On the Fisher Information Matrix in Type II Censored Data from the Exponentiated Exponential Family

In this note, we express, in a general setting, the Fisher information matrix under Type II censoring in terms of the hazard function and then obtain the Fisher information matrix under Type II censoring as a single integral for the exponentiated exponential family, which can be easily evaluated. The Fisher information under Type II censoring can also be used to characterize the exponential distribution among the exponentiated exponential family.     

Estimating the distribution of selection coefficients from phylogenetic data with applications to mitochondrial and viral DNA

The distribution of selection coefficients of new mutations is of key interest in population genetics. In this paper we explore how codon-based likelihood models can be used to estimate the distribution of selection coefficients of new amino acid replacement mutations from phylogenetic data. To obtain such estimates we assume that all mutations at the same site have the same selection coefficient. We first estimate the distribution of selection coefficients from two large viral data sets under the assumption that the viral population size is the same along all lineages of the phylogeny and that the selection coefficients vary among sites. We then implement several new models in which the lineages of the phylogeny may have different population sizes. We apply the new models to a data set consisting of the coding regions from eight primate mitochondrial genomes. The results suggest that there might be little power to determine the exact shape of the distribution of selection coefficient but that the normal and gamma distributions fit the data significantly better than the exponential distribution.     

长白山原始阔叶红松林径级结构模拟

以原始阔叶红松林为研究对象,设置了2块1hm2样地,利用Weibull分布函数、负指数函数和q值理论对其林分径级结构进行模拟。结果表明:2块样地林分的径级分布均呈倒"J"形,森林更新良好;Weibull分布函数和负指数函数都取得了较好的模拟效果,但Weibull分布函数更好地预测了大径阶株数,因此,Weibull分布函数比负指数函数更适合原始阔叶红松林径级结构的模拟;2块样地的q值偏小(分别为1.74和1.45),说明径级结构曲线较平缓,大径材数量多。q值法则对于阔叶红松林径阶株数密度的表达效果较好,可以被用于描述阔叶红松林的胸径分布。     

Effect of Porosity on Strength Distribution of Microcrystalline Cellulose

Fracture strength of pharmaceutical compacts varies even for nominally identical samples, which directly affects compaction, comminution, and tablet dosage forms. However, the relationships between porosity and mechanical behavior of compacts are not clear. Here, the effects of porosity on fracture strength and fracture statistics of microcrystalline cellulose compacts were investigated through diametral compression tests. Weibull modulus, a key parameter in Weibull statistics, was observed to decrease with increasing porosity from 17 to 56 vol.%, based on eight sets of compacts at different porosity levels, each set containing ∼50 samples, a total of 407 tests. Normal distribution fits better to fracture data for porosity less than 20 vol.%, whereas Weibull distribution is a better fit in the limit of highest porosity. Weibull moduli from 840 unique finite element simulations of isotropic porous materials were compared to experimental Weibull moduli from this research and results on various pharmaceutical materials. Deviations from Weibull statistics are observed. The effect of porosity on fracture strength can be described by a recently proposed micromechanics-based formula.Key words: diametral compression test, finite element simulations, normal distribution, reliability, Weibull modulus     

Shapes and functions of species–area curves: a review of possible models

Aim This paper reviews possible candidate models that may be used in theoretical modelling and empirical studies of species–area relationships (SARs). The SAR is an important and well‐proven tool in ecology. The power and the exponential functions are by far the models that are best known and most frequently applied to species–area data, but they might not be the most appropriate. Recent work indicates that the shape of species–area curves in arithmetic space is often not convex but sigmoid and also has an upper asymptote. Methods Characteristics of six convex and eight sigmoid models are discussed and interpretations of different parameters summarized. The convex models include the power, exponential, Monod, negative exponential, asymptotic regression and rational functions, and the sigmoid models include the logistic, Gompertz, extreme value, Morgan–Mercer–Flodin, Hill, Michaelis–Menten, Lomolino and Chapman–Richards functions plus the cumulative Weibull and beta‐P distributions. Conclusions There are two main types of species–area curves: sample curves that are inherently convex and isolate curves, which are sigmoid. Both types may have an upper asymptote. A few have attempted to fit convex asymptotic and/or sigmoid models to species–area data instead of the power or exponential models. Some of these or other models reviewed in this paper should be useful, especially if species–area models are to be based more on biological processes and patterns in nature than mere curve fitting. The negative exponential function is an example of a convex model and the cumulative Weibull distribution an example of a sigmoid model that should prove useful. A location parameter may be added to these two and some of the other models to simulate absolute minimum area requirements.     

Use of the Likelihood Ratio Test on the Uniform Distribution

In this article we give a simple procedure to determine the exact distribution of the likelihood ratio test of a statistical hypothesis regarding the parameter of the uniform distribution. The resulting distribution will be shown to serve as an approximation to the distribution of the likelihood ratio statistic for testing the equality of scale parameters of k independent Exponential populations.     

Analytical Quantile Solution for the S‐distribution,Random Number Generation and Statistical Data Modeling

The selection of a specific statistical distribution as a model for describing the population behavior of a given variable is seldom a simple problem. One strategy consists in testing different distributions (normal, lognormal, Weibull, etc.), and selecting the one providing the best fit to the observed data and being the most parsimonious. Alternatively, one can make a choice based on theoretical arguments and simply fit the corresponding parameters to the observed data. In either case, different distributions can give similar results and provide almost equivalent models for a given data set. Model selection can be more complicated when the goal is to describe a trend in the distribution of a given variable. In those cases, changes in shape and skewness are difficult to represent by a single distributional form. As an alternative to the use of complicated families of distributions as models for data, the S‐distribution [Voit, E. O. (1992) Biom. J. 7 , 855–878] provides a highly flexible mathematical form in which the density is defined as a function of the cumulative. S‐distributions can accurately approximate many known continuous and unimodal distributions, preserving the well known limit relationships between them. Besides representing well‐known distributions, S‐distributions provide an infinity of new possibilities that do not correspond with known classical distributions. Although the utility and performance of this general form has been clearly proved in different applications, its definition as a differential equation is a potential drawback for some problems. In this paper we obtain an analytical solution for the quantile equation that highly simplifies the use of S‐distributions. We show the utility of this solution in different applications. After classifying the different qualitative behaviors of the S‐distribution in parameter space, we show how to obtain different S‐distributions that accomplish specific constraints. One of the most interesting cases is the possibility of obtaining distributions that acomplish P(X ≤ X_c) = 0. Then, we demonstrate that the quantile solution facilitates the use of S‐distributions in Monte‐Carlo experiments through the generation of random samples. Finally, we show how to fit an S‐distribution to actual data, so that the resulting distribution can be used as a statistical model for them.     

Goodness of Fit of Probability Distributions for Sightings as Species Approach Extinction

Estimating the probability that a species is extinct and the timing of extinctions is useful in biological fields ranging from paleoecology to conservation biology. Various statistical methods have been introduced to infer the time of extinction and extinction probability from a series of individual sightings. There is little evidence, however, as to which of these models provide adequate fit to actual sighting records. We use L-moment diagrams and probability plot correlation coefficient (PPCC) hypothesis tests to evaluate the goodness of fit of various probabilistic models to sighting data collected for a set of North American and Hawaiian bird populations that have either gone extinct, or are suspected of having gone extinct, during the past 150 years. For our data, the uniform, truncated exponential, and generalized Pareto models performed moderately well, but the Weibull model performed poorly. Of the acceptable models, the uniform distribution performed best based on PPCC goodness of fit comparisons and sequential Bonferroni-type tests. Further analyses using field significance tests suggest that although the uniform distribution is the best of those considered, additional work remains to evaluate the truncated exponential model more fully. The methods we present here provide a framework for evaluating subsequent models.     

Tests for an Epidemic Change in a Sequence of Exponentially Distributed Random Variables

Consider a sequence of independent exponential random variables that is susceptible to a change in the means. We would like to test whether the means have been subjected to an epidemic change after an unknown point, for an unknown duration in the sequence. The likelihood ratio statistic and a likelihood ratio type statistic are derived. The distribution theories and related properties of the test statistics are discussed. Percentage points and powers of the tests are tabulated for selected values of the parameters. The powers of these two tests are then compared to the two statistics proposed by Aly and Bouzar. The tests are applied to find epidemic changes in the set of Stanford heart transplant data and air traffic arrival data.     

A Comparison of Models to Infer the Distribution of Fitness Effects of New Mutations

Knowing the distribution of fitness effects (DFE) of new mutations is important for several topics in evolutionary genetics. Existing computational methods with which to infer the DFE based on DNA polymorphism data have frequently assumed that the DFE can be approximated by a unimodal distribution, such as a lognormal or a gamma distribution. However, if the true DFE departs substantially from the assumed distribution (e.g., if the DFE is multimodal), this could lead to misleading inferences about its properties. We conducted simulations to test the performance of parametric and nonparametric discretized distribution models to infer the properties of the DFE for cases in which the true DFE is unimodal, bimodal, or multimodal. We found that lognormal and gamma distribution models can perform poorly in recovering the properties of the distribution if the true DFE is bimodal or multimodal, whereas discretized distribution models perform better. If there is a sufficient amount of data, the discretized models can detect a multimodal DFE and can accurately infer the mean effect and the average fixation probability of a new deleterious mutation. We fitted several models for the DFE of amino acid-changing mutations using whole-genome polymorphism data from Drosophila melanogaster and the house mouse subspecies Mus musculus castaneus. A lognormal DFE best explains the data for D. melanogaster, whereas we find evidence for a bimodal DFE in M. m. castaneus.     

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。     

Force-Clamp Analysis Techniques Give Highest Rank to Stretched Exponential Unfolding Kinetics in Ubiquitin

Force-clamp spectroscopy reveals the unfolding and disulfide bond rupture times of single protein molecules as a function of the stretching force, point mutations, and solvent conditions. The statistics of these times reveal whether the protein domains are independent of one another, the mechanical hierarchy in the polyprotein chain, and the functional form of the probability distribution from which they originate. It is therefore important to use robust statistical tests to decipher the correct theoretical model underlying the process. Here, we develop multiple techniques to compare the well-established experimental data set on ubiquitin with existing theoretical models as a case study. We show that robustness against filtering, agreement with a maximum likelihood function that takes into account experimental artifacts, the Kuiper statistic test, and alignment with synthetic data all identify the Weibull or stretched exponential distribution as the best fitting model. Our results are inconsistent with recently proposed models of Gaussian disorder in the energy landscape or noise in the applied force as explanations for the observed nonexponential kinetics. Because the physical model in the fit affects the characteristic unfolding time, these results have important implications on our understanding of the biological function of proteins.     

The Distribution of Ocular Chlamydia Prevalence across Tanzanian Communities Where Trachoma Is Declining

BackgroundMathematical models predict an exponential distribution of infection prevalence across communities where a disease is disappearing. Trachoma control programs offer an opportunity to test this hypothesis, as the World Health Organization has targeted trachoma for elimination as a public health concern by the year 2020. Local programs may benefit if a single survey could reveal whether infection was headed towards elimination. Using data from a previously-published 2009 survey, we test the hypothesis that Chlamydia trachomatis prevalence across 75 Tanzanian communities where trachoma had been documented to be disappearing is exponentially distributed.Methods/FindingsWe fit multiple continuous distributions to the Tanzanian data and found the exponential gave the best approximation. Model selection by Akaike Information Criteria (AIC_c) suggested the exponential distribution had the most parsimonious fit to the data. Those distributions which do not include the exponential as a special or limiting case had much lower likelihoods of fitting the observed data. 95% confidence intervals for shape parameter estimates of those distributions which do include the exponential as a special or limiting case were consistent with the exponential. Lastly, goodness-of-fit testing was unable to reject the hypothesis that the prevalence data came from an exponential distribution.ConclusionsModels correctly predict that infection prevalence across communities where a disease is disappearing is best described by an exponential distribution. In Tanzanian communities where local control efforts had reduced the clinical signs of trachoma by 80% over 10 years, an exponential distribution gave the best fit to prevalence data. An exponential distribution has a relatively heavy tail, thus occasional high-prevalence communities are to be expected even when infection is disappearing. A single cross-sectional survey may be able to reveal whether elimination efforts are on-track.     

Times from Infection to Disease-Induced Death and their Influence on Final Population Sizes After Epidemic Outbreaks

For epidemic models, it is shown that fatal infectious diseases cannot drive the host population into extinction if the incidence function is upper density-dependent. This finding holds even if a latency period is included and the time from infection to disease-induced death has an arbitrary length distribution. However, if the incidence function is also lower density-dependent, very infectious diseases can lead to a drastic decline of the host population. Further, the final population size after an epidemic outbreak can possibly be substantially affected by the infection-age distribution of the initial infectives if the life expectations of infected individuals are an unbounded function of infection age (time since infection). This is the case for lognormal distributions, which fit data from infection experiments involving tiger salamander larvae and ranavirus better than gamma distributions and Weibull distributions.     

麻城黄山松天然林直径分布

为研究黄山松天然次生林直径分布特征,以麻城市黄山松天然次生林为研究对象,采用标准样地调查,计算林分直径的偏度、峰度,林分直径分布的Shannon-Weiner和Simpson指数,运用负指数分布、normal分布、lognormal分布、logistic分布和Weibull分布等5种概率密度函数对黄山松天然次生林林分直...     

Allometric analysis using the multivariate shifted exponential normal distribution

In allometric studies, the joint distribution of the log-transformed morphometric variables is typically elliptical and with heavy tails. To account for these peculiarities, we introduce the multivariate shifted exponential normal (MSEN) distribution , an elliptical heavy-tailed generalization of the multivariate normal (MN). The MSEN belongs to the family of MN scale mixtures (MNSMs) by choosing a convenient shifted exponential as mixing distribution. The probability density function of the MSEN has a simple closed-form characterized by only one additional parameter, with respect to the nested MN, governing the tail weight. The first four moments exist and the excess kurtosis can assume any positive value. The membership to the family of MNSMs allows us a simple computation of the maximum likelihood (ML) estimates of the parameters via the expectation-maximization (EM) algorithm; advantageously, the M-step is computationally simplified by closed-form updates of all the parameters. We also evaluate the existence of the ML estimates. Since the parameter governing the tail weight is estimated from the data, robust estimates of the mean vector of the nested MN distribution are automatically obtained by downweighting; we show this aspect theoretically but also by means of a simulation study. We fit the MSEN distribution to multivariate allometric data where we show its usefulness also in comparison with other well-established multivariate elliptical distributions.     

Distributions of Autocorrelated First-Order Kinetic Outcomes: Illness Severity

Many complex systems produce outcomes having recurring, power law-like distributions over wide ranges. However, the form necessarily breaks down at extremes, whereas the Weibull distribution has been demonstrated over the full observed range. Here the Weibull distribution is derived as the asymptotic distribution of generalized first-order kinetic processes, with convergence driven by autocorrelation, and entropy maximization subject to finite positive mean, of the incremental compounding rates. Process increments represent multiplicative causes. In particular, illness severities are modeled as such, occurring in proportion to products of, e.g., chronic toxicant fractions passed by organs along a pathway, or rates of interacting oncogenic mutations. The Weibull form is also argued theoretically and by simulation to be robust to the onset of saturation kinetics. The Weibull exponential parameter is shown to indicate the number and widths of the first-order compounding increments, the extent of rate autocorrelation, and the degree to which process increments are distributed exponential. In contrast with the Gaussian result in linear independent systems, the form is driven not by independence and multiplicity of process increments, but by increment autocorrelation and entropy. In some physical systems the form may be attracting, due to multiplicative evolution of outcome magnitudes towards extreme values potentially much larger and smaller than control mechanisms can contain. The Weibull distribution is demonstrated in preference to the lognormal and Pareto I for illness severities versus (a) toxicokinetic models, (b) biologically-based network models, (c) scholastic and psychological test score data for children with prenatal mercury exposure, and (d) time-to-tumor data of the ED₀₁ study.     

A beta‐binomial mixed‐effects model approach for analysing longitudinal discrete and bounded outcomes

Patient‐reported outcomes (PROs) are currently being increasingly used as primary outcome measures in observational and experimental studies since they inform clinicians and researchers about the health‐status of patients and generate data to facilitate improved care. PROs usually appear as discrete and bounded with U, J, or inverse J shapes, and hence, exponential family members offer inadequate distributional fits. The beta‐binomial distribution has been proposed in the literature to fit PROs. However, the fact that the beta‐binomial distribution does not belong to the exponential family limits its applicability in the regression model context, and classical estimation approaches are not straightforward. Moreover, PROs are usually measured in a longitudinal framework in which individuals are followed up for a certain period. Hence, each individual obtains several scores of the PRO over time, which leads to the repeated measures and defines the correlation structure in the data. In this work, we have developed and proposed an estimation procedure for the analysis of correlated discrete and bounded outcomes, particularly PROs, by a beta‐binomial mixed‐effects model. Additionally, we have implemented the methodology in the PROreg package in R. Because there are similar approaches in the literature to address the same issue, this work also incorporates a comparison study between our proposal and alternative methodologies commonly implemented in R and shows the superior performance of our estimation procedure. This paper was motivated by the analysis of the health‐status of patients with chronic obstructive pulmonary disease, where the main objective is the assessment of risk factors that may affect the evolution of the disease. The application of the proposed approach in the study leads to clinically relevant results.     

Representing Cumulative Germination.: 2. The Use of the Weibull Function and Other Empirically Derived Curves

The Weibull, Morgan–Mercer–Flodin, Richards, Mitscherlich,Gompertz and logistic functions were each fitted to a wide rangeof cumulative germinations of non-dormant seed. The Weibullproved the most suitable for describing cumulative germinationas it provided a consistently close fit to the data and wasinsensitive to choice of starting values, thus making it fairlyeasy to fit. The others provided either an inferior fit or elsea similar fit but with a greater sensitivity to starting values. The four parameters of the Weibull function reflect maximumgermination, germination rate, the lag in the onset of germinationand the shape of the cumulative distribution. A comparison between non-linear and linear fits of the Mitscherlich,Gompertz and logistic functions showed the clear superiorityof non-linear methods. Cumulative germination, Weibull function, Richards function, Morgan–Mercer–Flodin function, Mitscherlich (monomolecular) function, Gompertz function, logistic (autocatalytic) function, modelling, simulation     

Correlation Goodness‐of‐fit Test for the Logarithmically‐Decreasing Survival Distribution

In this paper, we use the correlation‐type goodness‐of‐fit test for the logarithmically‐decreasing survival distribution. This model was intoduced by Sultan , Balakrishnan and Childs (2001) as a special case of Type‐I truncated logistic distribution. The power of the test based on normal, Weibull and gamma distributions is also calculated. We also give application to real example.