Showing papers on "Probability-generating function published in 2001"

Estimation in integer‐valued moving average models

Abstract: The paper presents new characterizations of the integer-valued moving average model. For four model variants, we give moments and probability generating functions. Yule-Walker and conditional least ...

Asymptotic ruin probabilities for risk processes with dependent increments

Abstract: In this paper, we derive a Lundberg type result for asymptotic ruin probabilities in the case of a risk process with dependent increments We only assume that the probability generating functions exist, and that their logarithmic average converges Under these assumptions we present an elementary proof of the Lundberg limiting result, which only uses simple exponential inequalities, and does not rely on results from large deviation theory Moreover, we use dependence orderings to investigate, how dependencies between the claims affect the Lundberg coefficient The results are illustrated by several examples, including Gaussian and AR(1)-processes, and a risk process with adapted premium rules

Random Walks on Regular Languages and Algebraic Systems of Generating Functions

Abstract: A random walk on a regular language is a Markov chain on the set of all finite words from a finite alphabet A whose transition probabilities obey the following rules: (1) Only the last two letters of a word may be modified in one jump, and at most one letter may be adjoined or deleted. (2) Probabilities of modification, deletion, and/or adjunction depend only on the last two letters of the current word. Special cases include (a) reflecting random walks on the nonnegative integers; (b) LIFO queues; (c) finite-range random walks on homogeneous trees; and (d) random walks on the modular group PSL2(Z). It is shown that the n−step transition probabilities of a random walk on a regular language must obey one of three different types of power laws. The analysis is based on the study of an algebraic system of generating functions related to the Green’s function.

Bias calculations for adaptive urn designs

Abstract: A clinical trial model is considered in which two treatments with immediate binary responses are to be compared. An adaptive urn design is used to assign patients to the treatments. The bias and variance of the maximum likelihood estimators of the probabilities of success are derived by differentiating the fundamental identity of sequential analysis. By embedding the design in a continuous-time process, probability generating functions are then calculated to obtain approximations for the bias and variance. Simulation is used to assess the accuracy of the approximations. It is shown that the bias cannot be ignored, and that the adaptive rules which are subcritical in nature have the most mathematically tractable bias and are the least variable. Methods for correcting for the bias are also addressed.

Lyapunov-based controller design for bounded dynamic stochastic distribution control

Abstract: Following developments in the modelling and control of the output probability density function of dynamic stochastic systems, an approach for the controller design is presented using square root approximation, where a set of B-spline functions are used to approximate the square root of the measured output probability density function to guarantee its positiveness. A performance function is defined which measures the tracking error of the output probability density function with respect to a given distribution. Instead of finding an optimal control which minimises this performance function and then analysing the stability of the closed loop system, the new approach directly uses the performance function as a Lyapunov function to design the required controller. As a result, the controller obtained not only guarantees the decreasing of the performance function with respect to time, but also stabilises the closed-loop system, realising an asymptotically tracking performance of the output probability density function with respect to its target distribution. The algorithm described has been tested on a simulated example and desired results have been achieved.

Prediction of molecular weight distributions by probability generating functions. Application to industrial autoclave reactors for high pressure polymerization of ethylene and ethylene-vinyl acetate

Abstract: We develop a mathematical model able to describe the complete molecular weight distributions of polyethylene and elhylene-vinyl acetate copolymers obtained in high pressure autoclave reactors. We apply probability generating function definitions to the mass balances of radical and polymer species in the reacting medium. We use three different definitions of probability generating functions, each one directly applicable either to the number, weight or chromatographic distributions. These probability generating functions are numerically inverted to obtain the corresponding calculated molecular weight distribution. The capabilities of two different inversion methods are compared. Predictions are compared with experimental data obtained in an industrial reactor; good agreement is obtained. The approach presented here is applicable to other types of polymerization reactors and post-polymerization processes.

Continuously variable survival exponent for random walks with movable partial reflectors.

Abstract: We study a one-dimensional lattice random walk with an absorbing boundary at the origin and a movable partial reflector. On encountering the reflector at site x, the walker is reflected (with probability r) to x-1 and the reflector is simultaneously pushed to x+1. Iteration of the transition matrix, and asymptotic analysis of the probability generating function show that the critical exponent delta governing the survival probability varies continuously between 1/2 and 1 as r varies between 0 and 1. Our study suggests a mechanism for nonuniversal kinetic critical behavior, observed in models with an infinite number of absorbing configurations.

Method, computer program, and storage medium for estimating randomness of function of representative value of random variable by the use of gradient of same function

Abstract: A method of estimating a measure of randomness of a function of at least one representative value of at least one random variable is constructed to have the steps of obtaining the at least one random variable; determining the at least one representative value of the obtained at least one random variable; determining a first statistic of the obtained at least one random variable; determining a gradient of the function with respect to the determined at least one representative value; and transforming the obtained first statistic into a second statistic of the function, using the determined gradient The step of transforming may be adapted to transform the first statistic into the second statistic, such that the second statistic responds to the first statistic more sensitively in the case of the gradient being steep than in the case of the gradient being gentle

On Bounds for the Concentration Function II

Abstract: We derive an upper bound for the concentration of the sum of i.i.d. random variables with values in \(\mathbb{Z}^d\) by appealing to functions of positive type and the structure theory of set addition.

Polya urn models under general replacement schemes

Abstract: In this paper, we consider a Polya urn model containing balls of m different labels under a general replacement scheme, which is characterized by an m × m ad- dition matrix of integers without constraints on the values of these m 2 integers other than non-negativity. This urn model includes some important urn models treated before. By a method based on the probability generating functions, we consider the exact joint distribution of the numbers of balls with particular labels which are drawn within n draws. As a special case, for m = 2, the univariate distribution, the prob- ability generating function and the expected value are derived exactly. We present methods for obtaining the probability generating functions and the expected values for all n exactly, which are very simple and suitable for computation by computer algebra systems. The results presented here develop a general workable framework for the study of Polya urn models and attract our attention to the importance of the exact analysis. Our attempts are very useful for understanding non-classical urn models. Finally, numerical examples are also given in order to illustrate the feasibility of our results.

On the convergence rate of ordinal comparisons of random variables

Abstract: The asymptotic exponential convergence rate of ordinal comparisons follows from well-known results in large deviations theory, where the critical condition is the existence of a finite moment generating function. In this note, we show that this is both a necessary and sufficient condition, and also show how one can recover the exponential convergence rate in cases where the moment generating function is not finite. In particular, by working with appropriately truncated versions of the original random variables, the exponential convergence rate can be recovered.

Exact non-null distributions of rank statistics

Abstract: In literature, the exact non-null distributions of one and two-sample rank statistics are known for small sample sizes (say N ≤ 12) only. To fill the gap between these small sample sizes and sample sizes for which approximations are accurate, new algorithms are developed. These algorithms are based on probability generating functions and we show that they give us the opportunity to compute the exact non-null distribution for much larger samples than the existing algorithms. Finally, we present some applications for nonparametric control charts and power curves.

Maximum Entropy Method and It's Application in Probability Density Function of Traffic Flow

Abstract: Probability density function of traffic flow varies with different survey data.It is difficult to find a probability density function fitting for all kinds of traffic survey data.So to get an ordinary generating method of probability density function is greatly important for traffic researcher and engineer. The authors analyze the problem from the view of maximum entropy principle,work out six probability density function which often appear in traffic practice,and derive a formula and it's practical algorithm from physicist's achievement.The method and program are proven to be really effective in theoretical simulation and practice examination.

Discrete-time queues with general service times and general server interruptions

Abstract: In this contribution, we investigate a discrete-time single- server queue subjected to server interruptions. Server interruptions are modeled as an on/off process with geometrically distributed on-periods and generally distributed off-periods. As message lengths can exceed one time-slot, different operation modes are considered depending on whether service of an interrupted message continues, partially restarts or completely restarts after an interruption. For all alternatives, we establish expressions for the steady-state probability generating functions of the buffer contents at message departure time and at random slot boundaries. From these results, closed- form expressions for various performance measures, such as mean and variance of the buffer occupancy, can be established. As an application, we show that this model is able to assess performance of low-priority traffic in a two- priority HOL scheduling discipline. We then illustrate our approach with some numerical examples.© (2001) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

A hazard function approximation used in reliability theory

Abstract: The cumulative hazard function H(n) should accumulate to infinity over the distribution support, because the survivor function is Sf(n)=exp(-H(n)). The widely used approximation for the cumulative hazard function, H(n)/spl ap//spl Sigma//sub k=1//sup n/h(k), for a small value of the hazard function, h(k), can be useful and reasonably accurate for computing the survivor function. For the continuous case, assuming that pdf exists, the H(n) diverges as it should. For the discrete case, two examples show the use of the hazard function approximation. In example A for the uniform probability mass function, the approximation diverges. In example B for the geometric probability mass function, the approximation converges to the finite value, 1.606695, when it should be diverging. The result is surprising in light of the difference between the continuous case, pdf, and the discrete case, pmf. Thus in practice, the approximation must be used with caution.

A nonlinear function for gradient based BSS

Abstract: Blind source separation (BSS) deals with separating independent signals form their linear mixtures observed at different sensors. In this paper a nonlinear function based on the cost function as a Kullback-Leibler divergence between the joint probability density function of the source vector and its parametric model, is proposed. This cost function is equivalent to maximization of the information transfer between inputs and outputs, and minimization of the mutual information between components of the output vector. Derivation process becomes extremely simple due to a simple approximation. Simulations with communication signals indicate that the proposed algorithm provides better accuracy.

A Note on Bisexual Galton-Watson Branching Processes with Inmigration

Abstract: Recently, from the branching model introduced in [1], new bisexual GaltonWatson branching processes allowing immigration have been developed in [2] and some probabilistical analysis about them has been obtained. In particular, for the bisexual Galton-Watson process allowing the immigration of females and males, it has been proved (see [3]) that, under certain conditions, the sequence representing the number of mating units per generation converges in distribution to a positive, finite and non-degenerate random variable. The aim of this paper is to provide, through a different methodology, an alternative proof of this limit result. In this new, and more technical proof, we make use of the underlying probability generating functions. In Section 2, a brief description of the probability model is considered and some basic definitions and results are given. Section 3 is devoted to prove the asymptotic result previously indicated.