有限混合Gamma分布的拓扑稠密性证明

首先给出了有限混合Erlang分布在正实数轴上所有概率分布中稠密的理论证明,进而给出了混合Gamma分布具有稠密性的结论,说明有限混合Gamma分布具有广泛的适用性,可以用来刻画正实数轴上的任意随机变量.     

Impulsive differential equations with Gamma distributed moments of impulses and p-moment exponential stability

Differential equations with impulses at random moments are set up and investigated. We study the case of Gamma distributed random moments of impulses. Several properties of solutions are studied based on properties of Gammma distributions. Some sufficient conditions for p-moment exponential stability of the solutions are given.     

A chaotic gas-like model for trading markets

This paper considers the ideal gas-like model for trading markets, where each individual interacts with others trading in money-conservative collisions. Traditionally this model introduces different rules of random selection and exchange between pairs of agents, what leads to different money distributions in the community. Real economic transactions are complex but obviously non-random. Therefore, unlike the traditional model, this work introduces chaotic elements in the evolution of the economic system. As a result, it is found that the chaotic gas-like model can reproduce the referenced wealth distributions observed in real economies, i.e. the Gamma, Exponential and Pareto distributions.     

A Comparison of Stochastic Systems with Different Types of Delays

In this article, we investigate the effects of different types of delays, a fixed delay and a random delay, on the dynamics of stochastic systems as well as their relationship with each other in the context of a just-in-time network model. The specific example on which we focus is a pork production network model. We numerically explore the corresponding deterministic approximations for the stochastic systems with these two different types of delays. Numerical results reveal that the agreement of stochastic systems with fixed and random delays depend on the population size and the variance of the random delay, even when the mean value of the random delay is chosen the same as the value of the fixed delay. When the variance of the random delay is sufficiently small, the histograms of state solutions to the stochastic system with a random delay are similar to those of the stochastic model with a fixed delay regardless of the population size. We also compared the stochastic system with a Gamma distributed random delay to the stochastic system constructed based on the Kurtz's limit theorem from a system of deterministic delay differential equations with a Gamma distributed delay. We found that with the same population size the histogram plots for the solution to the second system appear more dispersed than the corresponding ones obtained for the first case. In addition, we found that there is more agreement between the histograms of these two stochastic systems as the variance of the Gamma distributed random delay decreases.     

Stochastic processes with orthogonal polynomial eigenfunctions

Markov processes which are reversible with either Gamma, Normal, Poisson or Negative Binomial stationary distributions in the Meixner class and have orthogonal polynomial eigenfunctions are characterized as being processes subordinated to well-known diffusion processes for the Gamma and Normal, and birth and death processes for the Poisson and Negative Binomial. A characterization of Markov processes with Beta stationary distributions and Jacobi polynomial eigenvalues is also discussed.     

Near-exact distributions for the independence and sphericity likelihood ratio test statistics

In this paper we show how, based on a decomposition of the likelihood ratio test for sphericity into two independent tests and a suitably developed decomposition of the characteristic function of the logarithm of the likelihood ratio test statistic to test independence in a set of variates, we may obtain extremely well-fitting near-exact distributions for both test statistics. Since both test statistics have the distribution of the product of independent Beta random variables, it is possible to obtain near-exact distributions for both statistics in the form of Generalized Near-Integer Gamma distributions or mixtures of these distributions. For the independence test statistic, numerical studies and comparisons with asymptotic distributions proposed by other authors show the extremely high accuracy of the near-exact distributions developed as approximations to the exact distribution. Concerning the sphericity test statistic, comparisons with formerly developed near-exact distributions show the advantages of these new near-exact distributions.     

非均匀随机数产生

本文详细介绍了产生非均匀随机数的一般方法,常用连续分布(正态分布,Gamma分布,Beta分布,χ2分布和F分布)的抽样法及利用R软件产生随机数的方法。     

Robust estimation for nonparametric generalized regression

This paper focuses on nonparametric regression estimation for the parameters of a discrete or continuous distribution, such as the Poisson or Gamma distributions, when anomalous data are present. The proposal is a natural extension of robust methods developed in the setting of parametric generalized linear models. Robust estimators bounding either large values of the deviance or of the Pearson residuals are introduced and their asymptotic behaviour is derived. Through a Monte Carlo study, for the Poisson and Gamma distributions, the finite properties of the proposed procedures are investigated and their performance is compared with that of the classical ones. A resistant cross-validation method to choose the smoothing parameter is also considered.     

Determining <Emphasis Type="Italic">p</Emphasis>-values for systems cointegration tests with a prior adjustment for deterministic terms

Following Doornik (J Econ Surv 12:573–593, 1998) I present a procedure to approximate the asymptotic distributions of systems cointegration tests with a prior adjustment for deterministic terms suggested by Lütkepohl (Econometrica 72:647–662, 2004), Saikkonen and Lütkepohl (Econometric Theory 16:373–406, 2000a, J Business Econ Stat 18:451–464, 2000b, Time Series Anal 21:435–456, 2000c) and Saikkonen and Luukkonen (J Econ 81:93–126, 1997). These tests rely upon different assumptions as to the inclusion of deterministic components such as a constant, a linear trend or a level shift. The asymptotic distributions, which are functions of Brownian motions, are approximated by Gamma distributions. Only estimates of the mean and variance of the asymptotic test distributions are needed to fit the Gamma distributions. Such estimates are obtained from response surfaces. The required coefficients to compute the asymptotic moments are presented in this paper. Via the fitted Gamma distributions one can, then, easily derive p-values or arbitrary percentiles.     

Generalized Inv-Log-Gamma-G processes

Abstract

Gamma processes belong to subordinators for which very small jumps occurs infinitely many times in any finite time interval but their sums are finite. Here we consider their novel and important modifications with a nice application potential. A generalization of fractional kth lower record value process defined in Bieniek and Szynal, called Inverse-Log-Gamma-G process is investigated. Explicit relation with the Gamma process is presented and conditional, posterior and finite dimensional distributions are derived. The results are obtained by appropriate transformations of known stochastic processes. In contrast with the regression this allows us to describe the finite dimensional distributions of the processes of interest and in this way to make their full characterization.     

Testing Departures from Gamma, Rayleigh and Truncated Normal Distributions

This paper provides necessary and sufficient conditions for a solution to likelihood equations for an exponential family of distributions, which includes Gamma, Rayleigh and singly truncated normal distributions. Furthermore, the maximum likelihood estimator is obtained as a limit case when the equations have no solution. These results provide a way to test departures from Rayleigh and singly truncated normal distributions using the likelihood ratio test. A new easy way to test departures from a Gamma distribution is also introduced.     

A Comparison of Restricted and Unrestricted Estimators in Estimating Linear Functions of Ordered Scale Parameters of Two Gamma Distributions

The problem of estimating linear functions of ordered scale parameters of two Gamma distributions is considered. A necessary and sufficient condition on the ratio of two coefficients is given for the maximum likelihood estimator (MLE) to dominate the crude unbiased estimator (UE) in terms of mean square error. A modified MLE which satisfies the restriction is also suggested, and a necessary and sufficient condition is also given for it to dominate the admissible estimator based solely on one sample. The estimation of linear functions of variances in two sample problem and also of variance components in a one-way random effect model is mentioned.     

The generalized near-integer Gamma distribution: a basis for ‘near-exact’ approximations to the distribution of statistics which are the product of an odd number of independent Beta random variables

In this paper the concept of near-exact approximation to a distribution is introduced. Based on this concept it is shown how a random variable whose exponential has a Beta distribution may be closely approximated by a sum of independent Gamma random variables, giving rise to the generalized near-integer (GNI) Gamma distribution. A particular near-exact approximation to the distribution of the logarithm of the product of an odd number of independent Beta random variables is shown to be a GNI Gamma distribution. As an application, a near-exact approximation to the distribution of the generalized Wilks Λ statistic is obtained for cases where two or more sets of variables have an odd number of variables. This near-exact approximation gives the exact distribution when there is at most one set with an odd number of variables. In the other cases a near-exact approximation to the distribution of the logarithm of the Wilks Lambda statistic is found to be either a particular generalized integer Gamma distribution or a particular GNI Gamma distribution.     

The robustness of mean and variance approximations in risk analysis

This paper examines further the problem of estimating the mean and variance of a continuous random variable from estimates of three points within the distribution, typically the median or mode and two extreme fractiles. The problem arises most commonly in PERT and risk analysis where it can usually be assumed that the distribution in question is bell-shaped and positively skewed, often typified by a Beta distribution. Over the years, a number of alternative approximations have been proposed, usually as modifications to the original PERT formulae. The accuracy of a number of these approximations is investigated based not only on a Beta distribution, but also for three other commonly used bell-shaped, positively skewed distributions, namely the Gamma, Lognormal and F distributions. It is shown that a balanced weighted average of the median and the 4% fractiles provides a consistent estimator of the distribution mean across all four distributions. Furthermore, reasonably accurate estimates of the variance can also be obtained by treating the three fractiles as defining an equivalent discrete distribution with the same probability weight as in the formula for the mean.     

Modeling clustered binary data with nonparametric unobserved heterogeneity: An application to stock crash analysis

Various random effects models have been developed for clustered binary data; however, traditional approaches to these models generally rely heavily on the specification of a continuous random effect distribution such as Gaussian or beta distribution. In this article, we introduce a new model that incorporates nonparametric unobserved random effects on unit interval (0,1) into logistic regression multiplicatively with fixed effects. This new multiplicative model setup facilitates prediction of our nonparametric random effects and corresponding model interpretations. A distinctive feature of our approach is that a closed-form expression has been derived for the predictor of nonparametric random effects on unit interval (0,1) in terms of known covariates and responses. A quasi-likelihood approach has been developed in the estimation of our model. Our results are robust against random effects distributions from very discrete binary to continuous beta distributions. We illustrate our method by analyzing recent large stock crash data in China. The performance of our method is also evaluated through simulation studies.     

Selfdecomposable Fields

In the present paper, we study selfdecomposability of random fields, as defined directly rather than in terms of finite-dimensional distributions. The main tools in our analysis are the master Lévy measure and the associated Lévy-Itô representation. We give the dilation criterion for selfdecomposability analogous to the classical one. Next, we give necessary and sufficient conditions (in terms of the kernel function) for a Volterra field driven by a Lévy basis to be selfdecomposable. In this context, we also study the so-called Urbanik classes of random fields. We follow this with the study of existence and selfdecomposability of integrated Volterra fields. Finally, we introduce infinitely divisible field-valued Lévy processes, give the Lévy-Itô representation associated with them and study stochastic integration with respect to such processes. We provide examples in the form of Lévy semistationary processes with a Gamma kernel and Ornstein–Uhlenbeck processes.     

Bilateral gamma distributions and processes in financial mathematics

We present a class of Lévy processes for modelling financial market fluctuations: bilateral Gamma processes. Our starting point is to explore the properties of bilateral Gamma distributions, and then we turn to their associated Lévy processes. We treat exponential Lévy stock models with an underlying bilateral Gamma process as well as term structure models driven by bilateral Gamma processes, and apply our results to a set of real financial data (DAX 1996–1998).     

First passage probability distributions in Markov models and the HIV incubation distribution under treatment

In this paper, I derive some first passage probability distributions for some continuous time Markov chains. By using these results, I then derive some exact HIV incubation distributions under treatment by anti-viral drugs such as AZT under some general conditions. It is shown that under some special conditions, these distributions are mixtures of generalized Gamma distributions.     

双边定数截尾下Pareto分布的可靠性分析

基于双边定数截尾样本,选取未知参数的先验分布为无信息先验和Gamma分布,分别在平方损失和LINEX损失下,研究了Pareto分布的形状参数和可靠性指标(可靠度和失效率)的Bayes估计.为了研究估计的精度,采用Monte-Carlo模拟的方法给出了数值检验的例子.结果表明在LINEX损失下并选用Gamma先验分布时,参数的Bayes估计是最优的.     

Fractal Activity Time Models for Risky Asset with Dependence and Generalized Hyperbolic Distributions

Risky asset models with the dependence through fractal activity time are described. The construction of the fractal activity time is implemented via superpositions of Ornstein-Uhlenbeck type processes driven by Lévy noise. The model features both tractable dependence structure and desired marginal distributions of the returns from the generalized hyperbolic class: the Variance Gamma and normal inverse Gaussian. These distributions provide good fit to real financial data. Pricing formulae for the proposed models are derived.