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We study the classical multicomponent disordered 3D spin system taking into account the temperature of the medium in the framework of the model of nearest neighbors. The latter allows the 3D problem with a cubic lattice to reduce to the 1D Heisenberg spin glass problem with a random environment. Using the Hamilton equations of motion, a recurrent equation is obtained that connects three spins in successive nodes of 1D lattice, taking into account the influence of a random environment. This equation, together with the corresponding conditions of a local minimum energy in nodes, allows to construct node-by-node a stable spin chains and, accordingly, to calculate all parameters of statistical ensemble from the first principles of classical mechanics, without using any additional assumptions, in particular, the main axiom of statistical mechanics – the equiprobability of statistical states. Using the example of 1D Heisenberg spin glass model, the features of the new approach are studied in detail and the statistical mechanics of the system are constructed without using the standard representation of the partition function (PF).

The paper shows that, for large number of particles and for distinguishable and non-interacting identical particles, convergence to equiprobability of the $W$ microstates of the famous Boltzmann-Planck entropy formula $S=k \log(W)$ is proved by the Shannon-McMillan theorem, a cornerstone of information theory. This result further strengthens the link between information theory and statistical mechanics., Comment: 5 pages

Starting from seminal neglected work by Rappeport (Rappeport 1968 Algorithms and computational procedures for the application of order statistics to queuing problems. PhD thesis, New York University), we revisit and expand on the exact algorithms to compute the distribution of the maximum, the minimum, the range and the sum of the J largest order statistics of a multinomial random vector under the hypothesis of equiprobability. Our exact results can be useful in all those situations in which the multinomial distribution plays an important role, from goodness-of-fit tests to the study of Poisson processes, with applications spanning from biostatistics to finance. We describe the algorithms, motivate their use in statistical testing and illustrate two applications. We also provide the codes and ready-to-use tables of critical values.

Starting from a neglected work by Rappeport (1968), we re-propose an exact algorithm to compute the distribution of the maximum of a multinomial random vector under the hypothesis of equiprobability. We then show how to compute the exact probabilities of the sum of the J largest order statistics of the vector, following the suggestions and correcting the errors in the same article. Finally, we introduce brand new ways of computing the exact probabilities of the multinomial minimum and of the multinomial range. The exact probabilities we derive can be used in all those situations in which the multinomial distribution plays an important role, from goodness-of-fit tests to the study of Poisson processes, with applications spanning from biostatistics to finance. For all algorithms, we provide Matlab codes and ready-to-use tables of critical values.

Summary. The winners of many lotteries are determined by selecting at random some numbered balls from an urn. This paper discusses the use of Pearson’s standard goodness-of-fit statistic to test for the equiprobability of occurrence of such lottery numbers, whether taken individually, in pairs or in larger subsets. Because the numbers are drawn without replacement, Pearson’s statistic does not follow a simple χ 2 -distribution, even for large samples of draws. In fact, it can be shown that its asymptotic distribution is that of a weighted sum of χ 2 random variables. An explicit formula is given for the weights, and this result is used to check the uniformity of winning numbers in Canada’s Lotto 6/49 over a period of nearly 20 years.

Many geological bodies or events including various types of mineral deposits, earthquakes and landslides can be represented as points on small-scale maps. Various methods exist to express probability of occurrence of such events in terms of various map patterns based on geological, geophysical and geochemical data (Agterberg 1989a). The machine-based approach was greatly facilitated by the development of Geographic Information systems (GIS, cf. Bonham-Carter 1994). Weights-of-Evidence modeling and weighted logistic regression are two powerful methods useful for estimating probabilities of occurrence of an event within a small unit area. Weights-of-Evidence (WofE) consists of first assuming that the event can occur anywhere within the study area according to a completely random Poisson distribution model. This equiprobability assumption provides the prior probability that only depends on size of an arbitrarily small unit area. Various indicator map patterns commonly reduced to binary (presence-absence) or ternary (presence-absence-unknown) form are used to update this prior probability by means of Bayes’ rule in order to create a map of posterior probabilities that is useful for selecting target areas for further exploration for undiscovered mineral deposits or for the prediction of occurrence of other discrete events such as earthquakes or landslides. If probabilities are transformed into logits, Bayes’ rule is simplified: the posterior logit simply is equal to the sum of the prior logit and the weights of which there is only one for each map layer at the same point. These weights are either positive (W+) or negative (W−) depending on presence or absence of the indicator, or zero for missing data. An important consideration in WofE applications is that the indicator patterns should be approximately conditionally independent (CI). WofE will be illustrated by applications to gold deposits in Meguma Terrain, Nova Scotia, and to flowing wells in the Greater Toronto area. Weighted logistic regression (WLR) also can be used to estimate probabilities of occurrence of discrete events. Both WofE and WLR are applied to gold occurrences in the Gowganda area on the Canadian Shield, northern Ontario, and to occurrences of hydrothermal vents on the East Pacific Rise. Indicator patterns used include favorable rock types, proximity to anticlinal structures or contacts between rock units, indices representing various geochemical elements, proximity to lineaments and igneous intrusives, aeromagnetic data, relative age, and topographic elevation. The Kolmogorov-Smirnov test is used for testing goodness of fit.

We study a symmetric generalization $\mathfrak{p}^{(N)}_k(\eta, \alpha)$ of the binomial distribution recently introduced by Bergeron et al, where $\eta \in [0,1]$ denotes the win probability, and $\alpha$ is a positive parameter. This generalization is based on $q$-exponential generating functions ($e_{q^{gen}}^z \equiv [1+(1-q^{gen})z]^{1/(1-q^{gen})};\,e_{1}^z=e^z)$ where $q^{gen}=1+1/\alpha$. The numerical calculation of the probability distribution function of the number of wins $k$, related to the number of realizations $N$, strongly approaches a discrete $q^{disc}$-Gaussian distribution, for win-loss equiprobability (i.e., $\eta=1/2$) and all values of $\alpha$. Asymptotic $N\to \infty$ distribution is in fact a $q^{att}$-Gaussian $e_{q^{att}}^{-\beta z^2}$, where $q^{att}=1-2/(\alpha-2)$ and $\beta=(2\alpha-4)$. The behavior of the scaled quantity $k/N^\gamma$ is discussed as well. For $\gamma, Comment: 13 pages, 14 figures

We investigated the statistical properties of the centre of pressure (COP) vector during quiet standing, as measured on a force platform. To derive a statistical model for the data, the COP vector, locally averaged COP, and deviations from the average COP were analysed. Whereas indefinite multimodal distributions for the components of the COP were observed, the density functions of deviation from the averaged COP had reproducible properties. The anterior-posterior and mediolateral components of the deviations had in common that they were correlated variables with a non-circular statistic. The main axes of the equiprobability density curves were rotated with respect to the natural anterior-posterior and mediolateral directions. By means of more detailed statistical analysis of the data, two distinct populations of subjects were identified. In some cases we have found Gaussian distributions of the components of deviation from the average COP. In most cases the statistics was non-Gaussian. The average contrast, a new dimensionless characteristic quantity, independent of platform calibration, was introduced as a useful indicator of non-Gaussian effects in the postural control system.

In this communication, complex systems with a near trivial dynamics are addressed. First, under the hypothesis of equiprobability in the asymptotic equilibrium, it is shown that the (hyper) planar geometry of an N-dimensional multi-agent economic system implies the exponential (Boltzmann-Gibss) wealth distribution and that the spherical geometry of a gas of particles implies the Gaussian (Maxwellian) distribution of velocities. Moreover, two non-linear models are proposed to explain the decay of these statistical systems from an out-of-equilibrium situation toward their asymptotic equilibrium states.

According to the movement of an equiprobability density ellipse in the complex plane of the speckle amplitude, a phase angle is obtained to theoretically evaluate the so-called free statistical properties of the phase difference for partially developed Gaussian speckle fields. The statistical properties of the phase difference between two points in the partially developed speckle fields have been investigated in the diffraction region. The theoretical results have been discussed with respect to Kadono's experimental results (Opt. Quantum Electron. 19 (1987) 59). The validity of this theory has been discussed.

In this paper, we discuss Bayes’ postulate and its interpretation. We extend the binomial trial method proposed by de Finetti [1] to trinomial trials, for which we argue that the consideration of equiprobability a priori for the possible outcomes of the trinomial trials implies that the parameter vector has Dirichlet(1,1) as prior. Based on this result, we agree with Stigler [2] in that the notion in Bayes’ postulate stating “absolutely know nothing” is related to the possible outcomes of an experiment and not to “non-information” about the parameter.

In this chapter, a statistical measure of complexity and the Fisher-Shannon information product are introduced and their properties are discussed. These measures are based on the interplay between the Shannon information, or a function of it, and the separation of the set of accessible states to a system from the equiprobability distribution, i.e. the disequilibrium or the Fisher information, respectively. Different applications in discrete and continuous systems are shown. Some of them are concerned with quantum systems, from prototypical systems such as the H-atom, the harmonic oscillator and the square well to other ones such as He-like ions, Hooke’s atoms or just the periodic table. In all of them, these statistical indicators show an interesting behavior able to discern and highlight some conformational properties of those systems.

Limiting dilution assays (LDA) are used to estimate an unknown cell fraction of interest within a sample. This paper discusses a method for designing an LDA using the distribution of the cell fraction of interest, examining three different design approaches: geometric progression, equally spaced log, and equiprobability. Two common estimation methods, minimum chi-square and maximum likelihood, also are investigated. These designs and estimation methods, coupled with varying numbers of wells per dilution and dilutions per design, are compared quantitatively through computer simulation. Performance measures computed were mean relative bias and mean squared error.

The distribution of the chi-square goodness-of-fit statistic is studied in the equiprobable case. Tables of exact critical values are given for a = .1, .05, .01, .005; k = 2(1)4, N = 26(1)50; k = 5, N = 26(1)40; k = 6(1)10, N = 26(1)30, where a is the desired significance level, k is the number of cells and N is the sample size. Methods of fitting the true distribution are compared. If k> 3, it is found that a simple additive adjustment to the asymptotic chi-square fit leads to high accuracy even for N between 10 and 20. For k = 2, the Yates corrected chi-square statistic is very accurately fitted by the usual chi-square distribution.

Boltzmann's hypothesis of the equiprobability of the various ways of combining microslates (giving rise to the different configurations of a system) is erroneous. This renders untenable the ergodic hypothesis according to which the configurations that prime the order in an isolated system are reached randomly and points to the need to postulate the existence of a hidden variable in order to explain the system's shift away from its state of maximum entropy.

This study presents an expository geometric derivation of the Pareto distribution with applications to conflict behavior. In contrast to typical stochastic models which assume equiprobability of access to scarce resources, the derivation here assumes sequential arrivals of colonists or other resource acquisitors. A log-exponential distribution is derived which turns out to be identical to the Pareto law of income distribution, the derivation here differing from others in its emphasis on sequential arrivals and resource scarcity. This model is then applied to the upper tail of land distribution in El Salvador and the distribution of colonial populations in 1914. In both cases, the log-exponential (Pareto) distribution can be accepted while the exponential itself as a plausible theoretical rival is rejected. Invariance properties of the Pareto distribution suggest reasons why political violence is likely if the model holds throughout the range of resource holdings, as was the case in 1914.

Measurements of the phase-quadrature components of the microwave field forward-scattered from random water waves have continued since the observance in 1963 of non-Rayleigh probability distributions of the incoherent scattered field. Measurements of the variances of the phase-quadrature components of the incoherent field at various phase angles with respect to the coherent field have yielded new results not explained by plane-wave theory: 1) The equiprobability ellipse of the phase-quadrature components of the incoherent field rotates us a function of apparent surface roughness. 2) The rotation of the equiprobability ellipse varies more rapidly with roughness for "narrow" surface illumination and less rapidly for "broad" illumination. 3) The greatest ratio of the two variances appears to rise from unity with increasing roughness to a peak value (maximum departure from Rayleigh), and presumably decreases at greater roughnesses. The data indicate that nonplanar illuminating and receiving wavefronts may be responsible for these divergences from plane-wave theory.

The general equations for the propagation of error with distance traveled for the case of fix-corrected Doppler navigation along a path which consists of one or more rhumb-line legs are derived and shown to consist of a combination of six types of error variances. The specific expressions when the position-finding system used for fix correction is either a rho-theta (i.e., VORTAC) or a hyperbolic (i.e., Loran) type are given. The expressions for the resulting error spreads in terms of RMS or arbitrary x and y coponents are shown to be straightforward and easy to apply in most cases. The exact expressions for the size and spatial orientation of the elliptical equiprobability error contours are shown to be more complex and more difficult to apply practically. However, it is pointed out that the RMS error specification method is easily applicable in all of the practical cases considered. The use of the RMS error circle approximation is always safe although in some cases it may lead to ?overdesign.?

A step-by-step method is presented herein for partitioning a certain kind of hypothesis H about the m-way contingency table into (a) a series of hypotheses about marginal tables formed from the m-way table by ignoring one or more of table's m dimensions; and (b) a hypothesis about independence, conditional independence, or conditional equiprobability in the m-way table. This step-by-step method facilitates both the testing of H and the calculation of , the estimated expected frequencies in the m-way table under H. The method introduced herein for calculating is easier to apply than the usual iterative-scaling method in many cases.

A probabilistic analysis is presented of certain pointer-based implementations of dictionaries, linear lists, and priority queues; in particular, simple list and d-heap implementations. Under the assumption of equiprobability of histories, i.e., of paths through the internal state space of the implementation, it is shown that the integrated space and time costs of a sequence of n supported operations converge as n → ∞ to Gaussian random variables. For list implementations the mean integrated spatial costs grow asymptotically as n2, and the standard deviations of the costs as n 3 2 . For d-heap implementations of priority queues the mean integrated space cost grows only as n2√log n, i.e., more slowly than the worst-case integrated cost. The standard deviation grows as n 3 2 . These asymptotic growth rates reflect the convergence as n → ∞ of the normalized structure sizes to datatype-dependent deterministic functions of time, as earlier discovered by Louchard. This phenomenon is clarified with the aid of path integrals. Path integral techniques, drawn from physics, greatly facilitate the computation of the cost asymptotics. This is their first application to the analysis of dynamic data structures.

Publisher Summary The analysis of data by log-linear models involves several distinct stages. In the first stage, a plausible model is proposed for the data under study. In the second stage, unknown parameters in the model are estimated from the data, generally by the method of maximum likelihood. This methods yields estimates available in closed form, as in the equiprobability model, or estimates computed by a version of the Newton–Raphson algorithm with analogies to weighted regression analysis, as in the log-linear time-trend model. In the third stage, these parameter estimates are used in the statistical tests of the model's adequacy. More specific insight into deviations between model and data are provided by the analysis of adjusted residuals and of selected linear combinations of frequencies. This chapter discusses maximum-likelihood equations and the Newton–Raphson algorithm for general log-linear models for both multinominal and Poisson data. The general analogies with weighted regression analysis are used to describe the Newton-Raphson algorithm and to develop large-sample approximations to the distributions of parameter estimates, residuals, and chi-square statistics.

A sample $(n_1, n_2,\cdots, n_t)$ is taken from a $t$-category multinomial population. The hypothesis of equiprobability, that the $t$ physical probabilities associated with the cells are all equal to $1/t$, is called the null hypothesis. Conditional on the non-null hypothesis, a symmetric Dirichlet prior of parameter $k$ is assumed $(k \geqq 0)$ and the Bayes factor against the null hypothesis, with this assumption, is denoted by $F(k)$. A conjecture made in 1965 is almost proved, namely that $F(k)$ has a unique local maximum and that this occurs for a finite value of $k$ if and only if Pearson's $X^2$ exceeds its number of degrees of freedom. The result is required for the calculation of $\max F(k)$, which provides a non-Bayesian significance criterion whose simple asymptotic distribution is good even in the extreme tail, and even for sample sizes less than $t$. This criterion arose from an attitude involving a Bayes/non-Bayes compromise.

Characteristics of the phase of Gaussian speckles are studied theoretically in both image and diffraction fields. The general analysis for these studies is given by introducing an "optical system function," and the results are applied to the analysis of the speckle phases in both fields. The speckle phase distributions are evaluated by a phase angle defined as the extent of the equiprobability density ellipse stretched from the origin in the complex plane of the speckle amplitude. In addition, several peculiar properties of the equiprobability density ellipse are summarized.