A note on a gamma distribution computer program /

Mode of access: Internet.

Locally asymptotically normal families of distributions; certain approximations to families of distributions and their use in the theory of estimation and testing hypotheses.

Bibliography: p. 98.

The moments of the non-central t-distribution,

"This work was supported by the Office of Naval Reseaarch under Contract Nonr. 404(16)."

The beta distribution in reliability and quality control /

Bibliography: leaves 41-43.

Asymptotic and structural properties of special cases of the Wright function arising in probability theory

This analysis paper presents previously unknown properties of some special cases of the Wright function whose consideration is necessitated by our work on probability theory and the theory of stochastic processes. Specifically, we establish new asymptotic properties of the particular Wright function 1Ψ1(ρ, k; ρ, 0; x) = X∞ n=0 Γ(k + ρn) Γ(ρn) x n n! (|x| < ∞) when the parameter ρ ∈ (−1, 0)∪(0, ∞) and the argument x is real. In the probability theory applications, which are focused on studies of the Poisson-Tweedie mixtures, the parameter k is a non-negative integer. Several representations involving well-known special functions are given for certain particular values of ρ. The asymptotics of 1Ψ1(ρ, k; ρ, 0; x) are obtained under numerous assumptions on the behavior of the arguments k and x when the parameter ρ is both positive and negative. We also provide some integral representations and structural properties involving the ‘reduced’ Wright function 0Ψ1(−−; ρ, 0; x) with ρ ∈ (−1, 0) ∪ (0, ∞), which might be useful for the derivation of new properties of members of the power-variance family of distributions. Some of these imply a reflection principle that connects the functions 0Ψ1(−−;±ρ, 0; ·) and certain Bessel functions. Several asymptotic relationships for both particular cases of this function are also given. A few of these follow under additional constraints from probability theory results which, although previously available, were unknown to analysts.

Reliability analysis with suspended items

Reliability analysis has several important engineering applications. Designers and operators of equipment are often interested in the probability of the equipment operating successfully to a given age - this probability is known as the equipment's reliability at that age. Reliability information is also important to those charged with maintaining an item of equipment, as it enables them to model and evaluate alternative maintenance policies for the equipment. In each case, information on failures and survivals of a typical sample of items is used to estimate the required probabilities as a function of the item's age, this process being one of many applications of the statistical techniques known as distribution fitting. In most engineering applications, the estimation procedure must deal with samples containing survivors (suspensions or censorings); this thesis focuses on several graphical estimation methods that are widely used for analysing such samples. Although these methods have been current for many years, they share a common shortcoming: none of them is continuously sensitive to changes in the ages of the suspensions, and we show that the resulting reliability estimates are therefore more pessimistic than necessary. We use a simple example to show that the existing graphical methods take no account of any service recorded by suspensions beyond their respective previous failures, and that this behaviour is inconsistent with one's intuitive expectations. In the course of this thesis, we demonstrate that the existing methods are only justified under restricted conditions. We present several improved methods and demonstrate that each of them overcomes the problem described above, while reducing to one of the existing methods where this is justified. Each of the improved methods thus provides a realistic set of reliability estimates for general (unrestricted) censored samples. Several related variations on these improved methods are also presented and justified. - i

Asymptotic properties of stationary sequences.

Bibliography: p. 146.

Multivariate analyses of variance and covariance for simulation studies involving normal time series

Photocopy.

On the empirical eigenvalue distribution of slotted amplify-and-forward relaying protocol model

In this paper we consider the case of large cooperative communication systems where terminals use the protocol known as slotted amplify-and-forward protocol to aid the source in its transmission. Using the perturbation expansion methods of resolvents and large deviation techniques we obtain an expression for the Stieltjes transform of the asymptotic eigenvalue distribution of a sample covariance random matrix of the type HH† where H is the channel matrix of the transmission model for the transmission protocol we consider. We prove that the resulting expression is similar to the Stieltjes transform in its quadratic equation form for the Marcenko-Pastur distribution.

The beta Laplace distribution

The Laplace distribution is one of the earliest distributions in probability theory. For the first time, based on this distribution, we propose the so-called beta Laplace distribution, which extends the Laplace distribution. Various structural properties of the new distribution are derived, including expansions for its moments, moment generating function, moments of the order statistics, and so forth. We discuss maximum likelihood estimation of the model parameters and derive the observed information matrix. The usefulness of the new model is illustrated by means of a real data set. (C) 2011 Elsevier B.V. All rights reserved.

An agent-based approach to value theory and wealth distribution in economics

Any attempt to model an economy requires foundational assumptions about the relations between prices, values and the distribution of wealth. These assumptions exert a profound influence over the results of any model. Unfortunately, there are few areas in economics as vexed as the theory of value. I argue in this paper that the fundamental problem with past theories of value is that it is simply not possible to model the determination of value, the formation of prices and the distribution of income in a real economy with analytic mathematical models. All such attempts leave out crucial processes or make unrealistic assumptions which significantly affect the results. There have been two primary approaches to the theory of value. The first, associated with classical economists such as Ricardo and Marx were substance theories of value, which view value as a substance inherent in an object and which is conserved in exchange. For Marxists, the value of a commodity derives solely from the value of the labour power used to produce it - and therefore any profit is due to the exploitation of the workers. The labour theory of value has been discredited because of its assumption that labour was the only ‘factor’ that contributed to the creation of value, and because of its fundamentally circular argument. Neoclassical theorists argued that price was identical with value and was determined purely by the interaction of supply and demand. Value then, was completely subjective. Returns to labour (wages) and capital (profits) were determined solely by their marginal contribution to production, so that each factor received its just reward by definition. Problems with the neoclassical approach include assumptions concerning representative agents, perfect competition, perfect and costless information and contract enforcement, complete markets for credit and risk, aggregate production functions and infinite, smooth substitution between factors, distribution according to marginal products, firms always on the production possibility frontier and firms’ pricing decisions, ignoring money and credit, and perfectly rational agents with infinite computational capacity. Two critical areas include firstly, the underappreciated Sonnenschein-Mantel- Debreu results which showed that the foundational assumptions of the Walrasian general-equilibrium model imply arbitrary excess demand functions and therefore arbitrary equilibrium price sets. Secondly, in real economies, there is no equilibrium, only continuous change. Equilibrium is never reached because of constant changes in preferences and tastes; technological and organisational innovations; discoveries of new resources and new markets; inaccurate and evolving expectations of businesses, consumers, governments and speculators; changing demand for credit; the entry and exit of firms; the birth, learning, and death of citizens; changes in laws and government policies; imperfect information; generalized increasing returns to scale; random acts of impulse; weather and climate events; changes in disease patterns, and so on. The problem is not the use of mathematical modelling, but the kind of mathematical modelling used. Agent-based models (ABMs), objectoriented programming and greatly increased computer power however, are opening up a new frontier. Here a dynamic bargaining ABM is outlined as a basis for an alternative theory of value. A large but finite number of heterogeneous commodities and agents with differing degrees of market power are set in a spatial network. Returns to buyers and sellers are decided at each step in the value chain, and in each factor market, through the process of bargaining. Market power and its potential abuse against the poor and vulnerable are fundamental to how the bargaining dynamics play out. Ethics therefore lie at the very heart of economic analysis, the determination of prices and the distribution of wealth. The neoclassicals are right then that price is the enumeration of value at a particular time and place, but wrong to downplay the critical roles of bargaining, power and ethics in determining those same prices.

The probability archive : from essence to uncertainty in the growth of knowledge

From a ‘cultural science’ perspective, this paper traces one aspect of a more general shift, from the realist representational regime of modernity to the productive DIY systems of the internet era. It argues that collecting and archiving is transformed by this change. Modern museums – and also broadcast television – were based on determinist or ‘essence’ theory; while internet archives like YouTube (and the internet as an archive) are based on ‘probability’ theory. The paper goes through the differences between modernist ‘essence’ and postmodern ‘probability’; starting from the obvious difference that in a museum each object is selected by experts for its intrinsic properties, while on the internet you don’t know what you will find. The status of individual objects is uncertain, although the productivity of the overall archive is unlimited. The paper links these differences with changes in contemporary culture – from a Newtonian to a quantum universe, progress to risk, institutional structure to evolutionary change, objectivity to uncertainty, identity to performance. Borrowing some of its methodology from science fiction, the paper uses examples from museums and online archives, ranging from the oldest stone tool in the world to the latest tribute vid on the net.

The probability archive : essence, uncertainty, and the ‘Olduvai imperative.’

Starting from a local problem with finding an archival clip on YouTube, this paper expands to consider the nature of archives in general. It considers the technological, communicative and philosophical characteristics of archives over three historical periods: 1) Modern ‘essence archives’ – museums and galleries organised around the concept of objectivity and realism; 2) Postmodern mediation archives – broadcast TV systems, which I argue were also ‘essence archives,’ albeit a transitional form; and 3) Network or ‘probability archives’ – YouTube and the internet, which are organised around the concept of probability. The paper goes on to argue the case for introducing quantum uncertainty and other aspects of probability theory into the humanities, in order to understand the way knowledge is collected, conserved, curated and communicated in the era of the internet. It is illustrated throughout by reference to the original technological 'affordance' – the Olduvai stone chopping tool.