Multivariate Polya and Inverse Polya Distributions of Order k

Multivariate Polya and inverse Polya distributions of order k are derived by means of generalized urn models and by compounding the type II multinomial and multivariate negative binomial distributions of order k of PHILIPPOU , ANTZOULAKOS and TRIPSIANNIS (1990, 1988), respectively, with the Dirichlet distribution. It is noted that the above two distributions include as special cases a multivariate hypergeometric distribution of order k, a negative one, an inverse one, a negative inverse one and a discrete uniform of the same order. The probability generating functions, means, variances and covariances of the new distributions are obtained and five asymptotic results are established relating them to the above-mentioned multinomial and multivariate negative binomial distributions of order k, and to the type II negative binomial and the type I multivariate Poisson distributions of order k of PHILIPPOU (1983), and PHILIPPOU , ANTZOULAKOS and TRIPSIAN-NIS (1988), respectively. Potential applications are also indicated. The present paper extends to the multivariate case the work of PHILIPPOU , TRIPSIANNIS and ANTZOULAKOS (1989) on Polya and inverse Polya distributions of order k..     

On Countable Mixtures of Bivariate Binomial Distributions

A unified treatment is given for mixtures of bivariate binomial distributions with respect to their index parameter(s). The use of probability generating functions is employed and a number of interesting properties including probabilities, factorial moments, factorial cumulants and conditional distributions are derived. Five classes of such mixtures are examined and several well known bivariate discrete distributions are used as illustrative examples. Biological applications are indicated including the fit of three bivariate distributions to an actual set of human family data.     

Some Approximations of the Borel-Tanner or the Generalized Poisson Distribution

This paper considers some approximations for the Borel-Tanner (Generalized Poisson) sums by using (i) Gram-Charlier Poisson expansion, (ii) Mixture of two Poisson distributions, (iii) Variance stabilizing technique, and (iv) negative binomial distribution. It has been found that the approximation obtained by using the negative binomial distribution seems to be more efficient than the other approximation.     

Efficient Simulation of Multivariate Binomial and Poisson Distributions

Power investigations, for example, in statistical procedures for the assessment of agreement among multiple raters often require the simultaneous simulation of several dependent binomial or Poisson distributions to appropriately model the stochastical dependencies between the raters' results. Regarding the rather large dimensions of the random vectors to be generated and the even larger number of interactions to be introduced into the simulation scenarios to determine all necessary information on their distributions' dependence stucture, one needs efficient and fast algorithms for the simulation of multivariate Poisson and binomial distributions. Therefore two equivalent models for the multivariate Poisson distribution are combined to obtain an algorithm for the quick implementation of its multivariate dependence structure. Simulation of the multivariate Poisson distribution then becomes feasible by first generating and then convoluting independent univariate Poisson variates with appropriate expectations. The latter can be computed via linear recursion formulae. Similar means for simulation are also considered for the binomial setting. In this scenario it turns out, however, that exact computation of the probability function is even easier to perform; therefore corresponding linear recursion formulae for the point probabilities of multivariate binomial distributions are presented, which only require information about the index parameter and the (simultaneous) success probabilities, that is the multivariate dependence structure among the binomial marginals.     

The Multiple Poisson Distribution,Its Characteristics and a Variety of Forms

The multiple Poisson distribution, known under different names, such as generalized Poisson, compound Poisson, composed Poisson, stuttering Poisson, Poisson power series, Poisson-stopped sum distribution, etc., plays an important role in discrete distribution theory. Here we want to show its basic characteristics, the variety of its forms and specify the generalizing distributions.     

The Negative Binomial Distribution of Order k and Some of Its Properties

The negative binomial distribution of order k is introduced and briefly studied. First it is shown that it is a proper probability distribution. Then its probability generating function, mean and variance are derived. Finally it is shown that the number of trials until the rth kth consecutive success (r ≧ 1, k ≧ 1) in independent trials with constant success probability p (0 < p < 1) is distributed as negative binomial distribution of order k. The present paper generalizes results of SHANE (1973), PHILIPPOU and MUWAFI (1982), and PHILIPPOU, GEORGHIOU and PHILIPPOU (1982).     

Distribution of Polymorphus minutus among its intermediate hosts

Hirsch R. P. 1979. Distribution of Polymorphus minutus among its intermediate hosts. International journal for Parasitology10: 243–248. In 1971, Crofton investigated patterns of distribution of Polymorphus minutus in the intermediate host, Gammarus pulex. Among his conclusions were: (1) P. minutus populations occur in patterns similar to negative binomial distributions, and (2) parasite-induced host mortality results in patterns similar to truncated (high end) negative binomial distributions. Those conclusions, however, were not tested by statistical analyses. To test Crofton's observations, Chi-square goodness of fit tests were applied to data used by Crofton and an additional two stations sampled by Hynes & Nicholas in 1963. Analyses were expanded to include five theoretical distributions, four patterns of host mortality and various rates of host mortality. Truncated forms of negative binomial, positive binomial and Poisson distributions were also investigated where nontruncated distributions failed to fit observed distributions. It was found that negative binomial distributions most frequently describe patterns of P. minutus distribution with the exception of one population described by Poisson and another by positive binomial distributions. Crofton's assumption that truncated distributions result from parasite-induced host mortality seems unlikely in light of those analyses.     

Characterization of the Mixtures of Gompertz Distributions by Conditional Expectation of Order Statistics

The mixtures of Gompertz random variables (Gompertz , 1825) X₁ and X₂ are identified in terms of relations between the conditional expectation of [exp (αX_2:2) — exp (αX_1:2)]^k given X_1:2 and the hazard rate function of the distribution, k is a positive integer and α < 0. Here X_1:2 and X_2:2 denote the corresponding order statistics. In addition, we also mention some related theorems to characterize the mixtures of Gompertz distributions. Finally, when the sample size is n, the above results are also valid and we also give an application to Multi-Hit models of carcinogenesis (Parallel Systems).     

The Distribution of the Product of Non-Central Beta Variates

This paper is concerned with the probability distribution of the product of independent noncentral beta variates. Using series expansion of incomplete beta function, it is shown that this distribution is poisson weighted sum of mixtures of central beta densities. Thus the result given here is more general than those reported by Steece (1976), Nandi (1980) and Onukogu and Gupta (1981).     

On Mohan's Property of the Poisson Distribution

Two interesting results encountered in the literature concerning the Poisson and the negative binomial distributions are due to Moran (1952) and Patil & Seshadri (1964), respectively. Morans result provided a fundamental property of the Poisson distribution. Roughly speaking, he has shown that if Y, Z are independent, non-negative, integer-valued random variables with X = Y | Z then, under some mild restrictions, the conditional distribution of Y | X is binomial if and only if Y, Z are Poisson random variables. Motivated by Morans result Patil & Seshadri obtained a general characterization. A special case of this characterization suggests that, with conditions similar to those imposed by Moran, Y | X is negative hypergeometric if and only if Y, Z are negative binomials. In this paper we examine the results of Moran and Patil & Seshadri in the case where the conditional distribution of Y | X is truncated at an arbitrary point k – 1 (k = 1, 2, …). In fact we attempt to answer the question as to whether Morans property of the Poisson distribution, and subsequently Patil & Seshadris property of the negative binomial distribution, can be extended, in one form or another, to the case where Y | X is binomial truncated at k – 1 and negative hypergeometric truncated at k – 1 respectively.     

Bayesian Estimation of the Distribution Function of the Poisson Model

This paper presents the Bayes estimators of the Poisson distribution function based on complete and truncated data under a natural conjugate prior. Laplace transform of the incomplete gamma function and the Gauss hypergeometric function have been employed in order to overcome the intractability of the integrals. Numerical examples from biosciences are given to illustrate the results. A Monte Carlo study has been carried out to compare Bayes estimators under complete data with the corresponding maximum liklihood estimators.     

Estimation of the Parameters of Compound Intervened Poisson Distribution

The Compound Intervened Poisson Distribution (CIPD) has been well discussed by Dhanavanthan (1998). In this paper the estimation of the parameters of the CIPD are studied.     

Characterization of the WEIBULL Distribution by Properties of Order Statistics

Let X_1:n, X_2:n, X_3:n…, X_n:n be the order statistics of n independent random variables with the common (absolutely continuous strictly increasing) distribution function F. The main results given in this article are:

• 1 For any fixed r and two distinct numbers s₁ and s₂ (1<r<s₁<s₂≦n) the distributions of V_i and W_i (defined in (1.11) and (1.12) are identical for i = 1,2 iff F(x) is WEIBULL (1.2).
• 2 The statistics D₁ and D₂ (as in (1.8) and (1.9)) are independent iff F(x) is WEIBULL (1.2).
• 3 The statistics U_i (1≦j≦n?1) and X_i:n (i ≦ j) are independent iff F(x) is WEIBULL (1.2).
• 4 Let X, X₁, X₂, …, X_k be random variables such that

These conditions are necessary and sufficient for F(x) to be WEIBULL .     

Statistical Significance of Mutation Frequencies,and the Power of Statistical Testing,Using the Poisson Distribution

The Poisson distribution may be employed to test whether mutation frequencies differ from control frequencies. This paper describes how this testing procedure may be used for either one-tailed or two-tailed hypotheses. It is also shown how the power of the statistical test can be calculated, the power being the probability of correctly concluding the null hypothesis to be false.     

ABCA1基因多态性R219K与帕金森症和阿尔兹海默症发病率的相关性研究

目的:探讨与研究三磷酸腺苷结合盒转运体A1 (Adenosine triphosphate (ATP)-binding cassette transporter A1)基因多态性R219K与帕金森症(Parkinson disease,PD)和阿尔兹海默症(Alzheimer disease,AD)发病率的相关性。方法:选择2016年2月到2019年8月在本院门诊与住院的帕金森症患者42例作为PD组,同期选择本院门诊与住院的阿尔兹海默症患者42例作为AD组,同期选择本院门诊健康体检者84例作为对照组。调查入选者的一般资料,检测三组血液样本的ABCA1基因多态性R219K情况并进行相关性分析。结果:AD组低密度脂蛋白(low-density lipoprotein,LDL-C)、总胆固醇(total cholesterol,TC)、甘油三酯(triglyceride,TG)与尿酸(Uric acid,UA)均低于对照组,而高密度脂蛋白(high-density lipoprotein,HDL-C)、同型半胱氨酸(homocysteine,Hcy)值高于对照组(P0.05);AD组TC均低于PD组,而HDL高于PD组。PD患者HDL-C均低于对照组,而LDL、TC和TG与对照组无差异(P0.05),三组空腹血糖(Fasting blood glucose,FBG)值对比差异无统计学意义(P0.05)。PD组与AD组的ABCA1 R219K GA基因型、A等位基因频率都显著高于对照组(P0.05),PD组与AD组对比差异无统计学意义(P0.05)。在168例入选者中,直线相关分析显示ABCA1 R219K GA基因型与A等位基因与帕金森症或阿尔兹海默症发生有显著相关性(P0.05)。结论:ABCA1基因多态性R219K在帕金森症和阿尔兹海默症患者中比较常见,ABCA1 R219K GA基因型与A等位基因可诱发帕金森症和阿尔兹海默症的发生。     

The Induction of Phagocytic Activation by Mixtures of the Water Chlorination By‐Products,Dichloroacetate‐ and Trichloroacetate,in Mice after Subchronic Exposure

In this study, groups of B6C3F1 male mice were treated with dichloroacetate (DCA), trichloroacetate (TCA), and mixtures of the compounds (Mix I, II, and III) daily by gavage, for 13 weeks. The tested doses were 7.5, 15, and 30 mg DCA/kg/day and 12.5, 25, and 50 mg TCA/kg/day. The DCA: TCA ratios in Mix I, II, and III were 7.5:12.5, 15:25, and 30:50 mg/kg/day, respectively. Peritoneal lavage cells were collected at the end of the treatment period and assayed for the biomarkers of phagocytic activation, including superoxide anion and tumor necrosis factor‐alpha production, and myeloperoxidase activity. The mixtures produced nonlinear effects on the biomarkers of phagocytic activation, with Mix I and II effects were found to be additive, but Mix III effects were found to be less than additive. © 2013 Wiley Periodicals, Inc. J BiochemMol Toxicol 27:237‐242, 2013; View this article online at wileyonlinelibrary.com . DOI 10.1002/jbt.21476     

Missing Data and the Mixtures of Discrete and Continuous Random Variables

In many practical applications we deal with a problem of estimation of a density function of a vector x some components of which are discrete, while the remaining ones are continuous. Among many models that can be used in this case the most useful are the location model and the kernel model. The problem arises when the observed data contain missing values i.e. on some individuals some of the variables have not been observed with no particular pattern of missingness. An application of the EM algorithm will allow us to estimate the parameters of the location model from incomplete data. The method is described in Section 2. In Section 3 some suggestions how to deal with incompleteness when the kernel model is used are made. Finally, Section 4 contains an example.     

一类重随机Poisson过程的等待时间的分布密度公式及其在害虫预测预报中的应用

对具有强度过程λｔ（Ｘ）＝Ｘ＾αＶ（ｔ）（α∈Ｒ＋）的重随机Ｐｏｉｓｓｏｎ过程「Ｎｔ：ｔ≥ｔ０」,推导出其等待时间Ｗｎ的分布密度函数ｆＷｎ（ｔ）的表达式;讨论了进一步推广所得结果的途径;利用所得结果解决了一个预测害虫最适防治时刻的问题。     

Direct Evaluation of Vapour-Liquid Equilibria of Mixtures by Molecular Dynamics Using Gibbs-Duhem Integration

Abstract

We present an extension of the Gibbs-Duhem integration method that permits direct evaluation of vapour-liquid equilibria of mixtures by molecular dynamics. The Gibbs-Duhem integration combines the best elements of the Gibbs ensemble Monte Carlo technique and thermodynamic integration. Given conditions of coexistence of pure substances, simultaneous but independent molecular dynamics simulations of each phase at constant number of particles, constant pressure, constant temperature and constant fugacity fraction of species 2 are carried out in succession along coexistence lines. In each simulation, the coexistence pressure is adjusted to satisfy the Clapeyron-type equation. The Clapeyron-type equation is a first-order nonlinear differential equation that prescribes how the pressure must change with the fugacity fraction of species 2 to maintain coexistence at constant temperature. The Clapeyron-type equation is solved by the predictor-corrector method. Running averages of mole fraction and compressibility factor for the two phases are used to evaluate the right-hand side of the Clapeyron-type equation. The Gibbs-Duhem integration method is applied to three prototypes of binary mixtures of the two-centre Lennard-Jones fluid having various elongations. The starting points on the coexistence curve were taken from published data.