Two New Discrete Distributions

CONSUL and JAIN (1973a) introduced a generalized Poisson distribution, which has applications in reliability theory and many biometric studies, and described some of its properties. Here we obtain two new distributions treating two of the parameters of the above distribution as random variables having gamma and absolute value distributions. One of the new distributions is related to the negative binomial distribution. Their moments also have been obtained.     

Symmetric Bernoulli Distributions and Generalized Binomial Distributions

The generalized binomial distribution is defined as the distribution of a sum of symmetrically distributed Bernoulli random variates. Several two-parameter families of generalized binomial distributions have received attention in the literature, including the Polya urn model, the correlated binomial model and the latent variable model. Some properties and limitations of the three distributions are described. An algorithm for maximum likelihood estimation for two-parameter generalized binomial distributions is proposed. The Polya urn model and the latent variable model were found to provide good fits to sub-binomial data given by Parkes. An extension of the latent variable model to incorporate heterogeneous response probabilities is discussed.     

Incorporation of inter-individual heterogeneity into the multistage carcinogenesis model: approach to the analysis of cancer incidence data

We investigate a multistage carcinogenesis frailty model to incorporate inter-individual heterogeneity into carcinogenic response. Attention is focused on inference concerning the effects of different sources of population heterogeneity on cancer rates. The authors consider unobserved variability arising from either carcinogen exposure or background characteristics. Gamma and Inverse-Gaussian distributions are selected for frailty models, and the baseline hazard function is the generalized Armitage-Doll model (i.e. non-frailty model) in which exposure effects shift the age scale instead of acting multiplicatively on cancer rates. For illustration, we apply the method to solid cancer data from a cohort of atomic bomb survivors to examine some features of proposed models. The results show that the Gamma frailty model for the heterogeneity of baseline rates provides the best goodness-of-fit of the model and a non-zero frailty variance. Parameter estimates are, for the most part, comparable between the Gamma and Inverse-Gaussian frailty models. In a heterogeneous population the exposure effects on young adulthood cancer rates might be underestimated for the non-frailty model. Meaningful information regarding each source of heterogeneity has been provided by the proposed method. Therefore, the multistage carcinogenesis frailty model approach is useful for analyses of epidemiological cancer data to assess population heterogeneity and heterogeneity-influenced exposure effects.     

Evaluations of Interventions Using Mathematical Models with Exponential and Non-exponential Distributions for Disease Stages: The Case of Ebola

Many mathematical models for the disease transmission dynamics of Ebola have been developed and studied, particularly during and after the 2014 outbreak in West Africa. Most of these models are systems of ordinary differential equations (ODEs). One of the common assumptions made in these ODE models is that the duration of disease stages, such as latent and infectious periods, follows an exponential distribution. Gamma distributions have also been used in some of these models. It has been demonstrated that, when the models are used to evaluate disease control strategies such as quarantine or isolation, the models with exponential and Gamma distribution assumptions may generate contradictory results (Feng et al. in Bull Math Biol 69(5):1511–1536, 2007). Several Ebola models are considered in this paper with various stage distributions, including exponential, Gamma and arbitrary distributions. These models are used to evaluate control strategies such as isolation (or hospitalization) and timely burial and to identify potential discrepancies between the results from models with exponential and Gamma distributions.     

Generalized Poisson distribution: the property of mixture of Poisson and comparison with negative binomial distribution

We prove that the generalized Poisson distribution GP(theta, eta) (eta > or = 0) is a mixture of Poisson distributions; this is a new property for a distribution which is the topic of the book by Consul (1989). Because we find that the fits to count data of the generalized Poisson and negative binomial distributions are often similar, to understand their differences, we compare the probability mass functions and skewnesses of the generalized Poisson and negative binomial distributions with the first two moments fixed. They have slight differences in many situations, but their zero-inflated distributions, with masses at zero, means and variances fixed, can differ more. These probabilistic comparisons are helpful in selecting a better fitting distribution for modelling count data with long right tails. Through a real example of count data with large zero fraction, we illustrate how the generalized Poisson and negative binomial distributions as well as their zero-inflated distributions can be discriminated.     

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.     

武夷山景区主要景观类型斑块大小分布规律及其等级尺度效应分析

以武夷山风景名胜区为研究对象,运用景观分类生态原则和群落生态学原则,将武夷山风景名胜区划分为10类景观类型,并提取各类型斑块的面积、周长、数量等信息。在此基础上,运用正态分布、对数正态分布、Weibull分布、г-分布、Bata-分布及负指数分布等6个常见的概率分布,分别在3种等级划分条件下,对各景观类型斑块大小分布规律进行研究．结果表明,多数斑块大小服从对数正态分布。少数服从r．分布和Weibull分布,而所有类型的斑块大小均不服从正态分布;对于同一景观类型而言,不同划分等级对其斑块大小的分布规律产生一定的影响,即斑块大小分布存在着等级效应,其中以b等级划分相对比较理想;对不同的景观类型而言,斑块大小分布规律依类型的不同而存在差异,而且就整个景区所有斑块大小分布来说,尚未有一种概率分布能理想地加以刻画;由具有相同分布规律的不同景观类型的斑块构成的整体,其斑块大小分布规律基本保持不变,而由不同分布规律的不同景观类型的斑块构成的整体,其斑块大小分布规律发生变化。     

南海北部近海渔业资源密度概率分布特征

根据2014-2017年南海北部近海8个调查航次渔获量数据,结合统计方法分析该海域渔业资源密度分布特征并探索其适宜概率分布类型,进而估算区域平均资源密度.结果表明:各时期资源密度变异系数(CV)在0.67~1.03,说明该海域渔业资源密度呈较高程度的不均匀空间分布,且渔获资源密度频率分布呈现明显的右偏特征,总体以0~1000 kg·km-2资源密度为主导;单样本Kolmogorov-Smirnov检验结果表明,对数正态、伽玛和韦伯分布是该区域资源密度的适宜分布类型;在海域平均资源密度估算方面,对数正态所得结果与另两个分布类型在统计学上无显著差异,而伽玛和韦伯分布的估计值有显著差异.与1960-1970年代相比,该海域渔业资源密度适宜概率分布型已从单一类型转变为多类型,这主要归于渔业资源结构、捕捞强度以及气候变化等引起的低渔获量比例变化.     

Skew‐normal/independent linear mixed models for censored responses with applications to HIV viral loads

Often in biomedical studies, the routine use of linear mixed‐effects models (based on Gaussian assumptions) can be questionable when the longitudinal responses are skewed in nature. Skew‐normal/elliptical models are widely used in those situations. Often, those skewed responses might also be subjected to some upper and lower quantification limits (QLs; viz., longitudinal viral‐load measures in HIV studies), beyond which they are not measurable. In this paper, we develop a Bayesian analysis of censored linear mixed models replacing the Gaussian assumptions with skew‐normal/independent (SNI) distributions. The SNI is an attractive class of asymmetric heavy‐tailed distributions that includes the skew‐normal, skew‐t, skew‐slash, and skew‐contaminated normal distributions as special cases. The proposed model provides flexibility in capturing the effects of skewness and heavy tail for responses that are either left‐ or right‐censored. For our analysis, we adopt a Bayesian framework and develop a Markov chain Monte Carlo algorithm to carry out the posterior analyses. The marginal likelihood is tractable, and utilized to compute not only some Bayesian model selection measures but also case‐deletion influence diagnostics based on the Kullback–Leibler divergence. The newly developed procedures are illustrated with a simulation study as well as an HIV case study involving analysis of longitudinal viral loads.     

北京地区植被景观中斑块大小的分布特征

利用GIS软件ARC／INFO将北京地区的1：20万植被图数字化,并提取各斑块的面积信息。该图包含72个基本的斑块类型,它们又分属于森林、灌丛、草地、果园、农田和水体6大类型。这些大类又分别包含20、28、4、7、11和2个基本类型。选用斑块个数、总面积、平均斑块面积、标准差、变异系数、中值、最大斑块面积、最小斑块面积、极差和偏态系数等几个描述统计量,以及厂一分布、对数正态分布、Weibull分布、指数分布和正态分布等5个概率分布来刻画斑块大小的分布特征。结果表明：除具有很少斑块的少数基本类型以外,其它基本类型以及所有6个大类的斑块大小的分布都不是对称的,而是右偏的。因此,普通的正态分布不能对它们加以刻画;其他4种概率分布也只能刻画部分类型,并且服从对数正态分布的类型最多,服从负指数分布的斑块类型最少。     

混合位置尺度分布中分量参数的统计推断

本文讨论混合位置尺度分布当混合比已知时分量参数的假设检验和区间估计．本文所提出的方法基于广义枢轴模型．当只有感兴趣的参数未知时,检验的实际水平等于名义水平,且各置信域的实际覆盖率等于名义覆盖率．在更一般的场合,检验是相合的,并且各置信域的实际覆盖率趋于名义覆盖率．模拟显示我们的方法是令人满意的．     

Characterization and Simulation of Uncertain Frequency Distributions: Effects of Distribution Choice,Variability, Uncertainty,and Parameter Dependence

We use bootstrap simulation to characterize uncertainty in parametric distributions, including Normal, Lognormal, Gamma, Weibull, and Beta, commonly used to represent variability in probabilistic assessments. Bootstrap simulation enables one to estimate sampling distributions for sample statistics, such as distribution parameters, even when analytical solutions are not available. Using a two-dimensional framework for both uncertainty and variability, uncertainties in cumulative distribution functions were simulated. The mathematical properties of uncertain frequency distributions were evaluated in a series of case studies during which the parameters of each type of distribution were varied for sample sizes of 5, 10, and 20. For positively skewed distributions such as Lognormal, Weibull, and Gamma, the range of uncertainty is widest at the upper tail of the distribution. For symmetric unbounded distributions, such as Normal, the uncertainties are widest at both tails of the distribution. For bounded distributions, such as Beta, the uncertainties are typically widest in the central portions of the distribution. Bootstrap simulation enables complex dependencies between sampling distributions to be captured. The effects of uncertainty, variability, and parameter dependencies were studied for several generic functional forms of models, including models in which two-dimensional random variables are added, multiplied, and divided, to show the sensitivity of model results to different assumptions regarding model input distributions, ranges of variability, and ranges of uncertainty and to show the types of errors that may be obtained from mis-specification of parameter dependence. A total of 1,098 case studies were simulated. In some cases, counter-intuitive results were obtained. For example, the point value of the 95th percentile of uncertainty for the 95th percentile of variability of the product of four Gamma or Weibull distributions decreases as the coefficient of variation of each model input increases and, therefore, may not provide a conservative estimate. Failure to properly characterize parameter uncertainties and their dependencies can lead to orders-of-magnitude mis-estimates of both variability and uncertainty. In many cases, the numerical stability of two-dimensional simulation results was found to decrease as the coefficient of variation of the inputs increases. We discuss the strengths and limitations of bootstrap simulation as a method for quantifying uncertainty due to random sampling error.     

A note on permutation tests for variance components in multilevel generalized linear mixed models

In many applications of generalized linear mixed models to multilevel data, it is of interest to test whether a random effects variance component is zero. It is well known that the usual asymptotic chi-square distribution of the likelihood ratio and score statistics under the null does not necessarily hold. In this note we propose a permutation test, based on randomly permuting the indices associated with a given level of the model, that has the correct Type I error rate under the null. Results from a simulation study suggest that it is more powerful than tests based on mixtures of chi-square distributions. The proposed test is illustrated using data on the familial aggregation of sleep disturbance.     

The Coalescence of Intrahost HIV Lineages Under Symmetric CTL Attack

Cytotoxic T lymphocytes (CTLs) are immune system cells that are thought to play an important role in controlling HIV infection. We develop a stochastic ODE model of HIV-CTL interaction that extends current deterministic ODE models. Based on this stochastic model, we consider the effect of CTL attack on intrahost HIV lineages assuming that CTLs attack several epitopes with equal strength. In this setting, we introduce a limiting version of our stochastic ODE under which we show that the coalescence of HIV lineages can be described through Poisson-Dirichlet distributions. Through numerical experiments, we show that our results under the limiting stochastic ODE accurately reflect HIV lineages under CTL attack when the HIV population size is on the low end of its hypothesized range. Current techniques of HIV lineage construction depend on the Kingman coalescent. Our results give an explicit connection between CTL attack and HIV lineages.     

Root length density and water uptake distributions of winter wheat under sub-irrigation

As the critical information to study flow transport in soil–plant systems, root distributions and root-water-uptake (RWU) patterns have been studied extensively. However, most root distribution data in the past were collected under surface irrigation. Less research has been conducted to characterize root distributions under sub-irrigation. The objectives of this study were to (1) test if the generalized function of normalized root length density (NRLD) in the literature was applicable to root distributions of winter wheat under natural sub-irrigation, which provides water from subsurface by capillary rise from the water table, and (2) estimate RWU distributions of winter wheat under natural sub-irrigation. Column experiments were conducted to study the distributions of root length density (RLD) and RWU of winter wheat (Triticum aestivum L. cv. Nongda 189) during a growing period of 57 days from planting to tillering stages under surface irrigation and natural sub-irrigation. Data of root distributions and soil water content were collected in the experiments with different treatments of irrigation levels. Results showed that the RLD distributions of winter wheat under both surface irrigation and natural sub-irrigation were of similar patterns. The NRLD distributions under sub-irrigation were adequately characterized by the generalized function. An inverse method was employed to estimate the average RWU rate distributions of winter wheat. In addition, based on the potential RWU coefficient and the NRLD function, a simple approach was developed to predict RWU rates at different depths. The predicted RWU rates had a good agreement with the estimated RWU rate distributions using the inverse method.Section editor: R. E. Munns     

New class of Johnson distributions and its associated regression model for rates and proportions

By starting from the Johnson distribution pioneered by Johnson ( 1949 ), we propose a broad class of distributions with bounded support on the basis of the symmetric family of distributions. The new class of distributions provides a rich source of alternative distributions for analyzing univariate bounded data. A comprehensive account of the mathematical properties of the new family is provided. We briefly discuss estimation of the model parameters of the new class of distributions based on two estimation methods. Additionally, a new regression model is introduced by considering the distribution proposed in this article, which is useful for situations where the response is restricted to the standard unit interval and the regression structure involves regressors and unknown parameters. The regression model allows to model both location and dispersion effects. We define two residuals for the proposed regression model to assess departures from model assumptions as well as to detect outlying observations, and discuss some influence methods such as the local influence and generalized leverage. Finally, an application to real data is presented to show the usefulness of the new regression model.     

A Class of Distributions in the First Emptiness of a Semi-Infinite Reservoir

This paper considers the probability distribution of the volume of a certain substance (e.g. river discharge, rainfall, deposites of clay, organism, etc.) that flows into a semi-infinite reservoir before its first emptiness for continuous and homogeneous input process when the substance is released at unit rate per unit of time. A few moments of the distribution have been computed. A generalized gamma, and a generalized exponential distributions as particular cases are also discussed. Some possible applications of the generalized negative exponential distribution have been mentioned. These distributions are in fact the continuous analogues of the discrete LAGRANGE distributions recently considered by JAIN and others.     

Assessing Biases in the Evaluation of Classification Assays for HIV Infection Recency

Identifying recent HIV infection cases has important public health and clinical implications. It is essential for estimating incidence rates to monitor epidemic trends and evaluate the effectiveness of interventions. Detecting recent cases is also important for HIV prevention given the crucial role that recently infected individuals play in disease transmission, and because early treatment onset can improve the clinical outlook of patients while reducing transmission risk. Critical to this enterprise is the development and proper assessment of accurate classification assays that, based on cross-sectional samples of viral sequences, help determine infection recency status. In this work we assess some of the biases present in the evaluation of HIV recency classification algorithms that rely on measures of within-host viral diversity. Particularly, we examine how the time since infection (TSI) distribution of the infected subjects from which viral samples are drawn affect performance metrics (e.g., area under the ROC curve, sensitivity, specificity, accuracy and precision), potentially leading to misguided conclusions about the efficacy of classification assays. By comparing the performance of a given HIV recency assay using six different TSI distributions (four simulated TSI distributions representing different epidemic scenarios, and two empirical TSI distributions), we show that conclusions about the overall efficacy of the assay depend critically on properties of the TSI distribution. Moreover, we demonstrate that an assay with high overall classification accuracy, mainly due to properly sorting members of the well-represented groups in the validation dataset, can still perform notoriously poorly when sorting members of the less represented groups. This is an inherent issue of classification and diagnostics procedures that is often underappreciated. Thus, this work underscores the importance of acknowledging and properly addressing evaluation biases when proposing new HIV recency assays.     

k-长DNA子序列频数分布研究

在详细阐述了生成DNA序列分形图像的Hao方法后,提出一种能够直观显示k-长DNA子序列频数分布差异性的三维频数分布图生成方法。把3D频数分布图转化为1D对数频谱图,突出显示了频数分布的局部特征,提出k-长DNA子序列频数区划分准则,并详细研究了甚高频数区的n阶零间隔现象,指出n阶零间隔分布就是基因组进化过程所留痕迹的假设,并给出对数频谱图特征的生物学解释。实验发现许多DNA序列频数概率分布近似服从非中心F分布,对于分布呈多峰现象的基因组序列,可采用多个非中心F分布的叠加来拟合。在比较非中心F分布与Gamma分布后,提出一种结合二者在拟合方面具有互补优势的新分布,实验证明这种新分布能够更好地吻合实际DNA序列的频数分布。最后研究了两种特异出现频数(最高出现频数与出现频数为1的k-长子序列个数)与k值的关系,发现不同物种的这两种关系具有良好的一致性。     

Determination of Number and Size of Spherical Particles from Extraction Replicas

For spherical particles randomly dispersed in the space of a specimen the estimators of the parameters of the space structure from the measurements obtained from extraction replicas are given. First an arbitrary form of the probability density function f(x) of the diameter X and then the generalized RAYLEIGH and lognormal distributions of X are considered. Unbiased estimators of the space parameters and of parameters of these distributions are found. The variances of these estimators are given and unbiased estimators of these variances are determined.