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Volume 30 - Issue 1 - June 2005
• A brief history of The Mathematical Scientist, 1976–2004
• Markov and life insurance
This article outlines Markov's involvement with life insurance, particularly in the areas of retirement funds and juvenile insurance.
• In defence of the reverse gambler's belief
B. O'Neill, B. D. Puza
We investigate the problem of predicting the outcomes of a sequence of discrete random variables that are almost uniform, in the sense that they are generated from a random process that is designed to produce independent uniform outcomes, but may not do so exactly. Using assumptions based around this notion, we derive a useful stochastic ordering for prediction. This leads us to reject the gambler's belief as unsound and conclude that the reverse gambler's belief is the optimal prediction method.
• Cooperative effects in multi-server queueing systems
We discuss the cooperative effect of n single-server queues aggregated into a single queueing system. The performances of two such systems, with and without competition, are compared.
• A dynamical, self-thinning approach to scaling vegetation systems
Barnes and Roderick (2004) proposed a model incorporating a self-thinning mechanism to scale between individual plants and systems of vegetation. Their generic method took a dynamical systems approach through a mass balance formulation, which facilitated theoretical analysis while maintaining simplicity for ease of application. The present paper provides an overview of this model, as well as an outline of its application and comparison with datasets. The framework embodies empirically observed characteristics of ecosystems, such as those of competing species and the intermediate disturbance hypothesis, and it also accommodates the interactions between plant components, providing a means of scaling up plant partitioning.
• Another posterior paradox
B. D. Puza, K. R. W. Brewer, T. J. O'Neill
An inference problem is presented in which the only datum is the sum of n independent and identically distributed normal random variables with mean μ, and where n itself has a known probability distribution. The choice of a flat prior for μ over the whole real line, or even over a long finite interval, leads to a posterior for n which favours small values of that parameter. Consequently, the estimation of μ is biased upwards. This paradox is resolved by adopting a proper prior for μ that approximates f(μ) / 1 / |μ| in its middle ranges. The chosen prior has some empirical support from Benford's law of numbers. The connection with this law is discussed, as is the choice of a suitable prior in a number of related situations. This article warns of the possible dangers in blindly using a flat prior distribution to represent a priori ignorance.
• On the linear combination and ratio of Laplace random variables
Saralees Nadarajah, Samuel Kotz
The distributions of the linear combination αX + βY and the ratio | X / Y | are derived, where X and Y are independent Laplace random variables.
• Moment properties of some time series models
S. S. Appadoo, M. Ghahramani, A. Thavaneswaran
An ARMA representation was used in Thavaneswaran, Appadoo and Samanta (2005) to derive the kurtosis of various classes of time series models. In this paper, moment properties of various random coefficient autoregressive models are considered. It is shown that random coefficient autoregressive models have larger kurtosis than the usual linear time series models.