Probability <Emphasis Type="Italic">Distributome</Emphasis>: a web computational infrastructure for exploring the properties,interrelations, and applications of probability distributions

Probability distributions are useful for modeling, simulation, analysis, and inference on varieties of natural processes and physical phenomena. There are uncountably many probability distributions. However, a few dozen families of distributions are commonly defined and are frequently used in practice for problem solving, experimental applications, and theoretical studies. In this paper, we present a new computational and graphical infrastructure, the Distributome, which facilitates the discovery, exploration and application of diverse spectra of probability distributions. The extensible Distributome infrastructure provides interfaces for (human and machine) traversal, search, and navigation of all common probability distributions. It also enables distribution modeling, applications, investigation of inter-distribution relations, as well as their analytical representations and computational utilization. The entire Distributome framework is designed and implemented as an open-source, community-built, and Internet-accessible infrastructure. It is portable, extensible and compatible with HTML5 and Web2.0 standards (http://Distributome.org). We demonstrate two types of applications of the probability Distributome resources: computational research and science education. The Distributome tools may be employed to address five complementary computational modeling applications (simulation, data analysis and inference, model-fitting, examination of the analytical, mathematical and computational properties of specific probability distributions, and exploration of the inter-distributional relations). Many high school and college science, technology, engineering and mathematics (STEM) courses may be enriched by the use of modern pedagogical approaches and technology-enhanced methods. The Distributome resources provide enhancements for blended STEM education by improving student motivation, augmenting the classical curriculum with interactive webapps, and overhauling the learning assessment protocols.     

Tailweight, quantiles and kurtosis: A study of competing distributions

This paper studies analytically and numerically the tail behavior of the symmetric variance-gamma (VG), t, and exponential-power (EP) distributions. Special emphasis is on the VG, which is a direct competitor of the t in the financial context of modeling the distribution of log-price increments.     

The Transform Likelihood Ratio Method for Rare Event Simulation with Heavy Tails

We present a novel method, called the transform likelihood ratio (TLR) method, for estimation of rare event probabilities with heavy-tailed distributions. Via a simple transformation (change of variables) technique the TLR method reduces the original rare event probability estimation with heavy tail distributions to an equivalent one with light tail distributions. Once this transformation has been established we estimate the rare event probability via importance sampling, using the classical exponential change of measure or the standard likelihood ratio change of measure. In the latter case the importance sampling distribution is chosen from the same parametric family as the transformed distribution. We estimate the optimal parameter vector of the importance sampling distribution using the cross-entropy method. We prove the polynomial complexity of the TLR method for certain heavy-tailed models and demonstrate numerically its high efficiency for various heavy-tailed models previously thought to be intractable. We also show that the TLR method can be viewed as a universal tool in the sense that not only it provides a unified view for heavy-tailed simulation but also can be efficiently used in simulation with light-tailed distributions. We present extensive simulation results which support the efficiency of the TLR method.     

Optimizing fuzzy p-hub center problem with generalized value-at-risk criterion

In this study, we reduce the uncertainty embedded in secondary possibility distribution of a type-2 fuzzy variable by fuzzy integral, and apply the proposed reduction method to p-hub center problem, which is a nonlinear optimization problem due to the existence of integer decision variables. In order to optimize p-hub center problem, this paper develops a robust optimization method to describe travel times by employing parametric possibility distributions. We first derive the parametric possibility distributions of reduced fuzzy variables. After that, we apply the reduction methods to p-hub center problem and develop a new generalized value-at-risk (VaR) p-hub center problem, in which the travel times are characterized by parametric possibility distributions. Under mild assumptions, we turn the original fuzzy p-hub center problem into its equivalent parametric mixed-integer programming problems. So, we can solve the equivalent parametric mixed-integer programming problems by general-purpose optimization software. Finally, some numerical experiments are performed to demonstrate the new modeling idea and the efficiency of the proposed solution methods.     

Structure of Stieltjes classes of moment-equivalent probability laws

A Stieltjes class is a one-parameter family of moment-equivalent distribution functions constructed by modulation of a given indeterminate distribution function F, called the center of the class. Members of a Stieltjes class are mutually absolutely continuous, and conversely, any pair of moment-equivalent and mutually absolutely continuous distribution functions can be joined by a Stieltjes class. The center of a Stieltjes class is an equally weighted mixture of its extreme members, and this places restrictions on which distributions can belong to a Stieltjes class with a given center. The lognormal law provides interesting illustrations of the general ideas. In particular, it is possible for two moment equivalent infinitely divisible distributions to be joined by a Stieltjes class, and random scaling can be used to construct new Stieltjes classes from a given Stieltjes class.     

A transformed geometric distribution in stochastic modelling

The analytical concepts of infinite divisibility and (0) unimodality are fundamental to the study of probability distributions in general and to discrete distributions in particular. In this paper, a one-one correspondence is established between these two important properties which will permit any infinitely divisible discrete distribution (with finite mean value) to be transformed into a (0) unimodal discrete distribution. When this transformation is applied specifically to the geometric distribution, the result is a novel distribution, which can be fully and explicitly specified and whose factorial moments can be expressed in closed forms. This transformed geometric distribution is found to apply to underreported geometrically distributed decision processes, embedded renewal processes with logarithmically distributed components, and M/M/1 queues in which the service mechanism has been uniformly improved.     

Prediction of the evolution of the dispersed phase in bubbly flow problems

For modeling multi-phase where the dispersed phase plays a major role in determining the flow structure and inter phase transfer quantities, the size distribution of the bubbles has to be considered. This can be done by extension of the mass balance equation to a population balance equation. In this work, a least squares spectral method is tested for predicting the evolution of the dispersed phase in a vertical two-phase bubbly flow. The least squares spectral method consists in minimizing the L² norm of the residual over the simulation domain. The results are compared with experimental data obtained for two different initial bubble distributions.     

A new approach for parameter estimation of finite Weibull mixture distributions for reliability modeling

The aim of this paper is to model lifetime data for systems that have failure modes by using the finite mixture of Weibull distributions. It involves estimating of the unknown parameters which is an important task in statistics, especially in life testing and reliability analysis. The proposed approach depends on different methods that will be used to develop the estimates such as MLE through the EM algorithm. In addition, Bayesian estimations will be investigated and some other extensions such as Graphic, Non-Linear Median Rank Regression and Monte Carlo simulation methods can be used to model the system under consideration. A numerical application will be used through the proposed approach. This paper also presents a comparison of the fitted probability density functions, reliability functions and hazard functions of the 3-parameter Weibull and Weibull mixture distributions using the proposed approach and other conventional methods which characterize the distribution of failure times for the system components. GOF is used to determine the best distribution for modeling lifetime data, the priority will be for the proposed approach which has more accurate parameter estimates.     

A characterization of the Behrens—Fisher distribution with applications to Bayesian inference

The Behrens-Fisher distribution is generally defined as the convolution of two Student t distributions and it is well known that it can be represented as a scale mixture of normals. By extending this standardized distribution to a location-scale family in the usual way we prove that this generalised Behrens-Fisher distribution can also be represented as a location mixture of t distributions when the mixing distribution is, in turn, a Student t. This characterization is applied to the computation of certain predictive distributions appearing in the Bayesian analysis of two sample problems.     

Newsvendor-type models with decision-dependent uncertainty

Models for decision-making under uncertainty use probability distributions to represent variables whose values are unknown when the decisions are to be made. Often the distributions are estimated with observed data. Sometimes these variables depend on the decisions but the dependence is ignored in the decision maker??s model, that is, the decision maker models these variables as having an exogenous probability distribution independent of the decisions, whereas the probability distribution of the variables actually depend on the decisions. It has been shown in the context of revenue management problems that such modeling error can lead to systematic deterioration of decisions as the decision maker attempts to refine the estimates with observed data. Many questions remain to be addressed. Motivated by the revenue management, newsvendor, and a number of other problems, we consider a setting in which the optimal decision for the decision maker??s model is given by a particular quantile of the estimated distribution, and the empirical distribution is used as estimator. We give conditions under which the estimation and control process converges, and show that although in the limit the decision maker??s model appears to be consistent with the observed data, the modeling error can cause the limit decisions to be arbitrarily bad.     

A new look at Bergström’s theorem on convergence in distribution for sums of dependent random variables

In his 1972Periodica Mathematica Hungarica paper, H. Bergström stated a theorem on convergence in distribution for triangular arrays of dependent random variables satisfying, a ?-mixing condition. A gap in his proof of this theorem is explained and a more general version is proved under weakened hypotheses. The method used consists of comparisons between the given array and associated arrays which are parameterized by a truncation variable. In addition to the main theorem, this method yields a proof of equality of limiting finite-dimensional distributions for processes generated by the given associated arrays as well as the result that if a limit distribution for the centered row sums does exist, it must be infinitely divisible. Several corollaries to the main theorem specialize this result for convergence to distributions within certain subclasses of the infinitely divisible laws.     

Quasi-Steady-State particle distributions for an equation of the Landau-Fokker-Planck type with sources

The formation of nonequilibrium particle distributions for the power-law interaction potentials V = α/r ^β, where 1 ≤ β < 4, is considered. The analytical and numerical studies are based on a one-dimensional (in the velocity space) kinetic equation of the Landau-Fokker-Planck type with energy (particle) sources (sinks) localized in the high-energy range. Fully conservative finite-difference schemes are used for numerical modeling. The resulting asymptotic estimates are confirmed by numerical computations. The results can be used to predict the behavior of intrinsic and extrinsic semiconductors influenced by particle fluxes or electromagnetic radiation.     

Latent models for cross-covariance

We consider models for the covariance between two blocks of variables. Such models are often used in situations where latent variables are believed to present. In this paper we characterize exactly the set of distributions given by a class of models with one-dimensional latent variables. These models relate two blocks of observed variables, modeling only the cross-covariance matrix. We describe the relation of this model to the singular value decomposition of the cross-covariance matrix. We show that, although the model is underidentified, useful information may be extracted. We further consider an alternative parameterization in which one latent variable is associated with each block, and we extend the result to models with r-dimensional latent variables.     

Portfolio value-at-risk estimation in energy futures markets with time-varying copula-GARCH model

This paper combines copula functions with GARCH-type models to construct the conditional joint distribution, which is used to estimate Value-at-Risk (VaR) of an equally weighted portfolio comprising crude oil futures and natural gas futures in energy market. Both constant and time-varying copulas are applied to fit the dependence structure of the two assets returns. The findings show that the constant Student t copula is a good compromise for effectively fitting the dependence structure between crude oil futures and natural gas futures. Moreover, the skewed Student t distribution has a better fit than Normal and Student t distribution to the marginal distribution of each asset. Asymmetries and excess kurtosis are found in marginal distributions as well as in dependence. We estimate VaR of the underlying portfolio to be 95% and 99%, by using the Monte Carlo simulation. Then using backtesting, we compare the out-of-sample forecasting performances of VaR estimated by different models.     

Multivariate distributions having Weibull properties

Random variables X₁ ,…, X_n are said to have a joint distribution with Weibull minimums after arbitrary scaling if min_i(a_iX_i) has a one dimensional Weibull distribution for arbitrary constants a_i > 0, i = 1,…, n. Some properties of this class are demonstrated, and some examples are given which show the existence of a number of distributions belonging to the class. One of the properties is found to be useful for computing component reliability importance. The class is seen to contain an absolutely continuous Weibull distribution which can be generated from independent uniform and gamma distributions.     

Normal theory likelihood ratio statistic for mean and covariance structure analysis under alternative hypotheses

The normal distribution based likelihood ratio (LR) statistic is widely used in structural equation modeling. Under a sequence of local alternative hypotheses, this statistic has been shown to asymptotically follow a noncentral chi-square distribution. In practice, the population mean vector and covariance matrix as well as the model and sample size are always fixed. It is hard to justify the validity of the noncentral chi-square distribution for the resulting LR statistic even when data are normally distributed and sample size is large. By extending results in the literature, this paper develops normal distributions to describe the behavior of the LR statistic for mean and covariance structure analysis. A sequence of local alternative hypotheses is not necessary for the proposed distributions to be asymptotically valid. When the effect size is medium and above or when the model is not trivially misspecified, empirical results indicate that a refined normal distribution describes the behavior of the LR statistic better than the commonly used noncentral chi-square distribution, as measured by the Kolmogorov-Smirnov distance. Quantile-quantile plots are also provided to better understand the different distributions.     

Probabilistic analysis of the Davis Putnam procedure for solving the satisfiability problem

We look at the instance distributions used by Goldberg [3] for showing that the Davis Putnam Procedure has polynomial average complexity and show that, in a sense, all these distributions are unreasonable. We then present a ‘reasonable’ family of instance distributions F and show that for each distribution in F a variant of the Davis Putnam Procedure without the pure literal rule requires exponential time with probability 1. In addition, we show that adding subsumption still results in exponential complexity with probability 1.     

The matrix-t distribution and its applications in predictive inference

The predictive distributions of the future responses and regression matrix under the multivariate elliptically contoured distributions are derived using structural approach. The predictive distributions are obtained as matrix-t which are identical to those obtained under matrix normal and matrix-t distributions. This gives inference robustness with respect to departures from the reference case of independent sampling from the matrix normal or dependent but uncorrelated sampling from matrix-t distributions. Some successful applications of matrix-t distribution in the field of spatial prediction have been addressed.     

Discretization of distributions in the maximum domain of attraction

In this paper we discuss the discretization of distributions belonging to some max-domain of attraction. Given a random variable X its discretization is defined as the minimal integer not less than X. Our first interest is on distributions that preserve the max-domain property after discretization. Secondly, we characterize the distributions which are regarded as the discretization of the distribution in the Gumbel max-domain of attraction. Lastly the correspondence of distribution in Gumbel max-domain of attraction is investigated.     

Principal points of a multivariate mixture distribution

A set of n-principal points of a distribution is defined as a set of n points that optimally represent the distribution in terms of mean squared distance. It provides an optimal n-point-approximation of the distribution. However, it is in general difficult to find a set of principal points of a multivariate distribution. Tarpey et al. [T. Tarpey, L. Li, B. Flury, Principal points and self-consistent points of elliptical distributions, Ann. Statist. 23 (1995) 103-112] established a theorem which states that any set of n-principal points of an elliptically symmetric distribution is in the linear subspace spanned by some principal eigenvectors of the covariance matrix. This theorem, called a “principal subspace theorem”, is a strong tool for the calculation of principal points. In practice, we often come across distributions consisting of several subgroups. Hence it is of interest to know whether the principal subspace theorem remains valid even under such complex distributions. In this paper, we define a multivariate location mixture model. A theorem is established that clarifies a linear subspace in which n-principal points exist.