Showing papers on "Symmetric probability distribution published in 1998"

Influence Function and Efficiency of the Minimum Covariance Determinant Scatter MAtrix Estimator

Abstract: The minimum Covariance Determinant (MCD) scatter estimator is a highly robust estimator for the dispersion matrix of a multivariate, elliptically symmetric distribution. It is fast to compute and intuitively appealing. In this note we derive its influence function and compute the asymptotic variances of its elements. A comparison with the one step reweighted MCD and with S-estimators is made. Also finite-sample results are reported.

Ruin probabilities for Erlang(2) risk processes

Abstract: In this paper we consider a risk process in which claim inter-arrival times have an Erlang(2) distribution. We consider the infinite time survival probability as a compound geometric random variable and give expressions from which both the survival probability from initial surplus zero and the ladder height distribution can be calculated. We consider explicit solutions for the survival/ruin probability in the case where the individual claim amount distribution is phase-type, and show how the survival/ruin probability can be calculated for other individual claim amount distributions.

A Class of Quantile Measures for Kurtosis

Abstract: Several authors have concluded that the standardized fourth central moment of a symmetric distribution is not a good measure of the shape of a distribution. Here we consider the properties of a class of parameters of a distribution based on its percentiles, as alternative measures of kurtosis. It is shown that this scale and location invariant measure maintains the symmetric ordering of van Zwet. Influence functions are used to show how this measure reflects the kurtosis of a distribution. Results of a simulation study indicate that the power of the Shapiro-Wilk test, for a large number of symmetric distributions alternative to the normal distribution, is almost linear as a function of appropriate functionals in this class. This suggests the use of this functional as a kurtosis measure.

Model-based probability density function estimation

Abstract: Noting that the probability density function of a continuous random variable has similar properties to a power spectral density, a new class of probability density function estimators is described. The specific model examined is the autoregressive model, although the extension to other time series models is evident. An example is given to illustrate the approach.

Spin States and Probability Distribution Functions

Abstract: Formulation of the conventional quantum mechanics in which a state is described by probability instead of wave function and density matrix is presented. We consider the possibility of constructing the invertable map of spinors onto positive probability distributions. For any value of spin, the basis of the irreducible representation of a rotation group is realized by a family of probability distributions of the spin projection parametrized by points on a sphere. Quantum states of a symmetric top described by the probability distributions are discussed.

Probability density of the determinant of a random Hermitian matrix

Abstract: The probability density function for the determinant of a nn random Hermitian matrix taken from the Gaussian unitary ensemble is calculated. It is found to be a Meijer G- function or a linear combination of two Meijer G-functions, depending on the parity of n. The integer moments of this probability density are also given.

Point processes and the infinite symmetric group. Part I: The general formalism and the density function

Abstract: We study a 2-parametric family of probability measures on an infinite-dimensional simplex (the Thoma simplex). These measures originate in harmonic analysis on the infinite symmetric group (S.Kerov, G.Olshanski and A.Vershik, Comptes Rendus Acad. Sci. Paris I 316 (1993), 773-778). Our approach is to interpret them as probability distributions on a space of point configurations, i.e., as certain point stochastic processes, and to find the correlation functions of these processes. In the present paper we relate the correlation functions to the solutions of certain multidimensional moment problems. Then we calculate the first correlation function which leads to a conclusion about the support of the initial measures. In the appendix, we discuss a parallel but more elementary theory related to the well-known Poisson-Dirichlet distribution. The higher correlation functions are explicitly calculated in the subsequent paper (A.Borodin, math.RT/9804087). In the third part (A.Borodin and G.Olshanski, math.RT/9804088) we discuss some applications and relationships with the random matrix theory. The goal of our work is to understand new phenomena in noncommutative harmonic analysis which arise when the irreducible representations depend on countably many continuous parameters.

Exact results on Sinai's diffusion

Abstract: We study the continuum version of Sinai's problem of a random walker in a random force field in one dimension. A method of stochastic representation is used to represent various probability distributions in this problem (mean probability density function and first passage time distributions). This method reproduces already known rigorous results and also confirms directly some recent results derived using approximation schemes. We demonstrate clearly, in the Sinai scaling regime, that the disorder dominates the problem and that the thermal distributions tend to zero-one laws.

Statistical properties of localisation--delocalisation transition in one dimension

Abstract: We study a one-dimensional model of disordered electrons (also relevant for random spin chains), which exhibits a delocalisation transition at half-filling. Exact probability distribution functions for the Wigner time and transmission coefficient are calculated. We identify and distinguish those features of probability densities that are due to rare, trapping configurations of the random potential from those which are due to the proximity to the delocalisation transition.

Convergence of simulated annealing using the generalized transition probability

Abstract: We prove weak ergodicity of the inhomogeneous Markov process generated by the generalized transition probability of Tsallis and Stariolo under power-law decay of the temperature. We thus have a mathematical foundation to conjecture convergence of simulated annealing processes with the generalized transition probability to the minimum of the cost function. An explicitly solvable example in one dimension is analyzed in which the generalized transition probability leads to a fast convergence of the cost function to the optimal value. We also investigate how far our arguments depend upon the specific form of the generalized transition probability proposed by Tsallis and Stariolo. It is shown that a few requirements on analyticity of the transition probability are sufficient to assure fast convergence in the case of the solvable model in one dimension.

Describing spinors using probability distribution functions

Abstract: Irreducible representations of the rotation group are realized using a family of positive probability distributions of the spin projections for an arbitrary value of the spin. The family is parametrized by the points on the sphere. An invertible mapping of the spinors onto the probability distribution functions is constructed. Examples of probability distributions for the well-known states with the spins 1/2 and 1 are presented.

On Maximum Non-averaged Differential Probability

Abstract: Maximum average of differential probability is one of the security measures used to evaluate block ciphers such as the MISTY cipher. Here average means the average for all keys. Thus, there are keys which yield larger maximum differential probability even if the maximum average of differential probability is sufficiently small. This paper presents the cases in which the maximum differential probability is larger than the maximum average of differential probability for some keys, and we try to determine the maximum differential probability considering the key effect.

Alternating kinetics of annihilating random walks near a free interface

Abstract: The kinetics of annihilating random walks in one dimension, with the half-line x>0 initially filled, is investigated. The survival probability of the nth particle from the interface exhibits power-law decay, , with for n = 1 and all odd values of n; for all n even, a faster decay with is observed. From consideration of the eventual survival probability in a finite cluster of particles, the rigorous bound is derived, while a heuristic argument gives . Numerically, this latter value appears to be a lower bound for . The average position of the first particle moves to the right approximately as , with a relatively sharp and asymmetric probability distribution.

Bayesian Probability and Scalar Performance Measures in Gaussian Models

Abstract: The transformation of a real, continuous variable into an event probability is reviewed from the Bayesian point of view, after which a Gaussian model is employed to derive an explicit expression for the probability. In turn, several scalar (one-dimensional) measures of performance quality and reliability diagrams are computed. It is shown that if the optimization of scalar measures is of concern, then prior probabilities must be treated carefully, whereas no special care is required for reliability diagrams. Specifically, since a scalar measure gauges only one component of performance quality—a multidimensional entity—it is possible to find the critical value of prior probability that optimizes that scalar measure; this value of “prior probability” is often not equal to the “true” value as estimated from group sample sizes. Optimum reliability, however, is obtained when prior probability is equal to the estimate based on group sample sizes. Exact results are presented for the critical value of “p...

p-adic stochastic hidden variable model

Abstract: We propose stochastic hidden variables model in which hidden variables have a p-adic probability distribution ρ(λ) and at the same time conditional probabilistic distributions P(U,λ), U=A,A′,B,B′, are ordinary probabilities defined on the basis of the Kolmogorov measure-theoretical axiomatics. A frequency definition of p-adic probability is quite similar to the ordinary frequency definition of probability. p-adic frequency probability is defined as the limit of relative frequencies νn but in the p-adic metric. We study a model with p-adic stochastics on the level of the hidden variables description. But, of course, responses of macroapparatuses have to be described by ordinary stochastics. Thus our model describes a mixture of p-adic stochastics of the microworld and ordinary stochastics of macroapparatuses. In this model probabilities for physical observables are the ordinary probabilities. At the same time Bell’s inequality is violated.

Filtered-Variate Prior Distributions for Histogram Smoothing

Abstract: We develop prior distributions for histogram inference favoring smooth population frequencies; that is, probability vectors with small differences for neighboring categories. We give a theory of prior-random probability vectors representable as a linear transform, or “filter,” of a standard random probability vector, or equivalently, a random weighted average of nonrandom smooth probability vectors. Promising methods of prior assessment are given based on elicitation of a list of typically smooth probability vectors, the empirical moments of which can then be matched by the mean vector and variance matrix of a constructed continuous-type filtered-variate prior distribution.

Generalization of a maximum-likelihood approach to blind source separation

Abstract: In the two-source two-sensor blind source separation scenario, only an orthogonal transformation remains to be disclosed once the observations have been whitened. In order to estimate this matrix, a maximum-likelihood (ML) approach has been suggested in the literature, which is only valid for sources with the same symmetric distribution and kurtosis values lying in certain positive range. In the present contribution, the expression for this ML estimator is reviewed and generalized to include almost any source distribution.

Average-ion level-population correlations in off-equilibrium plasmas

Abstract: We present techniques to calculate statistical distributions in time-dependent off-equilibrium atomic physics. Starting from the master equation, a first method assumes a Gaussian distribution and deduces the time evolution of the means and correlations. Alternatively, the discrete probability distribution is written as the product of the known statistical factor with the exponential of an unknown function. This function of the electronic populations can then be fitted using a second-order polynomial. Another method sets up a continuous version of the master equation, then expands the probability around the most probable configuration. It is remarkable that the obtained equation set is the same as in the Gaussian approximation of the first method. A major property of all these models is that they recover the probability distribution of thermodynamical equilibrium, when external conditions make it possible. Numerical tests on a two-level system are presented. @S1063-651X~98!05701-8#

Estimating probability densities from short samples: A parametric maximum likelihood approach

Abstract: A parametric method similar to autoregressive spectral estimators is proposed to determine the probability density function (pdf) of a random set. The method proceeds by maximizing the likelihood of the pdf, yielding estimates that perform equally well in the tails as in the bulk of the distribution. It is therefore well suited for the analysis short sets drawn from smooth pdfs and stands out by the simplicity of its computational scheme. Its advantages and limitations are discussed.

Partly divisible probability distributions

Abstract: Given a probability distribution µ a set Λ(µ) of positive real numbersis introduced, so that Λ(µ) measures the ”divisibility” of µ. The basicproperties of Λ(µ) are described and examples of probability distributionsare given, which exhibit the existence of a continuum of situations inter-polating the extreme cases of infinitely and minimally divisible probabilitydistributions. Mathematics Subject Classification (1991): Primary 42A85;Secondary 60B15, 60E10Key Words and Phrases: admissible probability distributions, convolutionsemigroups, divisibility, Fourier transforms. Introduction In probability theory it is often interesting to know whether a random variableXcan be decomposed into a sum of nindependent identical distributed randomvariablesX= X 1 +··· +X n .In terms of the probability distribution µof Xthe above equation meansthatµ= µ 1n ∗···∗µ 1 | {z n } n times , (1)where µ 1n is the probability distribution of X 1 .The notion of infinitely divisible probability distribution characterizes a spe-cial case of such phenomena, where the process of dividing a probability distri-bution µinto n-fold convolutions of probability distributions µ

Absolute Gralniness Thresholds and Linear Probability Models.

Abstract: This paper describes an experiment to determine t absolute graininess (GS) threshold of uniform (solid) ar images. The psychometric experiment used a variation the Method of Constant Stimuli where the observers sort t stimuli (samples) depending on their ability to “see” o detect graininess. Since graininess is composed of at le two Physical Image Parameters, (PIP), the lightness opti density curve for the Human Visual System (HVS) and th Wiener Spectrum, a Visual Algorithm is used to specify th stimuli GS. This is a variation on the classical method absolute threshold where the stimuli are specified in term of a single PIP. Several psychometric models are discus and methods to fit the experimental data are describ Finally, a linear probability model is used to determine th absolute GS threshold in terms of a Density-Wiener Spec space. Introduction the Image Quality Circle Image quality and its components is a complex problem th are still active research topics. To simplify the unde standing of image quality we use a step-by-step approa called the "Image Quality Circle" (1) (IQC), which is shown in Fig. 1. Figure 1: The Image Quality Circle The goal of an imaging system designer is to relate t technology variables of the imaging system to the qual preferences of the customer. Figure 1 shows this fund mental objective via the large arrow. The link betwee customer quality preference and the imaging system a 23 he ea of he r ast cal e e of s sed ed. e tra

Fuzzy modelling of measurements acquired by an intelligent ultrasonic telemeter

Abstract: The measurement uncertainty in physical sensors is often represented by a probabilistic statistical approach, but such a representation is not always adapted to the treatment of information and a fuzzy representation, based on possibility theory, can sometimes be preferred. That is why we have proposed a truncated triangular probability-possibility transformation, to be applied to the four most encountered symmetric probability laws, and to any unimodal and symmetric probability distribution which can be assimilated to one of these four probability laws. In this paper, we propose to build a fuzzy model of measurement sensors by applying this transformation. For this, a minimum of knowledge about the probabilistic modeling of sensors is required. In fact, according to the measurement context of the sensor considered, a more or less spread probabilistic modeling of the measurement will be available. In this paper, three main situations will be considered and for each situation, an adapted fuzzy model will be proposed. Finally, an application to distance measurements provided by two ultrasonic sensors, a pulse telemeter and a frequency modulated telemeter, is presented.

Inequalities Between the Second and Fourth Moments

Abstract: Let σ 2 be the variance and μ 4 the fourth moment of a symmetric probability distribution. We will prove that for distributions with non-negative characteristic function the inequality μ 4 ≥ 2σ 4 holds and that μ 4 − 2σ 4 if and only if the characteristic function f is given by f(x) = cos2(ax). for some . For symmetric unimodal distributions we have μ 4 ≥ (9/5)σ 4 and μ 4 = (9/5)σ 4 if and only if the characteristic function f is given by f(x) = (sin(ax))/ax, for some . The products of variances of adjoint positive definite densities have a greatest lower bound A. There is a self-adjoint distribution such that σ 4 = Λ. We will prove that for such distributions the equality μ 4 ≤ 2 + σ 4 holds.

Schroedinger's interpolation problem through Feynman-Kac kernels

Abstract: We discuss the so-called Schr{\"o}dinger problem of deducing the microscopic (basically stochastic) evolution that is consistent with given positive boundary probability densities for a process covering a finite fixed time interval. The sought for dynamics may preserve the probability measure or induce its evolution, and is known to be uniquely reproducible, if the Markov property is required. Feynman-Kac type kernels are the principal ingredients of the solution and determine the transition probability density of the corresponding stochastic process. The result applies to a large variety of nonequilibrium statistical physics and quantum situations.

Density of the transition probability for one class of generalized diffusion processes

Abstract: The existence of the density of the transition probability is investigated for a generalized diffusion process with transport that satisfies a certain condition of integrability with respect to the Gaussian measure.