The Fourier-series method for inverting transforms of probability distributions

This paper reviews the Fourier-series method for calculating cumulative distribution functions (cdf's) and probability mass functions (pmf's) by numerically inverting characteristic functions, Laplace transforms and generating functions. Some variants of the Fourier-series method are remarkably easy to use, requiring programs of less than fifty lines. The Fourier-series method can be interpreted as numerically integrating a standard inversion integral by means of the trapezoidal rule. The same formula is obtained by using the Fourier series of an associated periodic function constructed by aliasing; this explains the name of the method. This Fourier analysis applies to the inversion problem because the Fourier coefficients are just values of the transform. The mathematical centerpiece of the Fourier-series method is the Poisson summation formula, which identifies the discretization error associated with the trapezoidal rule and thus helps bound it. The greatest difficulty is approximately calculating the infinite series obtained from the inversion integral. Within this framework, lattice cdf's can be calculated from generating functions by finite sums without truncation. For other cdf's, an appropriate truncation of the infinite series can be determined from the transform based on estimates or bounds. For Laplace transforms, the numerical integration can be made to produce a nearly alternating series, so that the convergence can be accelerated by techniques such as Euler summation. Alternatively, the cdf can be perturbed slightly by convolution smoothing or windowing to produce a truncation error bound independent of the original cdf. Although error bounds can be determined, an effective approach is to use two different methods without elaborate error analysis. For this purpose, we also describe two methods for inverting Laplace transforms based on the Post-Widder inversion formula. The overall procedure is illustrated by several queueing examples.     

A matrix perturbation method for computing the steady-state probability distributions of probabilistic Boolean networks with gene perturbations

Modeling genetic regulatory interactions is an important issue in systems biology. Probabilistic Boolean networks (PBNs) have been proved to be a useful tool for the task. The steady-state probability distribution of a PBN gives important information about the captured genetic network. The computation of the steady-state probability distribution involves the construction of the transition probability matrix of the PBN. The size of the transition probability matrix is 2ⁿ×2ⁿ where n is the number of genes. Although given the number of genes and the perturbation probability in a perturbed PBN, the perturbation matrix is the same for different PBNs, the storage requirement for this matrix is huge if the number of genes is large. Thus an important issue is developing computational methods from the perturbation point of view. In this paper, we analyze and estimate the steady-state probability distribution of a PBN with gene perturbations. We first analyze the perturbation matrix. We then give a perturbation matrix analysis for the captured PBN problem and propose a method for computing the steady-state probability distribution. An approximation method with error analysis is then given for further reducing the computational complexity. Numerical experiments are given to demonstrate the efficiency of the proposed methods.     

A characterization of probability distributions by mean values

Let (X) be a measurable complex function onR;X, Y, Z be i.i.d. random variables; and (t, u, v)=E(tX+uY+nZ), wheret, u, vR. In this paper we describe a class of function (x) such that the distribution ofX, Y, Z is determined by the funetion (t, u, v). The main result is a generalization of the author's characterization of normal and stable distributions.     

Some general divergence measures for probability distributions

Summary General divergence measures for probability distributions are introduced and their main properties established. Connections with f-divergence corresponding to a convex function fare explored.     

Decompositions of multivariate probability distributions

The divisors of multivariate probability distributions are considered that are decreasing at infinity not more slowly than normal distributions and that satisfy various symmetry conditions (in particular, the condition of spherical symmetry).Translated from Problemy Ustoichivosti Stokhasticheskikh Modelei — Trudy Seminara, pp. 72–75, 1980.The author expresses her gratitude to A. A. Zinger with whom she had a discussion during which these problems arose.     

Asymptotic quantization of probability distributions

We give a brief introduction to results on the asymptotics of quantizatlon errors. The topics discussed in-clude the quantization dimension, asymptotic distributions of sets of prototypes, asymptotically optimalquantizations, approximations and random quantizations.     

Quantifying expert opinion for modelling fauna habitat distributions

Summary  The practical elicitation of expert beliefs about logistic regression models is considered. An experiment is reported in which ecologists quantified their prior beliefs about the relationship between various environmental attributes and the habitat distribution of certain rare and endangered fauna. Prior distributions were elicited from the ecologists and combined with sample data to form posterior distributions. The elicitation method was proposed by Garthwaite and Al-Awadhi (2004) and is implemented through an interactive graphical computer program. Classical stepwise logistic regression and alternative forms of prior distribution are compared using cross validation. Data on the environmental attributes have been mapped and stored in a GIS database and the posterior distributions can be used to predict the probability of a species' presence/absence at any site in the database.     

Stationary probability distributions for multiresponse linear learning models

A necessary and sufficient condition is given for the existence of stationary probability distributions of a non-Markovian model with linear transition rule. Similar to the Markovian case, stationary probability distributions are characterized as eigenvectors of nonnegative matrices. The model studied includes as special cases the Markovian model as well as the linear learning model and has applications in psychological and biological research, in control theory, and in adaption theory.     

Subset selection procedures for restricted families of probability distributions

In this paper we are interested in studying multiple decision procedures fork (k≧2) populations which are themselves unknown but which one assumed to belong to a restricted family. We propose to study a selection procedure for distributions associated with these populations which are convex-ordered with respect to a specified distributionG assuming that there exists a best one. The procedure described here is based on a statistic which is a linear function of the firstr order statistics and which reduces to the total life statistics whenG is exponential. The infimum of the probability of a correct selection and an asymptotic expression for this probability are obtained using the subset selection approach. Some other properties of this procedure are discussed. Asymptotic relative efficiencies of this rule with respect to some selection procedures proposed by Barlow and Gupta [3] for the star-ordered distributions and by Gupta [8] for the gamma populations with known shape parameters are obtained. A selection procedure for selecting the best population using the indifference zone approach is also studied. This research was supported by the Office of Naval Research Contract N00014-75-C-0455 at Purdue University. Reproduction in whole or in part is permitted for any purpose of the United States Government. Ming-Wei Lu is now at the Department of Vital and Health Statistics, Michigan.     

On a class of probability distributions

The application of a three parameter class of one-sided probability distributions is being discussed. For specific parameter values, this class contains as special cases a number of well-known distributions of statistics and statistical physics, namely, Gauss, Weibull, exponential, Rayleigh, Gamma, chi-square, Maxwell, and Wien (limiting case of Planck's distribution). One of the three parameters represents scale; the other two represent initial and terminal shape of the associated probability density function. A fourth parameter, shift, may be introduced. The distribution class discussed in this paper was introduced by L. Amoroso [2] in 1924. It is closely connected with a family of linear Fokker-Planck equations (generalized Feller equation). In fact, the class of probability density functions associated with the distribution class considered here is a special case of the set of all delta function initial condition solutions of the generalized Feller equation for a fixed value of the time variable. It will be shown that, as a function of the logarithm of the independent variable, the logarithm of the cumulative distribution function is asymptotically linear as the independent variable approaches zero from above. This fact leads to a general criterion for the applicability of the presented distribution family relative to given empirical data. The applicability criterion can be used to determine approximate values for the two shape parameters. They can subsequently be used as initial values in any of the established parameter estimation techniques.