Nullity distributions associated to Cartan connection

The Klein-Grifone approach to global Finsler geometry is adopted. The nullity distributions of the three curvature tensors of Cartan connection are investigated. Nullity distributions concerning certain relevant special Finsler spaces are considered. Concrete examples are given whenever the situation needs.     

On generalized binomial and multinomial distributions and their relation to generalized Poisson distributions

Summary The binomial and multinomial distributions are, probably, the best known distributions because of their vast number of applications. The present paper examines some generalizations of these distributions with many practical applications. Properties of these generalizations are studied and models giving rise to them are developed. Finally, their relationship to generalized Poisson distributions is examined and limiting cases are given.     

Irregularities of distribution with respect to strips

Summary A lower bound is established for the strip discrepancy of a broad class of point distributions. The bound implies unbounded strip discrepancy for equally weighted point distributions under favorable conditions. The methods of proof use notions from integral geometry.     

An algebraic approach to tempered distributions

A space of pseudoquotients is introduced that is shown to be isomorphic to the space of tempered distributions on RN. The Fourier transform is defined as a map from the space of pseudoquotients to the space of tempered distributions and as a transformation on pseudoquotients.     

From Archimedean to Liouville copulas

We use a recent characterization of the d-dimensional Archimedean copulas as the survival copulas of d-dimensional simplex distributions (McNeil and Nešlehová (2009) [1]) to construct new Archimedean copula families, and to examine the relationship between their dependence properties and the radial parts of the corresponding simplex distributions. In particular, a new formula for Kendall’s tau is derived and a new dependence ordering for non-negative random variables is introduced which generalises the Laplace transform order. We then generalise the Archimedean copulas to obtain Liouville copulas, which are the survival copulas of Liouville distributions and which are non-exchangeable in general. We derive a formula for Kendall’s tau of Liouville copulas in terms of the radial parts of the corresponding Liouville distributions.     

Characterization by truncated moments and its application to Pearson-type distributions

Summary A characterization theorem based on the proportional relation between two truncated moments is proved for both continuous and discrete distributions. The results are applied for characterizing distributions of Pearson's system and its discrete analogon.     

An excursion approach to maxima of the Brownian bridge

Distributions of functionals of Brownian bridge arise as limiting distributions in non-parametric statistics. In this paper we will give a derivation of distributions of extrema of the Brownian bridge based on excursion theory for Brownian motion. The idea of rescaling and conditioning on the local time has been used widely in the literature. In this paper it is used to give a unified derivation of a number of known distributions, and a few new ones. Particular cases of calculations include the distribution of the Kolmogorov–Smirnov statistic and the Kuiper statistic.     

Game-theoretic probability combination with applications to resolving conflicts between statistical methods

In the typical analysis of a data set, a single method is selected for statistical reporting even when equally applicable methods yield very different results. Examples of equally applicable methods can correspond to those of different ancillary statistics in frequentist inference and of different prior distributions in Bayesian inference. More broadly, choices are made between parametric and nonparametric methods and between frequentist and Bayesian methods. Rather than choosing a single method, it can be safer, in a game-theoretic sense, to combine those that are equally appropriate in light of the available information. Since methods of combining subjectively assessed probability distributions are not objective enough for that purpose, this paper introduces a method of distribution combination that does not require any assignment of distribution weights. It does so by formalizing a hedging strategy in terms of a game between three players: nature, a statistician combining distributions, and a statistician refusing to combine distributions. The optimal move of the first statistician reduces to the solution of a simpler problem of selecting an estimating distribution that minimizes the Kullback–Leibler loss maximized over the plausible distributions to be combined. The resulting combined distribution is a linear combination of the most extreme of the distributions to be combined that are scientifically plausible. The optimal weights are close enough to each other that no extreme distribution dominates the others. The new methodology is illustrated by combining conflicting empirical Bayes methods in the context of gene expression data analysis.     

Fitting the extended poisson process model to grouped binary data

Summary  Extended Poisson process modelling allows the construction of a broad class of distributions, including distributions over-dispersed or under-dispersed relative to the binomial distribution, with the binomial distribution being a special case. In this paper an iteratively re-weighted least squares algorithm for fitting such generalised binomial distributions is presented, and is illustrated with an example.     

Spare parts management: Linking distributional assumptions to demand classification

Spare parts are known to be associated with intermittent demand patterns and such patterns cause considerable problems with regards to forecasting and stock control due to their compound nature that renders the normality assumption invalid. Compound distributions have been used to model intermittent demand patterns; there is however a lack of theoretical analysis and little relevant empirical evidence in support of these distributions. In this paper, we conduct a detailed empirical investigation on the goodness of fit of various compound Poisson distributions and we develop a distribution-based demand classification scheme the validity of which is also assessed in empirical terms. Our empirical investigation provides evidence in support of certain demand distributions and the work described in this paper should facilitate the task of selecting such distributions in a real world spare parts inventory context. An extensive discussion on parameter estimation related difficulties in this area is also provided.     

Transient and periodic solution to the time-inhomogeneous quasi-birth death process

We derive the transient distribution and periodic family of asymptotic distributions and the transient and periodic moments for the quasi-birth-and-death processes with time-varying periodic rates. The distributions and moments are given in terms of integral equations involving the related random-walk process. The method is a straight-forward application of generating functions.      

Approximations to stochastic programs with complete fixed recourse

The probability distribution of the data entering a recourse problem is replaced by finite discrete distributions. It is proved that the convergence of the objective functions of the approximating problems to that one of the original problem can be achieved by choosing the discrete distributions in quite a natural way. For bounded feasible sets this implies the convergence of the optimal values. Finally some error bounds are derived.     

Empirical Hankel transforms and its applications to goodness-of-fit tests

We introduce a special Hankel transform for probability distributions on the nonnegative half-line and discuss some of its properties. Due to the uniqueness of the transform we suggest an integral type test statistic based on the empirical Hankel transform to treat simple and composite hypotheses goodness-of-fit problems. The special case of exponential distributions is studied in detail.     

The Sensitivity of the Number of Clusters in a Gaussian Mixture Model to Prior Distributions

One of the main advantages of Bayesian approaches is that they offer principled methods of inference in models of varying dimensionality and of models of infinite dimensionality. What is less widely appreciated is how the model inference is sensitive to prior distributions and therefore how priors should be set for real problems. In this paper prior sensitivity is considered with respect to the problem of inference in Gaussian mixture models. Two distinct Bayesian approaches have been proposed. The first is to use Bayesian model selection based upon the marginal likelihood; the second is to use an infinite mixture model which ‘side steps’ model selection. Explanations for the prior sensitivity are given in order to give practitioners guidance in setting prior distributions. In particular the use of conditionally conjugate prior distributions instead of purely conjugate prior distributions are advocated as a method for investigating prior sensitivity of the mean and variance individually.     

Investigation of a problem of an elastic half space subjected to stochastic temperature distribution at the boundary

The present work investigates the responses of stochastic type temperature distribution applied at the boundary of an elastic medium in the context of thermoelasticity without energy dissipation. We consider an one dimensional problem of half space and assume that the bounding surface of the half space is traction free and is subjected to two types of time dependent temperature distributions which are of stochastic types. In order to compare the results predicted by stochastic temperature distributions with the results of deterministic type temperature distribution, the stochastic type temperature distributions applied at the boundary are taken in such a way that they reduce to the cases of deterministic types as special cases. Integral transform technique along with stochastic calculus is used to solve the problem. The approximated solutions for physical fields like, stress, temperature, displacement etc. are derived for very small values of time where stochastic type boundary conditions are taken to be of white noise type. The problem is further illustrated with graphical representation of numerical solutions of the problem for a particular case. A detailed comparison of the results of stochastic temperature, displacement and stress distributions inside the half space with the corresponding results of deterministic distributions is presented and special features of the effects of stochastic type boundary conditions are highlighted.     

S-Bounded distributions,application to partial differential equations

The S-boundness at infinity of a distribution is defined to give some informations on the behaviour of a large class of distributions at infinity. The subspace A′ ? D′ of S-bounded distributions has been characterized and the properties of elements of A′ have been analyzed especially those interesting for partial differential equations. At the end, some propositions, concerning the S-boundness of solutions of linear partial differential equations have been proved.     

The input to and output from risk evaluation models

The work which has been done on analytic approaches to the evaluation of the risk in major capital investment opportunities suggests the following hypotheses: (a) The distributions of NPV and IRR which are output from risk evaluation models are often approximately normal. (b) The only important features of the distributions of the variables which are input to risk evaluation models are their means and standard deviations. The research which is described in this paper tests these hypotheses using five risk simulation case studies.     

Pitman closeness of record values to population quantiles

In this paper, we examine the Pitman closeness of record statistics to the population quantiles of a location-scale family of distributions and study its monotonicity properties. Even though in general it depends on the parent distribution, exact expressions are derived for the required probabilities in the case of Uniform (−1,1) and exponential distributions. For the population median, it is shown that the first upper record is the Pitman-closest among all upper record values. Moreover, for the population median, in the case of symmetric distributions, the Pitman closeness probabilities of records are shown to be distribution-free and explicit expressions are also derived for these probabilities.     

Fitting combinations of exponentials to probability distributions

Two techniques are described for approximating distributions on the positive half‐line by combinations of exponentials. One is based on Jacobi polynomial expansions, and the other on the logbeta distribution. The techniques are applied to some well‐known distributions (degenerate, uniform, Pareto, lognormal and others). In theory, the techniques yield sequences of combination of exponentials that always converge to the true distribution, but their numerical performance depends on the particular distribution being approximated. An error bound is given in the case the logbeta approximations. Copyright © 2006 John Wiley & Sons, Ltd.