Poisson calculus for spatial neutral to the right processes

Neutral to the right (NTR) processes were introduced by Doksum in 1974 as Bayesian priors on the class of distributions on the real line. Since that time there have been numerous applications to models that arise in survival analysis subject to possible right censoring. However, unlike the Dirichlet process, the larger class of NTR processes has not been used in a wider range of more complex statistical applications. Here, to circumvent some of these limitations, we describe a natural extension of NTR processes to arbitrary Polish spaces, which we call spatial neutral to the right processes. Our construction also leads to a new rich class of random probability measures, which we call NTR species sampling models. We show that this class contains the important two parameter extension of the Dirichlet process. We provide a posterior analysis, which yields tractable NTR analogues of the Blackwell--MacQueen distribution. Our analysis turns out to be closely related to the study of regenerative composition structures. A new computational scheme, which is an ordered variant of the general Chinese restaurant processes, is developed. This can be used to approximate complex posterior quantities. We also discuss some relationships to results that appear outside of Bayesian nonparametrics.Comment: Published at http://dx.doi.org/10.1214/009053605000000732 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Distributional Properties of means of Random Probability Measures

The present paper provides a review of the results concerning distributional properties of means of random probability measures. Our interest in this topic has originated from inferential problems in Bayesian Nonparametrics. Nonetheless, it is worth noting that these random quantities play an important role in seemingly unrelated areas of research. In fact, there is a wealth of contributions both in the statistics and in the probability literature that we try to summarize in a unified framework. Particular attention is devoted to means of the Dirichlet process given the relevance of the Dirichlet process in Bayesian Nonparametrics. We then present a number of recent contributions concerning means of more general random probability measures and highlight connections with the moment problem, combinatorics, special functions, excursions of stochastic processes and statistical physics.Bayesian Nonparametrics; Completely random measures; Cifarelli–Regazzini identity; Dirichlet process; Functionals of random probability measures; Generalized Stieltjes transform; Neutral to the right processes; Normalized random measures; Posterior distribution; Random means; Random probability measure; Two–parameter Poisson–Dirichlet process.

Distributional properties of means of random probability measures

The present paper provides a review of the results concerning distributional properties of means of random probability measures. Our interest in this topic has originated from inferential problems in Bayesian Nonparametrics. Nonetheless, it is worth noting that these random quantities play an important role in seemingly unrelated areas of research. In fact, there is a wealth of contributions both in the statistics and in the probability literature that we try to summarize in a unified framework. Particular attention is devoted to means of the Dirichlet process given the relevance of the Dirichlet process in Bayesian Nonparametrics. We then present a number of recent contributions concerning means of more general random probability measures and highlight connections with the moment problem, combinatorics, special functions, excursions of stochastic processes and statistical physics.Bayesian Nonparametrics; Completely random measures; Cifarelli-Regazzini identity; Dirichlet process; Functionals of random probability measures; Generalized Stieltjes transform; Neutral to the right processes; Normalized random measures; Posterior distribution; Random means; Random probability measure; Two-parameter Poisson-Dirichlet process

The Gamma-Rayleigh Distribution and Applications to Survival Data

Studies on probability distribution functions and their properties are needful as they are very important in modeling random phenomena. However, research has shown that some real life data can be modeled more adequately by distributions obtained as combination of two random variables with known probability distributions. This paper introduces the Gamma-Rayleigh distribution (GRD) as a new member of the Gamma-X family of generalized distributions. The Transformed-Transformer method is used to combine the Gamma and Rayleigh distributions. Various properties of the resulting twoparameter Gamma-Rayleigh distribution, including moments, moment generating function, survival function and hazard function are derived. Results of simulation study reveals that the distribution is unimodal, skewed and normal-type for some values of the shape parameter. The distribution is also found to relate with the Gamma, Rayleigh and Generalized-Gamma distributions. The method of maximum likelihood has been used to estimate the shape and scale parameters of the distribution. To illustrate its adequacy in modelling real life data the distribution is fitted to two survival data sets. The results show that the distribution produced fits that are competitive and compared better, in some cases, to the Gamma, Rayleigh, Weibull and Lognormal distributions.Keywords: Gamma-X family, Gamma-Rayleigh distribution, Maximum Likelihood estimators, Survival data

The ratio of independent generalized gamma random variables with applications

This paper originates from the interest in the distribution of a statistic defined as the ratio of independent generalized gamma random variables. It is shown that it can be represented as the product of independent generalized gamma random variables with some reparametrization. By decomposing the characteristic function of the negative natural logarithm of the statistic and by using the distribution of the difference of two independent generalized integer gamma random variables as a basis, accurate and computationally appealing near-exact distributions are derived for this statistic. In the process, a new parameter is introduced in the near-exact distributions, which allows to control the degree of precision of these approximations. Furthermore, the performance of the near-exact distributions is assessed using a measure of proximity between cumulative distribution functions and by comparison with the exact and empirical distributions. We illustrate the use of the proposed approximations on the distribution of the ratio of generalized variances in a multivariate multiple regression setting and with an example of application related with single-input single-output networks. The proposed results ensure less computing time and stability in results as well.National Research Fund and Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology).http://wileyonlinelibrary.com/journal/cmm4hj2022Statistic

Nonlinear synthetic unit hydrograph method that accounts for channel network type, A

2018 Spring.Includes bibliographical references.Stormflow hydrographs are commonly estimated using synthetic unit hydrograph (UH) methods, particularly for ungauged basins. Current synthetic UHs either consider very limited aspects of basin geometry or require explicit representation of the basin flow paths. None explicitly considers the channel network type (i.e., dendritic, parallel, pinnate, rectangular, and trellis). The goal of this study is to develop and test a nonlinear synthetic UH that explicitly accounts for the network type. The synthetic UH is developed using kinematic wave travel time expressions for hillslope and channel points in the basin. The effects of the network structure are then isolated into two random variables whose distributions are estimated based on the network type. The proposed method is applied to ten basins from each classification and compared to other related methods. The results suggest that considering network type improves the estimated UHs with the largest improvements seen for dendritic, parallel, and pinnate networks

Fourier image synthesis and slope spectrum analysis of deepwater, wind-wave scenes viewed at Brewster\u27s angle

A semi-empirical model for the Fourier synthesis of deepwater, wind-wave scenes has been constructed for the analysis of water-wave slope spectra. The main simplifying assumptions of this model are 1) fully-developed wind-wave surfaces are quasi-homogeneous, quasi -stationary and are therefore treatable by Fourier methods, 2) the subsurface is both optically and mechanically deep, and 3) the small range of spectral wave components defines a fetch-limited, small amplitude condition. A nonlinear transformation of wave slope to reflected and refracted radiance in both horizontal and vertical polarizations was effected under the special conditions of Brewster-angle viewing under clear skies at a spectral wavelength of 460 nanometers. Seventy two syntheses were varied with respect to six distinct solar positions, four distinct wind directions, and three distinct wind velocities. The synthetic wave scenes were analyzed via the forward Fourier transformation and their radiance magnitude spectra were compared with the original slope magnitude spectra of the initial synthesis in order to estimate the effects of the nonlinear radiance transformation on the recovery of the wave slope spectrum from imagery. Within the boundaries of this study, it was determined that 1) the limited results of Chapman and Irani [1981] have been generally verified, 2) the existence of an optimal imaging geometry for slope spec trum estimation is indicated, and 3) the presence of subresolution wave slopes creates a significant effect on wave slope spectra derived from imagery

A survey of random processes with reinforcement

The models surveyed include generalized P\'{o}lya urns, reinforced random walks, interacting urn models, and continuous reinforced processes. Emphasis is on methods and results, with sketches provided of some proofs. Applications are discussed in statistics, biology, economics and a number of other areas.Comment: Published at http://dx.doi.org/10.1214/07-PS094 in the Probability Surveys (http://www.i-journals.org/ps/) by the Institute of Mathematical Statistics (http://www.imstat.org