de transporte. Também estudamos um modelo para um sistema de partículas onde a deriva elétrica é descrita pelo mapa de frequência monotônica, e o raio de Larmor é uma variável aleatória que assume valores específicos para cada partícula do sistema. A função densidade de probabilidade para o raio de Larmor é obtida a partir da distribuição de Maxwell-Boltzmann, que caracteriza plasmas na condição de equilíbrio térmico. Um importante parâmetro neste modelo é a variável aleatória gama, definida pelo valor da função de Bessel de ordem zero avaliada no raio de Larmor da partícula. Resultados analíticos e numéricos descrevendo as principais propriedades estatísticas do parâmetro gama são apresentados. Tais resultados são então aplicados no estudo de duas medidas de transporte: a taxa de escape e a taxa de aprisionamento por ilhas de período um.

Large Distinct Part Sizes in a Random Integer Partition

A partition of a positive integer n is a way of writing it as the sum of positive integers without regard to order; the summands are called parts. The number of partitions of n, usually denoted by p(n), is determined asymptotically by the famous partition formula of Hardy and Ramanujan [5]. We shall introduce the uniform probability measure P on the set of all partitions of n assuming that the probability 1/p(n) is assigned to each n-partition. The symbols E and V ar will be further used to denote the expectation and variance with respect to the measure P . Thus, each conceivable numerical characteristic of the parts in a partition can be regarded as a random variable.

The transducer model for contrast detection and discrimination: formal relations, implications, and an empirical test.

The transducer function mu for contrast perception describes the nonlinear mapping of stimulus contrast onto an internal response. Under a signal detection theory approach, the transducer model of contrast perception states that the internal response elicited by a stimulus of contrast c is a random variable with mean mu(c). Using this approach, we derive the formal relations between the transducer function, the threshold-versus-contrast (TvC) function, and the psychometric functions for contrast detection and discrimination in 2AFC tasks. We show that the mathematical form of the TvC function is determined only by mu, and that the psychometric functions for detection and discrimination have a common mathematical form with common parameters emanating from, and only from, the transducer function mu and the form of the distribution of the internal responses. We discuss the theoretical and practical implications of these relations, which have bearings on the tenability of certain mathematical forms for the psychometric function and on the suitability of empirical approaches to model validation. We also present the results of a comprehensive test of these relations using two alternative forms of the transducer model: a three-parameter version that renders logistic psychometric functions and a five-parameter version using Foley's variant of the Naka-Rushton equation as transducer function. Our results support the validity of the formal relations implied by the general transducer model, and the two versions that were contrasted account for our data equally well.

Likelihood-based estimation of the regression model with scrambled responses

A significant problem in the collection of responses to potentially sensitive questions, such as relating to illegal, immoral or embarrassing activities, is non-sampling error due to refusal to respond or false responses. Eichhorn & Hayre (1983) suggested the use of scrambled responses to reduce this form of bias. This paper considers a linear regression model in which the dependent variable is unobserved but for which the sum or product with a scrambling random variable of known distribution, is known. The performance of two likelihood-based estimators is investigated, namely of a Bayesian estimator achieved through a Markov chain Monte Carlo (MCMC) sampling scheme, and a classical maximum-likelihood estimator. These two estimators and an estimator suggested by Singh, Joarder & King (1996) are compared. Monte Carlo results show that the Bayesian estimator outperforms the classical estimators in almost all cases, and the relative performance of the Bayesian estimator improves as the responses become more scrambled.

Alguns teoremas limites para sequências de variáveis aleatórias

O Teorema Central do Limite e a Lei dos Grandes Números estão entre os mais importantes resultados da teoria da probabilidade. O primeiro deles busca condições sob as quais [fórmula] converge em distribuição para a distribuição normal com parâmetros 0 e 1, quando n tende ao infinito, onde Sn é a soma de n variáveis aleatórias independentes. Ao mesmo tempo, o segundo estabelece condições para que [fórmula] convirja a zero, ou equivalentemente, para que [fórmula] convirja para a esperança das variáveis aleatórias, caso elas sejam identicamente distribuídas. Em ambos os casos as sequências abordadas são do tipo [fórmula], onde [fórmula] e [fórmula] são constantes reais. Caracterizar os possíveis limites de tais sequências é um dos objetivos dessa dissertação, já que elas não convergem exclusivamente para uma variável aleatória degenerada ou com distribuição normal como na Lei dos Grandes Números e no Teorema Central do Limite, respectivamente. Assim, somos levados naturalmente ao estudo das distribuições infinitamente divisíveis e estáveis, e os respectivos teoremas limites, e este vem a ser o objetivo principal desta dissertação. Para as demonstrações dos teoremas utiliza-se como estratégia principal a aplicação do método de Lyapunov, o qual consiste na análise da convergência da sequência de funções características correspondentes às variáveis aleatórias. Nesse sentido, faremos também uma abordagem detalhada de tais funções neste trabalho.

Efficiency Measures in the Agricultural Sector: The Beginning

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